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Volume 2010Hindawi Publishing Corporationhttp://www.hindawi.com

Fixed Point Theoryand ApplicationsEditor-in-Chief: Ravi P. Agarwal

Special Issue Impact of Kirk's Results on the Development of Fixed Point Theory

Guest Editors Mohamed A. Khamsi and Tomas Dominguez-Benavides

Impact of Kirk’s Results on theDevelopment of Fixed Point Theory

Fixed Point Theory and Applications

Impact of Kirk’s Results on theDevelopment of Fixed Point Theory

Guest Editors: Mohamed A. Khamsi and Tomas Dominguez-Benavides

Copyright q 2010 Hindawi Publishing Corporation. All rights reserved.

This is an issue published in volume 2010 of “Fixed Point Theory and Applications.” All articles are open access articlesdistributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited.

Editor-in-ChiefRavi P. Agarwal, Florida Institute of Technology, USA

Associate Editors

T. Benavides, SpainH. Ben-El-Mechaiekh, CanadaRobert F. Brown, USALjubomir B. Ciric, SerbiaM. de la Sen, SpainMarlene Frigon, CanadaM. Furi, ItalyL. Gorniewicz, PolandN. J. Huang, China

Jerzy Jezierski, PolandMohamed A. Khamsi, USAW. A. Kirk, USAV. Lakshmikantham, USAAnthony T. Lau, CanadaLai Jiu Lin, ChinaJuan J. Nieto, SpainDonal O’Regan, IrelandSimeon Reich, Israel

B. E. Rhoades, USASatit Saejung, ThailandD. R. Sahu, IndiaNaseer Shahzad, Saudi ArabiaBrailey Sims, AustraliaTomonari Suzuki, JapanAndrzej Szulkin, SwedenWataru Takahashi, JapanFabio Zanolin, Italy

Contents

Impact of Kirk’s Results on the Development of Fixed Point Theory, Mohamed A. Khamsi andTomas Dominguez-BenavidesVolume 2010, Article ID 821961, 2 pages

Ergodic Retractions for Families of Asymptotically Nonexpansive Mappings, Shahram SaeidiVolume 2010, Article ID 281362, 7 pages

Fixed Points of Discontinuous Multivalued Operators in Ordered Spaces with Applications,Shihuang Hong and Zheyong QiuVolume 2010, Article ID 745769, 13 pages

Some Variational Results Using Generalizations of Sequential Lower Semicontinuity,Ada Bottaro Aruffo and Gianfranco BottaroVolume 2010, Article ID 323487, 21 pages

Moduli and Characteristics of Monotonicity in Some Banach Lattices, Paweł Foralewski,Henryk Hudzik, Radosław Kaczmarek, and Miroslav KrbecVolume 2010, Article ID 852346, 22 pages

Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results,Ishak Altun and Dorel MihetVolume 2010, Article ID 782680, 11 pages

A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds,Massimo Furi, Maria Patrizia Pera, and Marco SpadiniVolume 2010, Article ID 845631, 11 pages

Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces,Abdul Rahim Khan, Hafiz Fukhar-ud-din, and Abdul Aziz DomloVolume 2010, Article ID 385986, 11 pages

Demiclosed Principle for Asymptotically Nonexpansive Mappings in CAT(0) Spaces,B. Nanjaras and B. PanyanakVolume 2010, Article ID 268780, 14 pages

On Some Properties of Hyperconvex Spaces, Marcin Borkowski, Dariusz Bugajewski,and Dev PhularaVolume 2010, Article ID 213812, 19 pages

Fixed Point Theorems for Set-Valued Contraction Type Maps in Metric Spaces,A. Amini-Harandi and D. O’ReganVolume 2010, Article ID 390183, 7 pages

Generalized IFSs on Noncompact Spaces, Alexandru Mihail and Radu MiculescuVolume 2010, Article ID 584215, 15 pages

Common Fixed Point of Multivalued Generalized ϕ-Weak Contractive Mappings,Behzad Djafari Rouhani and Sirous MoradiVolume 2010, Article ID 708984, 13 pages

Convergence of Inexact Iterative Schemes for Nonexpansive Set-Valued Mappings,Simeon Reich and Alexander J. ZaslavskiVolume 2010, Article ID 518243, 10 pages

Does Kirk’s Theorem Hold for Multivalued Nonexpansive Mappings?,T. Domınguez Benavides and B. GaviraVolume 2010, Article ID 546761, 20 pages

Convergence of the Sequence of Successive Approximations to a Fixed Point, Tomonari SuzukiVolume 2010, Article ID 716971, 14 pages

Fixed Point Theorems on Spaces Endowed with Vector-Valued Metrics,Alexandru-Darius Filip and Adrian PetruselVolume 2010, Article ID 281381, 15 pages

Normal Structure and Common Fixed Point Properties for Semigroups of NonexpansiveMappings in Banach Spaces, Anthony To-Ming LauVolume 2010, Article ID 580956, 14 pages

A Continuation Method for Weakly Kannan Maps, David Ariza-Ruiz andAntonio Jimenez-MeladoVolume 2010, Article ID 321594, 12 pages

Fixed Point Theorems for Nonlinear Operators with and without Monotonicity in PartiallyOrdered Banach Spaces, Hui-Sheng Ding, Jin Liang, and Ti-Jun XiaoVolume 2010, Article ID 108343, 11 pages

Weak and Strong Convergence of an Implicit Iteration Process for an AsymptoticallyQuasi-I-Nonexpansive Mapping in Banach Space, Farrukh Mukhamedov and Mansoor SaburovVolume 2010, Article ID 719631, 13 pages

A Kirk Type Characterization of Completeness for Partial Metric Spaces, Salvador RomagueraVolume 2010, Article ID 493298, 6 pages

Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings InvolvingOrbits, Adriana NicolaeVolume 2010, Article ID 458265, 19 pages

Halpern’s Iteration in CAT(0) Spaces, Satit SaejungVolume 2010, Article ID 471781, 13 pages

Equivalent Extensions to Caristi-Kirk’s Fixed Point Theorem, Ekeland’s Variational Principle,and Takahashi’s Minimization Theorem, Zili WuVolume 2010, Article ID 970579, 20 pages

Regularization and Iterative Methods for Monotone Variational Inequalities, Xiubin Xu andHong-Kun XuVolume 2010, Article ID 765206, 11 pages

Fixed Points for Discontinuous Monotone Operators, Yujun Cui and Xingqiu ZhangVolume 2010, Article ID 926209, 11 pages

Periodic Point, Endpoint, and Convergence Theorems for Dissipative Set-Valued DynamicSystems with Generalized Pseudodistances in Cone Uniform and Uniform Spaces,Kazimierz Włodarczyk and Robert PlebaniakVolume 2010, Article ID 864536, 32 pages

On Fixed Points of Maximalizing Mappings in Posets, S. HeikkilaVolume 2010, Article ID 634109, 8 pages

Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive NonselfMappings, Yan Hao, Sun Young Cho, and Xiaolong QinVolume 2010, Article ID 218573, 11 pages

Properties WORTH and WORTHH∗, (1 + δ) Embeddings in Banach Spaces with1-Unconditional Basis and wFPP, Helga Fetter and Berta Gamboa de BuenVolume 2010, Article ID 342691, 7 pages

Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed PointIterations, Teck-Cheong LimVolume 2010, Article ID 821928, 13 pages

Fixed Points for Pseudocontractive Mappings on Unbounded Domains, Jesus Garcıa-Falset andE. Llorens-FusterVolume 2010, Article ID 769858, 17 pages

Browder’s Convergence for Uniformly Asymptotically Regular Nonexpansive Semigroups inHilbert Spaces, Genaro Lopez Acedo and Tomonari SuzukiVolume 2010, Article ID 418030, 8 pages

Coincidence Theorems for Certain Classes of Hybrid Contractions, S. L. Singh and S. N. MishraVolume 2010, Article ID 898109, 14 pages

A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem andVariational Inequality Problem, Filomena Cianciaruso, Giuseppe Marino, Luigi Muglia,and Yonghong YaoVolume 2010, Article ID 383740, 19 pages

A New System of Generalized Nonlinear Mixed Variational Inclusions in Banach Spaces,Jian Wen PengVolume 2010, Article ID 908490, 15 pages

Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications,M. A. Ahmed and F. M. ZeyadaVolume 2010, Article ID 647085, 10 pages

Fixed Points of Single- and Set-Valued Mappings in Uniformly Convex Metric Spaces with NoMetric Convexity, Rafa Espınola, Aurora Fernandez-Leon, and Bozena PiatekVolume 2010, Article ID 169837, 16 pages

Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2010, Article ID 821961, 2 pagesdoi:10.1155/2010/821961

EditorialImpact of Kirk’s Results on the Development ofFixed Point Theory

Mohamed A. Khamsi1, 2 and Tomas Dominguez-Benavides3

1 Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, USA2 Department of Mathematics and Statistics, King Fahd University of Petroleum &Minerals, P.O. Box 411,Dhahran 31261, Saudi Arabia

3 Departamento de Analisis Matematico, Universidad de Sevilla, 41080 Sevilla, Spain

Correspondence should be addressed to Mohamed A. Khamsi, [email protected]

Received 31 December 2010; Accepted 31 December 2010

Copyright q 2010 M. A. Khamsi and T. Dominguez-Benavides. This is an open access articledistributed under the Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.

“The theory of Fixed Points is one of the most powerful tools of modern mathematics” is aquote by Felix Browder, who in the mid-nineteen sixties gave a new impetus to the modernfixed point theory via the development of nonlinear functional analysis as an active andvital branch of mathematics. The metric fixed point theory is a rather loose knit branch offixed point theory concerning methods and results that involve properties of an essentiallyisometric nature. The divide between the metric fixed point theory and the more generaltopological theory is often a vague one. The use of successive approximations to establishthe existence and uniqueness of solutions is at the origin of the metric theory. It goes back toCauchy, Liouville, Lipschitz, Peano, Fredholm, and especially Picard. However, it is the Polishmathematician S. Banach who is credited with placing the underlying ideas into an abstractframework suitable for broad applications well beyond the scope of elementary differentialand integral equations.

Kirk’s fixed point theorem published in 1965 had a profound impact on thedevelopment of the fixed point theory over the last 40 years. Through the concept ofgeneralized distance, Tarski’s classical fixed point theorem may be seen as a variant ofKirk’s fixed point theorem in discrete sets. This shows among other things the power of thistheorem.

This special issue focused on many types of applications of Kirk’s fixed point theorem.It included nonexpansive mappings in Banach and metric spaces, multivalued mappings inBanach and metric spaces, monotone mappings in ordered sets, multivalued mappings inordered sets, applications to nonmetric spaces like modular function spaces, and applicationsto logic programming and directed graphs.

2 Fixed Point Theory and Applications

The special issue contains 38 papers accepted. Most of the papers touched on allapplications of Kirk’s fixed point theorem.

Mohamed A. KhamsiTomas Dominguez-Benavides


Research ArticleErgodic Retractions for Families ofAsymptotically Nonexpansive Mappings

Shahram Saeidi1, 2

1 Department of Mathematics, University of Kurdistan, Sanandaj, P.O. Box 416, Kurdistan, Iran2 School of Mathematics, Institute for Research in Fundamental Sciences (IPM),P.O. Box 19395-5746, Tehran, Iran

Correspondence should be addressed to Shahram Saeidi, [email protected]

Received 2 October 2009; Revised 10 January 2010; Accepted 14 March 2010

Academic Editor: Mohamed Amine Khamsi

Copyright q 2010 Shahram Saeidi. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We prove some theorems for the existence of ergodic retractions onto the set of common fixedpoints of a family of asymptotically nonexpansive mappings. Our results extend correspondingresults of Benavides and Ramırez (2001), and Li and Sims (2002).

1. Introduction

Let E be a Banach space and C a nonempty closed and convex subset of E. We recall somedefinitions.

Definition 1.1. A mapping T : C → C is said to be

(i) nonexpansive if

∥∥Tx − Ty∥∥ ≤ ∥∥x − y∥∥, ∀x, y ∈ C; (1.1)

(ii) asymptotically nonexpansive if there exists a sequence {kn} of positive numberssatisfying the property limn→∞kn = 1 and

∥∥Tnx − Tny∥∥ ≤ kn∥∥x − y∥∥, ∀x, y ∈ C; (1.2)


(iii) of asymptotically nonexpansive type if for each x in C, we have

lim supn→∞

supy∈C

(∥∥Tnx − Tny∥∥ − ∥∥x − y∥∥) ≤ 0; (1.3)

(iv) weakly asymptotically nonexpansive if it satisfies the condition

lim supn→∞

∥∥Tnx − Tny∥∥ ≤ ∥∥x − y∥∥, ∀x, y ∈ C. (1.4)

(v) retraction if T2 = T . A subset F of C is called a nonexpansive retract of C if eitherF = ∅ or there exists a retraction of C onto F which is a nonexpansive mapping.

Definition 1.2. We say that a nonempty closed convex subset D of C satisfies property (ω)with respect to

(i) a mapping T : C → C if ωT (x) ⊂ D for every x ∈ D where

ωT (x) ={y ∈ C : y = w − lim

kTnk(x) for some nk −→ ∞

}, (1.5)

(ii) a semigroup of mappings ϕ = {T(t) : C → C : t ≥ 0} if ωϕ(x) ⊂ D for every x ∈ Dwhere

ωϕ(x) ={y ∈ C : y = w − lim

iT(ti)(x) for some ti ↑ ∞

}. (1.6)

Obviously, C itself verifies (ω).

Definition 1.3. (i) A mapping T : C → C is said to satisfy the (ω)-fixed point property ((ω)-fpp) if T has a fixed point in every nonempty closed convex subset D of C which satisfies (ω)with respect to T .

(ii) A semigroup ϕ = {T(t) : C → C : t ≥ 0} is said to satisfy the (ω)-fpp if ϕ has acommon fixed point in every nonempty closed convex subset D of C which satisfies (ω) withrespect to the semigroup ϕ.

(iii) A family ϕ = {Ti : C → C : i ∈ I} is said to satisfy the (ω)-fpp if ϕ has a commonfixed point in every nonempty closed convex subset D of C which satisfies (ω) with respectto each Ti.

In 1965, Kirk [1] proved that if C is a weakly compact convex subset of a Banach spacewith normal structure, then every nonexpansive mapping T : C → C has a fixed point. (Anonempty convex subset C of a normed linear space is said to have normal structure if eachbounded convex subset K of C consisting of more than one point contains a nondiametralpoint). Goebel and Kirk [2] proved that if E is assumed to be uniformly convex, then everyasymptotically nonexpansive self-mapping T of C has a fixed point. This was extended tomappings of asymptotically nonexpansive type by Kirk in [3]. However, whether normalstructure implies the existence of fixed points for mappings of asymptotically nonexpansive

Fixed Point Theory and Applications 3

type is a natural and still open question. Li and Sims [4] proved the following fixed pointresult in the case that E has uniform normal structure (It is known that a space with uniformnormal structure is reflexive and that all uniformly convex or uniformly smooth Banachspaces have uniform normal structure).

Theorem 1.4. Suppose E is a Banach space with uniform normal structure;C is a nonempty boundedsubset of E. Then

(i) every continuous and asymptotically nonexpansive type mapping T : C → C satisfies(ω)-fpp;

(ii) every semigroup ϕ = {T(t) : C → C : t ≥ 0} of asymptotically nonexpansive typemappings on C such that T(t) is continuous on C for each t ≥ 0 satisfies (ω)-fpp.

On the other hand, Bruck [5] initiated the study of the structure of the fixed point setF(T) = {x : Tx = x} in a general Banach space E: if C is a weakly compact convex subset ofE and T : C → C is nonexpansive and satisfies a conditional fixed point property, then F(T)is a nonexpansive retract of C. The same author [6] used this fact to derive the existence offixed points for a commuting family of nonexpansive mappings. See, for example, [7, 8] forsome related results.

Benavides and Ramırez [9] studied the structure of the set of fixed points for (weakly)asymptotically nonexpansive mappings.

Theorem 1.5. Let E be a Banach space and C a nonempty weakly compact convex subset of E.Assume that every asymptotically nonexpansive self-mapping of C satisfies the (ω)-fpp. Then for anycommuting family ϕ of asymptotically nonexpansive self-mappings of C, the common fixed point setof ϕ, F(ϕ), is a nonempty nonexpansive retract of C.

In this paper, we prove some theorems to guarantee the existence of nonexpansiveretractions onto the common fixed points of some families of (weakly) asymptoticallynonexpansive (type) mappings. The results obtained in this paper extend in some sense, forexample, Theorems 1.4 and 1.5, above.

2. Nonexpansive Retractions for Families ofWeakly Asymptotically Nonexpansive Mappings

Theorem 2.1. Let C be a nonempty weakly compact convex subset of a Banach space E, and ϕ = {Ti :i ∈ I} a family of weakly asymptotically nonexpansive mappings on C such that F(ϕ)/= ∅. Assumeone of the following assumptions is satisfied:

(a) ϕ satisfies the (ω)-fpp;

(b) F(ϕ) is a nonexpansive retract of C.

Then for each α ∈ I, there exists a nonexpansive retraction Pα from C onto F(ϕ), the commonfixed points of ϕ, such that PαTα = TαPα = Pα, and every closed convex ϕ-invariant subset of C is alsoPα-invariant.


Proof. Consider CC with the product topology induced by the weak topology on C. Now,consider an α ∈ I and define

R :={T ∈ CC : T is nonexpansive, T ◦ Tα = T,

and every closed convex ϕ-invariant subset of C is also T -invariant} (2.1)

By applying an argument similar to that in the proof of [9, Theorem 2], it follows that R iscompact (the topology on R is that of weak pointwise convergence) and there is a minimalelement Pα ∈ R in the following sense:

if T ∈ R and∥∥T(x) − T(y)∥∥ ≤ ∥∥Pα(x) − Pα(y)∥∥, ∀x, y ∈ C,

then∥∥T(x) − T(y)∥∥ =

∥∥Pα(x) − Pα(y)∥∥.( ∗)

First, we assume the case (a). We shall prove that Pα(x) ∈ F(ϕ) for all x ∈ C. For agiven x ∈ C, consider the set K = {T(Pα(x)) : T ∈ R}. Then K is a nonempty weakly compactconvex subset of C, because R is convex and compact. We will show that for all i ∈ I, Ksatisfies property (ω) with respect to Ti. Fix i ∈ I and take y ∈ K and z ∈ Csuch as Tnki y ⇀ z,

for some nk → ∞. There exists h ∈ R such that y = h(Pα(x)). Consider a subnet {Tnk(η)i } of

{Tnki } such that S(u) = ω − limηTnk(η)i (u) exists for every u ∈ C. Now, taking u = h(Pα(x)), we

have z = S(h(Pα(x))). Since S is nonexpansive, h ∈ R, and S ◦ h ◦ Tα = S ◦ h, it follows thatS ◦ h ∈ R and then z = S(h(Pα(x))) ∈ K. Thus K satisfies the property (ω) with respect to Ti.Since, ϕ satisfies the (ω)-fpp (by (a)), it follows that K ∩ F(ϕ)/= ∅. So, there exists h ∈ R withh(Pα(x)) ∈ F(ϕ). Let y = h(Pα(x)). Then Pα(y) = h(y) = y, and by using the minimality of Pα,we have

∥∥Pα(x) − y∥∥ =∥∥Pα(x) − Pα(y)∥∥ =

∥∥h(Pα(x)) − h(Pα(y))∥∥ =∥∥h(Pα(x)) − y∥∥ = 0. (2.2)

So, we get Pα(x) = y ∈ F(ϕ). Since this is so for each x ∈ C and Pα belongs to R, it followsthat P 2

α = Pα and PαTα = TαPα = Pα.Now, we assume the case (b). From (b), there is a nonexpansive retraction R from C

onto F(ϕ). Put ϕ′ := ϕ∪{R}. Since F(ϕ) = F(ϕ′), we can replace ϕ by ϕ′ in the above assertionsto obtain a minimal element Pα ∈ R in the sense (∗), where R ia defined here as

{T ∈ CC : T is nonexpansive, T ◦ Tα = T,

and every closed convex ϕ’-invariant subset of C is also T -invariant}.

(2.3)

We note that R ◦ T ◦ Tα = R ◦ T , (∀T ∈ R). Since R ∈ ϕ′, every closed convex ϕ′-invariantsubset of C is also R-invariant and consequently R ◦ T -invariant, (∀T ∈ R). So it is easy to seethat R ◦ T ∈ R, (∀T ∈ R). Therefore, for every x ∈ C, the set K = {T(Pα(x)) : T ∈ R} is anR-invariant subset of C. So, considering the fact that R(K) ⊆ K ∩R(C) = K ∩ F(ϕ), we obtainK ∩ F(ϕ)/= ∅. Now, we can repeat the argument used in the last paragraph to get the desiredresult.


A nonexpansive retraction satisfying the thesis of Theorem 2.1 is usually called anergodic retraction (see e.g., [10, 11]).

Combining Theorem 1.5 [9, Theorem 4] and Theorem 2.1(a), we get the followingimprovement of Theorem 1.5.

Corollary 2.2. Let E be a Banach space and C a nonempty weakly compact convex subset of E.Assume that every asymptotically nonexpansive self-mapping of C satisfies (ω)-fpp. Then for anycommuting family ϕ = {Ti : i ∈ I} of asymptotically nonexpansive self-mappings of C and for eachi ∈ I, there exists a nonexpansive retraction Pi from C onto F(ϕ), such that PiTi = TiPi = Pi, andevery closed convex ϕ-invariant subset of C is also Pi-invariant.

3. Ergodic Retractions for a Semigroup ofAsymptotically Nonexpansive Type

Assume that S is a semigroup and l∞(S) is the space of all bounded real-valued functionsdefined on S with supremum norm. For s ∈ S and f ∈ B(S), we define elements lsf and rsfin B(S) by (lsf)(t) = f(st) and (rsf)(t) = f(ts) for each t ∈ S, respectively. An element μ ofl∞(S)∗ is said to be a mean on X if ‖μ‖ = μ(1) = 1. We often write μt(f(t)) instead of μ(f) forμ ∈ l∞(S)∗ and f ∈ l∞(S). A mean μ is said to be invariant if μ(lsf) = μ(rsf) = μ(f) for eachs ∈ S and f ∈ l∞(S). S is said to be amenable if there is an invariant mean on l∞(S). As is wellknown, S is amenable when it is a commutative semigroup [12].

The following result which we need is well known (see [13]).

Lemma 3.1. Let f be a function of a semigroup S into E such that the weak closure of {f(t) : t ∈ S} isweakly compact. Then, for any μ ∈ l∞(S)∗, there exists a unique element fμ in E such that 〈fμ, x∗〉 =μt〈f(t), x∗〉 for all x∗ ∈ E∗. Moreover, if μ is a mean, then fμ ∈ co{f(t) : t ∈ S}.

We can write fμ by∫f(t)dμ(t). As a direct consequence of Lemma 3.1, we have the

following lemma.

Lemma 3.2. Let C be a nonempty closed convex subset of a Banach space E, ϕ = {T(t) : t ≥ 0} asemigroup of weakly asymptotically nonexpansive mappings on C such that weak closure of {T(t)x :t ≥ 0} is weakly compact for each x ∈ C, and μ a mean on l∞(R+).

If we write Tμx instead of∫T(t)x dμ(t), then the following hold.

(i) Tμx = x for each x ∈ F(ϕ).(ii) Tμx ∈ co{T(t)x : t ≥ 0} for each x ∈ C.(iii) If μ is invariant, then TμT(t) = Tμ for each t ≥ 0 and Tμ is a nonexpansive mapping from

C into itself.

Proof. We only need to prove that Tμ is nonexpansive: consider x, y ∈ C and x∗ ∈ J(Tμx−Tμy).Then for each s ≥ 0, we have

∥∥Tμx − Tμy∥∥2 =⟨Tμx − Tμy, x∗

⟩= μt

⟨T(t)x − T(t)y, x∗⟩ = μt⟨T(t + s)x − T(t + s)y, x∗⟩

≤ ∥∥Tμx − Tμy∥∥supt≥0

∥∥T(t + s)x − T(t + s)y∥∥. (3.1)


Therefore, ‖Tμx − Tμy‖ ≤ supt≥0‖T(t + s)x − T(t + s)y‖, for every s ≥ 0. Consequently, we get

∥∥Tμx − Tμy∥∥ ≤ infs≥0

supt≥0

∥∥T(t + s)x − T(t + s)y∥∥ ≤ ∥∥x − y∥∥. (3.2)

The following is our main result which is an improvement of Theorem 1.4 [4, Theorem2.2].

Theorem 3.3. Suppose E is a Banach space with uniform normal structure;C is a nonempty boundedclosed and convex subset of E; ϕ = {T(t) : t ≥ 0} is a semigroup of asymptotically nonexpansive typemappings on C such that T(t) is continuous on C for each t ≥ 0. Then there exists a nonexpansiveretraction P from C onto F(ϕ), such that PT(t) = T(t)P = P for each t ∈ S, and every closed convexϕ-invariant subset of C is also P -invariant.

Proof. Consider CC with the product topology induced by the weak topology on C. Now,define

R :={T ∈ CC : T is nonexpansive, T ◦ T(t) = T, ∀t ≥ 0

and every closed convex ϕ-invariant subset of C is also T -invariant}.

(3.3)

We note that R/= ∅, because the mapping Tμ in Lemma 3.2 belongs to R. By applying anargument similar to that in the proof of [9, Theorem 2] (see also the proof of [7, Lemma 3.1]),it follows that R is compact and there is a minimal element P ∈ R in the following sense:

if T ∈ R and∥∥T(x) − T(y)∥∥ ≤ ∥∥P(x) − P(y)∥∥, ∀x, y ∈ C,

then∥∥T(x) − T(y)∥∥ =

∥∥P(x) − P(y)∥∥. (3.4)

We will prove that P(x) ∈ F(ϕ) for all x ∈ C. For a given x ∈ C, consider the set K ={T(P(x)) : T ∈ R}. Then K is a nonempty weakly compact convex subset of C, because R

is convex and compact. Take y ∈ K and z ∈ Csuch as T(ti)y ⇀ z, for some ti ↑ ∞. Thereexists h ∈ R, such that y = h(P(x)). Consider a subnet {T(ti(η))} of {T(ti} such that S(u) =ω − limηT(ti(η))(u) exists for every u ∈ C. Now, taking u = h(P(x)), we have z = S(h(P(x))).Since S is nonexpansive, h ∈ R and S ◦h ◦T(t) = S ◦h for every t ≥ 0, it follows that S ◦h ∈ R

and then z = S(h(P(x))) ∈ K. ThusK satisfies the property (ω) with respect to the semigroupϕ. Now, from Theorem 1.4, it follows that K ∩ F(ϕ)/= ∅. From this and the argument used inthe proof of Theorem 2.1, we obtain P(x) ∈ F(ϕ). Since this holds for each x ∈ C, P 2 = P .

Acknowledgments

The author would like to thank the referee for useful comments and for pointing out anoversight regarding an earlier draft of this paper. This paper is dedicated to Professor WilliamArt Kirk. This research was in part supported by a Grant from IPM (no. 88470021).


References

[1] W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” Proceedings of theAmerican Mathematical Society, vol. 72, pp. 1004–1006, 1965.

[2] K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.

[3] W. A. Kirk, “Fixed point theorems for non-lipschitzian mappings of asymptotically nonexpansivetype,” Israel Journal of Mathematics, vol. 17, pp. 339–346, 1974.

[4] G. Li and B. Sims, “Fixed point theorems for mappings of asymptotically nonexpansive type,”Nonlinear Analysis: Theory, Methods & Applications, vol. 50, pp. 1085–1091, 2002.

[5] R. E. Bruck, “Properties of fixed-point sets of nonexpansive mappings,” Transactions of the AmericanMathematical Society, vol. 179, pp. 251–262, 1973.

[6] R. E. Bruck, “A common fixed point theorem for a commuting family of nonexpansive mappings,”Pacific Journal of Mathematics, vol. 53, pp. 59–71, 1974.

[7] S. Saeidi, “Ergodic retractions for amenable semigroups in Banach spaces with normal structure,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2558–2563, 2009.

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Research ArticleFixed Points of Discontinuous MultivaluedOperators in Ordered Spaces with Applications

Shihuang Hong and Zheyong Qiu

Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China

Correspondence should be addressed to Shihuang Hong, [email protected]

Received 24 September 2009; Accepted 3 March 2010

Academic Editor: T. Dominguez Benavides

Copyright q 2010 S. Hong and Z. Qiu. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Existence theorems of fixed points for multivalued increasing operators in partially ordered spacesare presented. Here neither the continuity nor compactness is assumed for multivalued operators.As an application, we lead to the existence principles for integral inclusions of Hammerstein typemultivalued maps.

1. Introduction

The influence of fixed point theorems for contractive and nonexpansive mappings (see [1, 2])on fixed point theory is so huge that there are many results dealing with fixed points ofmappings satisfying various types of contractive and nonexpansive conditions. On the otherhand, it is also huge that well-known Brouwer’s and Schauder’s fixed point theorems forset-contractive mappings exert an influence on this theory. However, if a mapping is notcompletely continuous, in general, it is difficult to verify that the mapping satisfies the set-contractive condition. In 1980, Monch [3] has obtained the following important fixed pointtheorem which avoids the above mentioned difficulty.

Theorem 1.1. Let E be a Banach space,K ⊂ E a closed convex subset. Suppose that (single) operatorF : K → K is continuous and satisfies that

(i) there exists x ∈ K such that if C ⊂ K ∩ co({x} ∪ F(C)) is countable, then C is relativelycompact,

then F has a fixed point in K.

It has been observed that continuity is an ideal and important property in the abovecited works, while in some applications the mapping under consideration may not be


continuous, yet at the same time it may be “not very discontinuous”. this idea has motivatedmany authors to study corresponding problems, for instance, the stability of Brouwer’sfixed point theorem [4], similar result for nonexpansive mappings [5], and existence andapproximation of the synthetic approaches to fixed point theorems [6]. Recently, fixed pointtheory for discontinuous multivalued mappings has attracted much attention and manyauthors studied the existence of fixed points for such mappings. We refer to [7–11]. Forexample, Hong [8] has extended Monch [3] to discontinuous multivalued operators inordered Banach spaces by using a quite weak compactness condition; that is, assuming thefollowing condition is satisfied.

(H) If C = {xn} is a countable totally ordered set and C ⊂ wcl({x1} ∪ A(C)), then C isweakly relatively compact. Here A is a multivalued operator and wcl(B) denotesthe weak closure of the set B.

The purpose of this paper is to present some results on fixed point theorems ofMonch type of multivalued increasing operators for which neither the continuity nor thecompactness is assumed in ordered topological spaces. However, we will use the followinghypothesis.

(H1) If C = {xn} ⊂ K is a countable totally ordered set and C ⊂ cl({x1} ∪ A(C)), then Chas a supremum.

E is a topological vector space endowed with partial ordering “≤”, cl(B) stands for the closureof the set B, and K = {x ∈ E | x ≥ u0}with u0 ∈ E is a given ordered set of E.

This paper is organized as follows. In Section 2, we introduce some definitions andpreliminary facts from partially ordered theory and multivalued analysis which are usedlater. In especial, we introduce a new partial ordering of sets which forms a basis to our mainresults. In Section 3, we state and prove existence of fixed points, also, maximal and minimalfixed point theorem is presented for discontinuous multivalued increasing operators whichare our main results. To illustrate the applicability of our theory, in Section 4, we discuss theexistence of solutions to the Hammerstein integral inclusions of the form

u(t) ∈∫T

0k(t, s)G(s, u(s))ds a.e. on [0, T]. (1.1)

2. Preliminaries

Let (E,≤) be a partially ordered topological vector space. By the notation “x < y” we alwaysmean that x ≤ y and x /=y. Let 2E stand for the collection of all nonempty subsets of E. Takeu0 ∈ E and let Ku0 = {x ∈ E | x ≥ u0} be a given ordered set of E. The ordered interval of E iswritten as [u, v] = {x ∈ E : u ≤ x ≤ v}.

For two subsets P,Q of E, we write P ≤ Q (or Q ≥ P ) if

∀p ∈ P, ∃q ∈ Q such that p ≤ q. (2.1)

Given a nonempty subsets Ω of E we say that A : Ω � 2E is increasing upwards ifu, v ∈ Ω, u ≤ v, and x ∈ A(u) imply that there exists y ∈ A(v) such that x ≤ y. A is increasingdownwards if u, v ∈ Ω, u ≤ v, and y ∈ A(v) imply an existence of x ∈ A(u) such that x ≤ y.If A is increasing upwards and downwards we say that A is increasing.


Let Γ ⊂ E be nonempty. The element y ∈ E is called an upper (lower) bound of Γ ifx ≤ y (x ≥ y) whenever x ∈ Γ. Γ is called upper (lower) bounded with respect to the orderingif its upper (lower) bounds exist. The element z ∈ E is called a supremum of Γ, written asz = sup Γ, if z is an upper bound and z ≤ y as long as y is another upper bound of Γ. Similarly,we can define the infimum inf Γ of Γ.

Throughout this paper, unless otherwise mentioned, the partial ordering of E alwaysintroduced by a closed cone if E is a Banach space. The following lemmas will be used in aftersections.

Lemma 2.1 (see [12]). Let E be an ordered Banach space and B a totally ordered and weakly relativelycompact subset of E, then there exists x∗ ∈ wcl(B) such that x ≤ x∗ foe all x ∈ B.

An ordered topological vector space E is said to have the limit ordinal property ifxn, yn ∈ E with xn ≤ yn for n = 1, 2, . . ., and xn → x∗, yn → y∗ for n → ∞ imply x∗ ≤ y∗. Byan analogy of the proof of Lemma 1.1.2 in [12], we have the following.

Lemma 2.2. If E has the limit ordinal property and {xn} is a relatively compact monotone sequence ofE, then {xn} is convergent. Moreover, xn ≤ x∗ if {xn} is increasing and x∗ ≤ xn if {xn} is decreasingfor n = 1, 2, . . . . Here limn→∞xn = x∗.

Remark 2.3. Under the assumptions of Lemma 2.2, it is evident that x∗ is the supremum(infimum) of increasing (decreasing) sequence {xn}.

Lemma 2.4. Let the increasing sequence {xn} have the supremum z. If {xni} is a infinity subsequenceof {xn}, then {xni} has the supremum z, too.

Proof. Evidently, z is an upper bound of {xni}. Let y be the other one, then xni ≤ y for i =1, 2, . . . . For any given n, since {xni} is infinity, there exists i0 such that xn ≤ xni0 , which impliesthat xn ≤ y for all n ≥ 1. From the definition of supremums it follows that z ≤ y, that is, z isthe supremum of {xni}.

Lemma 2.5. Suppose that every countable totally ordered subset of the partially ordered set Y has asupremum in Y . Let the operator F : Y → Y satisfy F(x) ≥ x for all x ∈ Y , then there exists x0 ∈ Ysuch that F(x0) = x0.

Proof. Take z0 ∈ Y any fixed and let zi+1 = F(zi) for i = 0, 1, . . ., then zi+1 ≥ zi that is, {zi} isincreasing. From our assumption it follows that {zi} has a supremum denoted by z1

0 = sup zi.Let

Γ1 = {z0, z1, . . .} ∪{z1

0

}. (2.2)

If z10 = F(z1

0), then the conclusion of the lemma is proved. Otherwise, take z1i = F(z1

i−1) fori = 1, 2, . . . .Again, the set {z1

0, z11, . . .} has the supremum z2

0 = sup z1i . Denote Γ2 = {z1

0, z11, . . .}∪

{z20}. If z2

0 = F(z20), then the conclusion of the lemma is proved. Otherwise, take z2

i = F(z2i−1)

for i = 1, 2, . . ., and let Γ3 = {z20, z

21, . . .} ∪ {z3

0} with z30 = sup z2

i . In general, having definedΓk = {zk−1

0 , zk−11 , . . .} ∪ {zk0} with zk−1

i = F(zk−1i ) and zk0 = sup zk−1

i , where z0i = zi and k, i =

1, 2, . . ., if zk0 = F(zk0), which completes the proof. Otherwise, repeating this process, either theconclusion of the lemma is proved, or we can obtain a set sequence Γ1,Γ2, . . . satisfying


(i) Γk = {zk−11 , zk−1

2 , . . .} ∪ {zk0} with zk0 = sup zk−1i and zki = F(zki−1), i, k = 1, 2, . . .;

(ii) zki−1 ≤ zki for i, k = 1, 2, . . .;

(iii) zk−1j ≤ zkt j, t = 0, 1, 2, . . ., and z0

i = zi.

Let Γ =⋃∞k=1 Γk, then Γ is a countable subset and

z0 ≤ x ∀x ∈ Γ. (2.3)

We claim that

F(Γ) ⊂ Γ. (2.4)

In fact, for any y ∈ F(Γ), there exists x ∈ Γ such that y = F(x). There exists Γk such thatx ∈ Γk. If x = zk−1

i for some nature number i, then y = F(x) = zk−1i+1 ∈ Γk which yields y ∈ Γ.

Otherwise, we have x = sup zk−1i = zk0 ∈ Γk+1. This implies that y = F(x) = F(k0) = zk1 ∈ Γk+1.

Consequently, y ∈ Γ. From the arbitrariness of y it follows that (2.4) is satisfied.Finally, combining (ii) and (iii) we see easily that Γ is totally ordered. Our hypothesis

guarantees that Γ has a supremum, written as x∗ = sup Γ. Note that (2.4) guarantees F(x∗) ∈Γ, we have F(x∗) ≤ x∗. On the other hand, the definition of F ensures that F(x∗) ≥ x∗. HenceF(x∗) = x∗. This proof is completed.

Let Ω be a nonempty subset ofKu0 . In this section we impose the following hypotheseson the increasing upwards multivalued operator A : Ω � 2E. Set

R = {x ∈ Ω | there exists u ∈ Ax such that x ≤ u} (2.5)

and for any x ∈ R define that

C(x) = {x, u1, u2, . . . , un, . . .}, D(x) = C(x) ∪ {w(x)}, (2.6)

where, w(x) = supC(x) and ui (i = 1, 2, . . .) is given as follows: since x ∈ R, there existsu1 ∈ Ax such that x ≤ u1. In virtue of the fact that A is increasing upwards, there existsu2 ∈ Au1 such that u1 ≤ u2. On the analogy of this process, there exists un ∈ Aun−1 suchthat un−1 ≤ un for n = 2, 3, . . . , Obviously, C(x) ⊂ cl({x} ∪ A(C)), thus, the condition (H1)guarantees that the supremum w(t) of C(x) exists.

Remark 2.6. In general, the sequences of these kinds, {un}, may not be unique, that is, every{un} corresponds to C(x), moreover, corresponds to D(x). For given x ∈ R, we denote withC(x) and D(x) the families of C(x) and D(x) as above, respectively.

In addition, if E has the limit ordinal property, D(x) is a closed set for any x ∈ R. Infact, let {uni} be any infinity subsequence of D for which

uni −→ x∗ for i −→ ∞. (2.7)

observing that {uni} is increasing, by Lemma 2.2 we get that x∗ is a supremum of {uni} andby Lemma 2.4 we get w(x) = x∗.


Definition 2.7. A set Γ is said to be sup-closed if the supremum of each countable subset ofΓ (provided that it exists) belongs to Γ. A multivalued operator A : Ω � 2E is said to havesup-closed values if Ax is sup-closed for each x ∈ Ω.

Defining

X(x) = {u : there exists D(x) ∈ D(x) such that u ∈ D(x)}. (2.8)

Lemma 2.8. Let E be an ordered topological space, Ω a nonempty subset of Ku0 with u0 ∈ E; letA : Ω � 2E have sup-closed values and satisfy hypothesis (H1). Moreover, assume that

(H2) A is increasing upwards and satisfies u0 ≤ Au0,

then for any C(x) ∈ C(x), C(x) has the supremum w(x) which belongs to R, that is,

w(x) ≤ x∗ for some x∗ ∈ A(w(x)). (2.9)

Proof. It is clear that C(x) has the supremum w(x) ∈ E. For any ui ∈ C(x)/{x}, fromui ∈ A(ui−1) and ui−1 ≤ w(x) there exists xi ∈ A(w(x)) such that ui ≤ xi. We can assumethat the sequence {xi} is increasing. Indeed, if xi ≤ xi+1 for i = 1, 2, . . ., our purpose isreached. Otherwise, there exists i0 such that xi0/≤xi0+1, then we take xi0+1 instead of xi0 . LetM = {w(x), x1, x2, . . . , xn, . . .}, then M ⊂ cl({w(x)} ∪A(M)). Condition (H1) guarantees thatM has a supremum x∗ = supM. Clearly,w(x) ≤ x∗. By virtue of the fact thatA has sup-closedvalues, we have x∗ ∈ A(w(x)). This proof is complete.

For the sake of convenience, in this paper, by w(x) we always stand for the supremumof C(x). For given x ∈ R, let W(x) be a set consisting of all w(x) given as in Lemma 2.8,thenW is an increasing map. Now for any un ∈ C(x) Lemma 2.4 shows D(un) ⊂ D(x), thus,W(un) ⊂ W(x). Define

Z = {D(x) : x ∈ R}. (2.10)

It is obvious that D(u0) ∈ Z. Hence, Z is nonempty. A relation “≤′′1 on Z is defined as follows(it is easy to see that (Z,≤1) is a partially ordered set):

D(x) = D(y)⇔ x = y, w(x) = w(y);

D(x)<1D(y)⇔ x < y and w(x) ≤ w(y).

Remark 2.9. It is clear we may assume that, for any u ∈ D(x), there exists v ∈ D(y) such thatu ≤ v if D(x)<1D(y).

Let us assume that there exists some u0 ∈ R such that

(H3) W(u0) ⊂ cl(A(X(u0))).

Define

S = {D(x) : x ∈ R, W(x) ⊂ cl(A(X(x)))}. (2.11)


Obviously, D(u0) ∈ S, that is, S is nonempty if A is increasing upwards. Now we denote ≤2

as a relation on S defined by, for any D(x),D(y) ∈ S,

(I) D(x) = D(y)⇔ x = y;

(II) D(x)<2D(y) ⇔ (a) for all D(x) ∈ D(x), there exists D(y) ∈ D(y) such thatD(x)<1D(y) and

(b) there exists a countable at most and totally ordered subset Q ⊂ R such that

(b1) x < q < y for any q ∈ Q.(b2) There exists D(x) ∈ D(x) such that

y ∈ cl

⎛⎝{w(x)} ∪

⋃q∈QW(q)

⎞⎠,

{y} ≥ W(q), w(x) ≤ q(∀q ∈ Q); (2.12)

(b3)⋃q∈Q X(q) is a totally ordered set and satisfies

⋃q∈Q X(q) ⊂ cl(W(x) ∪

A(⋃q∈Q X(q))). Q is called a link of linking D(x) with D(y).

Remark 2.10. (b2) may be satisfied. In fact, we can take empty set as a link of linking D(x)andD(y). Thus,D(x)<2D(y) implies that for any D(x) ∈ D(x) we can find D(y) ∈ D(y) suchthat D(x)<1D(y). In this case, we take w(x) = y. Besides, Q can be a finite set, for example,Q = {q1, q2, . . . , qm} with q1 < q2 < · · · < qm, then q1 = infW(x), y = supW(qm). (b3) and thecondition (H1) ensure

⋃q∈Q X(q) to exist the supremum, so, from Lemma 2.2 the element y

satisfying (b2) exists.

Lemma 2.11. The relation “≤2” satisfies that

(i) D(x)≤2D(x);(ii) D(x)≤2D(y) and D(y)≤2D(x) implies D(x) = D(y);(iii) D(x)≤2D(y), D(y)≤2D(z) implies D(x)≤2D(z).

Therefore, (S, ≤2) is a partially ordered set.

Proof. (i) and (ii) are satisfied. Trivial by (I) and (II)(a). To prove (iii), for any given D(x) ∈D(x) we take D(y) ∈ D(y) such that D(x)≤1D(y) and we can find D(z) ∈ D(z) such thatD(y)≤1D(z). It is sufficient to assume that at least one of the above equalities does not hold.The definition of <1 guarantees that

x ≤ y ≤ z, (2.13)

w(x) ≤ w(y) ≤ w(z), (2.14)

and at least one strictly inequality in (2.13) holds. The links linking D(x) with D(y) andlinking D(y) with D(z) are written, respectively, as Q′ and Q′′. Let Q = Q′ ∪Q′′ ∪ {y}, for anyq′ ∈ Q′, q′′ ∈ Q′′, if none of equalities in (2.13) holds, then by (b1) we have

x < q′ < y, y < q′′ < z. (2.15)


If at least one equality in (2.13) holds, for instance, x = y, then (b1) and (b2) show that

w(x) ≤ q′ ≤ w(y), w(y) ≤ q′′ < z. (2.16)

Hence, Q ⊂ R is a countable totally ordered subset and satisfies (b1).Next we will prove that Q satisfies (b2). It is clear that the following consequences are

true, that is, z ≥ w(y) ≥ w(x) and z ∈ cl({w(y)} ∪⋃q′′∈Q′′ W(q′′)) ⊂ cl({w(x)} ∪⋃q∈QW(q)). Itis easy to see that {z} ≥ W(q) for all q ∈ Q by {z} ≥ W(q′′) andW(q′′) ≥ W(q′) for all q′ ∈ Q′.Also, w(x) ≤ q for all q ∈ Q.

Finally, we prove that Q satisfies (b3). For all x1, x2 ∈⋃q∈Q X(q), there exist q′, q′′ ∈ Q

with q′ ≤ q′′ (because Q is totally ordered) and D(q′) ∈ D(q′), D(q′′) ∈ D(q′′) such thatx1 ∈ D(q′), x2 ∈ D(q′′). If q′, q′′ ∈ Q′ (or q′, q′′ ∈ Q′′), then x1 and x2 are ordered by (b3).If q′ ∈ Q′, q′′ ∈ Q′′, from (b1) and (b2) it follows that D(q′)<1D(q′′), which shows that x1 ≤w(q′) ≤ q′′ ≤ x2. To conclude,

⋃q∈Q X(q) is totally ordered. Noting that both Q′ and Q′′ have

supremums, by the definition of S, we have

W(y) ⊂ cl(A(X(y)))

. (2.17)

Therefore,

⋃q∈Q

X(q)=

⎛⎝⋃

q∈Q′X(q)⎞⎠⋃

⎛⎝⋃

q∈Q′′X(q)⎞⎠⋃X

(y)

⊂ cl

⎛⎝W(x) ∪A

⎛⎝⋃

q∈Q′X(q)⎞⎠⎞⎠⋃ cl

⎛⎝w

(y) ∪A

⎛⎝⋃

q∈Q′′X(q)⎞⎠⎞⎠⋃X

(y)

⊂ cl

⎛⎝W(x) ∪A

⎛⎝⋃

q∈Q′X(q)⎞⎠⎞⎠⋃

⎛⎝W(y) ∪A

⎛⎝⋃

q∈Q′′X(q)⎞⎠⎞⎠⋃ cl

(A(X(y)))

⊂⎛⎝W(x) ∪A

⎛⎝⋃

q∈QX(q)⎞⎠⎞⎠.

(2.18)

This shows that Q satisfies (b3). Consequently, D(x)<2D(z), which completes this proof.

3. Main Results

Now we can state and prove our main results.

Definition 3.1. u ∈ E is said to be a fixed point of the multivalued operator A if u ∈ A(u). Thefixed point x∗ of A is said to be a maximal fixed point of A if u = x∗ whenever u ∈ A(u) andx∗ ≤ u. If x∗ is a fixed point and if x∗ = u whenever u ∈ A(u) and u ≤ x∗, we say that x∗ is aminimal fixed point of A.


Theorem 3.2. Assume that E is an ordered topological space. Let u0 ∈ E, Ω ⊂ Ku0 be nonempty andthe multivalued operator A : Ω � 2E have sup-closed values such that hypotheses (H1)–(H3) hold.Then A admits at least one fixed point in Ku0 .

Proof. If S has a maximal element D(x∗), then x∗ is a fixed point of A. In fact, since x∗ ∈ R,we can find u ∈ A(x∗) such that x∗ ≤ u. From the definition of C(x∗) we can let u ∈ D(x∗) ∈D(x∗). This implies u ≤ w(x∗). We claim that x∗ = u. Suppose that x∗ < u, then x∗ < w(x∗)and D(x∗)<1D(w(x∗)). Take empty set as a link of linking D(x∗) with D(w(x∗)), we haveD(x∗)<2D(w(x∗)), which contradicts the definition of maximal element.

To prove the existence of maximal element of S, by Zorn’s lemma, is thus sufficientto show that every totally ordered subset of S has an upper bound. Let M be any such asubset of S. To this purpose, we consider the set N =

⋃D(x)∈MX(x). Obviously, N ⊂ Ku0 . We

claim that N is totally ordered. Indeed, for any y1, y2 ∈ N, there exist D(x1),D(x2) ∈M andD(x1) ∈ D(x1), D(x2) ∈ D(x2) such that y1 ∈ D(x1), y2 ∈ D(x2). If D(x1) = D(x2), theny1, y2 is ordered. Otherwise, we can assume that D(x1)<1D(x2), thus, from Lemma 2.8 and(b2) it follows that y1 ≤ w(x1) ≤ x2 ≤ y2. Conclusively, N is a totally ordered subset.

We will prove that any countable totally ordered subset of N has a supremum. It isenough to prove that any given strictly monotone sequence {yn} of N there is a supremum.From the definition of N, there exist D(xn) ∈M and D(xn) ∈ D(xn) such that yn ∈ D(xn) forn = 1, 2, . . . . For any x ∈ R, from the definition of D(x), it follows D(x) has a supremum.Moreover, Lemma 2.4 guarantees that {yn} has a supremum if yn ∈ D(xm) with n ≥ mfor some given m. It is suffices to consider the fact that there exists a subsequence of {yn}(without loss of generality, we may assume that it is {yn} itself) such that yn /∈D(xm) (n/=m).

Case 1. If {yn} is strictly increasing, then yi < yi+1 (i = 1, 2, . . .). We claim that

D(x1)<2D(x2)<2 · · ·<2D(xn)<2 · · · . (3.1)

If it is contrary, there exists some i such that D(xi+1)≤2D(xi). It is easy to know thatD(xi+1)/=D(xi). (b2) implies that w(xi+1) ≤ xi, therefore, yi+1 ≤ w(xi+1) ≤ xi ≤ yi. Thiscontradicts {yn} increasing. The claim follows.

Taking Qi as the link of linking D(xi+1) with D(xi) for i = 1, 2, . . . . Let

C =∞⋃i=1

⎛⎝X(xi) ∪

⎛⎝⋃

q∈Qi

X(q)⎞⎠⎞⎠, (3.2)

(b3) shows that C is countable totally ordered. For any z ∈ C \ {x1}, there exists j such thatz ∈ X(xj) ∪ (

⋃q∈Qj

X(q)). If z = xj , by means of (b2) and

W(xj−1) ∈ cl

(A(X(xj−1)))

, (3.3)


we have

z ∈ cl

⎛⎝W(xj−1

) ∪A⎛⎝ ⋃

q∈Qj−1

X(q)⎞⎠⎞⎠ ⊂ cl

⎛⎝A

(X(xj−1)) ∪A

⎛⎝ ⋃

q∈Qj−1

X(q)⎞⎠⎞⎠

= cl

⎛⎝A

⎛⎝X

(xj−1) ∪⎛⎝ ⋃

q∈Qj−1

X(q)⎞⎠⎞⎠⎞⎠ ⊂ cl(A(C)).

(3.4)

If z ∈ Xj with z/=xj , then, by (3.3), z ∈ cl(A(X(xj))) ⊂ cl(A(C)). If z ∈ ⋃q∈QjX(q) with

z/∈X(xj), then from condition (b3) it follows that z ∈ cl(A(⋃q∈Qj

X(q))) ⊂ cl(A(C)). To sumup, C ⊂ {x1} ∪ cl(A(C)) ⊂ cl({x1} ∪A(C)), which, combining the condition (H1), yields thatC has a supremum. Hence, by Lemma 2.4, {yn} has a supremum.

Case 2. It is clear that {yn} has a supremum when {yn} is decreasing.Now, we prove that N has a maximal element. Suppose, on the contrary, for any

y ∈ N, that there exists y1 ∈ N such that y ≤ y1 and y1 /=y. Let F(y) = y1, then F is anoperator mapping N into N and satisfies F(y) ≥ y and F(y)/=y for every y ∈N. In virtue ofLemma 2.5, there exists y∗ ∈ N such that F(y∗) = y∗. On the other hand, by the definition ofF, we have F(y∗) ≥ y∗ and F(y∗)/=y∗, a contradiction. Therefore, N has a maximal element,that is, there exists x∗ ∈N such that x ≤ x∗ for all x ∈N.

Finally, we shall prove that D(x∗) is an upper bound of M. Since x∗ ∈ N, there existsD(x) ∈ M and D(x) ∈ D(x) such that x∗ ∈ D(x), which implies that x∗ ≤ w(x). On theother hand, since w(x) ∈ N, we have w(x) ≤ x∗. This compels x∗ = w(x). Taking emptyset as a link of linking D(x) with D(w(x)), we have that D(x)≤2D(w(x)) = D(x∗). GivenD(u) ∈M, in virtue of M being totally ordered, or D(u)≤2D(x) which implies D(u)≤2D(x∗);or D(x)<2D(u), which, applying (b2), yields w(x) ≤ u. Therefore, x∗ = w(x) ≤ u. Noting thatu ∈ N, we have that u ≤ x∗. Conclusively, u = x∗, so, by (a) we have D(x∗) = D(u). Thisshows that D(x∗) is an upper bound of M. This proof is completed.

Remark 3.3. We observe that the result of Theorem 3.2 is true under assumptions ofTheorem 3.2 if all “ cl′′ are written as “wcl.′′ The following corollary shows that Theorem 3.2extends and improves the results of [8].

Corollary 3.4. Let E be an ordered Banach space,A : Ω ⊂ E → 2E be a multivalued operator havingnonempty and weakly closed values. Assume that there exists u0 ∈ E such that conditions (H2), (H3)and (H) hold, then A has at least a fixed point.

Proof. Lemma 2.1 shows that there exists x∗ ∈ wcl(C) such that x ≤ x∗ for all x ∈ C. By meansof Eberlein’s theorem and Lemma 2.2 we have that x∗ is the supremum of C, that is, (H1) issatisfied. Moreover, this implies that w(x) ∈ wcl(A(D)), that is, A is upper sequentially orderclosed in the sense of “weak.” Since A has weakly closed values, A has sup-closed values.From Remark 3.3 A has a fixed point.

Corollary 3.5. Let E be a weakly sequently completed ordered Banach space, P a normal cone. If theoperator A is bounded and satisfies conditions (H2) and (H3), then A has at least one fixed point inKu0 .


Proof. It is suffice to prove that condition (H1) holds. Under these hypotheses, everybounded subset is weakly relatively compact (see [4]), which implies that (H1) is true.

In what follows, we shall consider the existence of maximal and minimal fixed points.

Theorem 3.6. Under assumptions of Theorem 3.2, A has a minimal fixed point in Ku0 .

Proof. Let Fix(A) denote the set consisting of fixed points of A. From Theorem 3.2 it followsthat Fix(A) is nonempty. Set S1 = {D(x) : x ∈ R, x ≤ y for y ∈ Fix(A)}. Clearly, S1 ⊂ S and(S1,≤2) is a partially ordered set. By the same methods as to prove Theorem 3.2, we can provethat S1 has a maximal element D(x∗) and x∗ is a fixed point of A in S1. It is easy to see that x∗

is minimal fixed point of A. This completes the proof of Theorem 3.6.

The next result is dual to that of Theorem 3.6.

Theorem 3.7. Assume that E is an ordered topological space. Let v0 ∈ E, Ω ⊂ Kv0 =: {x ∈ E :x ≤ v0} be nonempty and the multivalued operator A : Ω � 2E have inf-closed values such that thefollowing hypotheses are satisfied.

(h1) If C = {xn} ⊂ Kv0 is a countable totally ordered set and C ⊂ cl({x1} ∪A(C)), then C hasa infimum.

(h2) A is increasing downwards and Av0 ≤ v0.

(h3) W(v0) ⊂ cl(A(X(v0))), where W(x) stands for a set which consists of all infimums ofC(x) (its definition is similar toW(x)).

Then A admits at least one fixed point in Kv0 .

Theorem 3.8. Assume that the operatorA is increasing and satisfies conditions (H1)–(H3) and (h1)–(h3), then A has maximal and minimal fixed points on [u0, v0].

Remark 3.9. If E has the limit ordinal property, A is increasing and has nonempty closedvalues. Assume that A([u0, v0]) is relatively sequentially compact and conditions (H3) and(h3) hold, then A has sup-closed and inf-closed values and satisfies conditions (H1) and (h1)on [u0, v0]. Thereby, A has maximal and minimal fixed points on [u0, v0]. In this sense, weextend and improve the corresponding results of Theorem 2.1 in [10].

Corollary 3.10. Let E be a partially ordered Banach space. If there exist u0, v0 ∈ E with u0 ≤ v0 suchthat u0 ≤ Au0, Av0 ≤ v0. Assume that A is increasing, has nonempty closed values, and satisfies oneof the following hypotheses, then A has maximal and minimal fixed points on [u0, v0].

(s1) P is a regular cone.

(s2) If C = {xn} ⊂ [u0, v0] is countable totally ordered subset and C ⊂ cl({x1} ∪A(C)), thenC is relatively compact subset.

(s3) A([u0, v0]) is a weakly relatively compact set.

(s4) [u0, v0] is a bounded ordered interval, and for any countable noncompact subset C ⊂[u0, v0] with α(C)/= 0, one has α(A(C)) < α(C), where α(·) denotes Kuratowskii’snoncompactness measure.

Proof. (s2) implies (H1) and (h1) holds. The rest is clear.


Remark 3.11. (s2) is main condition of [12] for single-valued operators, (s4) is main conditionof [13]. Hence the results presented here extend and improve the corresponding results of theabove mentioned papers.

4. Application

In this section we assume that (E, ‖ · ‖) is a Banach space with partial ordering derived by thecontinuous bounded function ϕ : E → R as follows (see [14]):

x ≤ y iff∥∥x − y∥∥ ≤ ϕ(x) − ϕ(y). (4.1)

To illustrate the ideas involved in Theorem 3.8 we discuss the Hammerstein integralinclusions of the form

u(t) ∈∫T

0k(t, s)G(s, u(s))ds on [0, T]. (4.2)

Here k is a real single-valued function, while G : [0, T] × E → 2E is a multivalued map withnonempty closed values.

Let 0 < T < ∞, I = [0, T], p ∈ [1,∞], q ∈ [0,∞] and r ∈ [1,∞] be the conjugate

exponent of q, that is, 1/q+1/r = 1. Let ‖u‖p = (∫T

0 ‖u(s)‖pds)1/p

denote the norm of the spaceLp(I, E). For u, v ∈ Lp(I, E) stipulate that u ≤ v if and only if u(t) ≤ v(t) with all t ∈ I.

In order to prove the existence of solutions to (4.2) in Lp(I, E) we assume thefollowing.

(S1) The function k : I2 → R+ satisfies that k(t, ·) ∈ Lr(I) and t → ‖k(t, ·)‖r belongs toLp(I).

(S2) G(t, u) is increasing with regard to u for fixed t ∈ [0, T].

(S3) There exist u0, v0 ∈ C(I, E) with u0 ≤ v0 such that u0(t) ≤ G(t, u0(t)) andG(t, v0(t)) ≤ v0(t) for every t ∈ I.

(S4) G(·, x) has a strongly measurable selection on I for each x ∈ E.

(S5) sup{‖u(t)‖ : u(t) ∈ G(t, x)} ≤ h(t) a.e. on I for all x ∈ E. Here h ∈ Lq(I,R+).

Theorem 4.1. Assume that conditions (S1)–(S5) hold, then (4.2) has maximal and minimal solutionsin [u0, v0].

Proof. Define a multivalued operator A as follows:

(Ax)(t) =∫T

0k(t, s)G(s, x(s))ds. (4.3)

(S4) guarantees that A makes sense. For any v ∈ Ax with x ∈ Lp(I, E), there exists u ∈ G(·, x)


such that v(t) =∫T

0k(t, s)u(s)ds. From (S5) and Holder inequality, it follows that

‖v(t)‖ ≤∫T

0k(t, s)‖u(s)‖ds ≤

∫T0h(s)k(t, s)ds ≤ ‖h‖q‖k(t, ·)‖r =: a(t). (4.4)

This implies that v ∈ Lp(I, E), that is, A maps Lp(I, E) into itself. We seek to applyTheorem 3.8. Note that (S1) and (S3) guarantee that u0 ≤ A(u0), A(v0) ≤ v0. For every givent ∈ I and any C(u0) ∈ C(u0), set C(u0)(t) = {xn(t)} with xn(t) ≤ xn+1(t) for n = 1, 2, . . . .Thus, {ϕ(xn(t))} is a decreasing sequence. Note that ϕ is a bounded function, we obtain thatthe sequence {ϕ(xn(t))} is convergent. Hence, for any ε > 0, there exists a natural number n0

such that

‖xm(t) − xn(t)‖ ≤ ϕ(xn(t)) − ϕ(xm(t)) < ε, (4.5)

whenever m > n ≥ n0. This shows that {xn(t)} is a Cauchy sequence, thereby {xn(t)} isconvergent. Lemma 2.2 guarantees sup(C(u0)(t)) = w(u0)(t) = limn→∞xn(t), which yieldsw(u0)(t) ∈ cl(A(C(u0))(t)) ⊂ cl(A(X(u0))(t)). From the arbitrariness of t it follows thatw(u0) is supremum of C(u0). From (4.4) and the dominated convergence theorem, it followsthat w(u0) ∈ Lp(I, E). Moreover, ‖xn −w(u0)‖p → 0 for n → ∞. Consequently, W(u0) ⊂cl(A(X(u0))). Similarly, we have W(v0) ⊂ cl(A(X(v0))). This shows that (H3) and (h3) aresatisfied for t ∈ I. (S2) guarantees that A is increasing. It is easy to see that A has closedvalues. This yields that A has sup-closed and inf-closed values.

Finally, we check conditions (H1) and (h1). Suppose that the set C = {xn} ⊂ Ku0 iscountable, totally ordered, and satisfies C(t) ⊂ cl({x1(t)} ∪ (AC)(t)) for all t ∈ I. We haveto prove that the set C(t) has a supremum. Since C(t) is countable totally ordered, we canassume C(t) = {xn(t) : n ≥ 1} with xn(t) ≤ xn+1(t) for n = 1, 2, . . . . This implies that thesequence {ϕ(xn(t))} is decreasing. In the same way, we can prove that the sequence {xn(t)}is convergent. Again, Lemma 2.2 guarantees that C(t) has a supremum, which implies thatcondition (H1) is satisfied. Similarly, we can prove that condition (h1) holds. All conditionsof Theorem 3.8 are satisfied, consequently, the operator A has minimum and maximum fixedpoints in [u0, v0] and this proof is completed.

Remark 4.2. By comparing the results of Theorem 4.2 in [15] in which Couchouron andPrecup have proved that (4.2) has at least one solution, we omit the conditions that G(t, x) iscontinuous and has compact values in Theorem 4.1.

Acknowledgments

This work was supported by the Natural Science Foundation of Zhejiang Province (Y607178)and the Natural Science Foundation of China (10771048).

References

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[3] H. Monch, “Boundary value problems for nonlinear ordinary differential equations of second orderin Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 4, no. 5, pp. 985–999, 1980.

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[6] W. A. Kirk and L. M. Saliga, “Some results on existence and approximation in metric fixed pointtheory,” Journal of Computational and Applied Mathematics, vol. 113, no. 1, pp. 141–152, 2000.

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[10] S. Carl and S. Heikkila, “Fixed point theorems for multifunctions with applications to discontinuousoperator and differential equations,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1,pp. 56–69, 2004.

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[12] D. Guo, J. Sun, and Z. Liu, Functional Method of Nonlinear Ordinary Differential Equations, ShandongScience and Technology Press, Jinan, China, 1995.

[13] S. Chang and Y. Ma, “Coupled fixed points for mixed monotone condensing operators andan existence theorem of the solutions for a class of functional equations arising in dynamicprogramming,” Journal of Mathematical Analysis and Applications, vol. 160, no. 2, pp. 468–479, 1991.

[14] J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions,” Transactions of theAmerican Mathematical Society, vol. 215, pp. 241–251, 1976.

[15] J.-F. Couchouron and R. Precup, “Existence principles for inclusions of Hammerstein type involvingnoncompact acyclic multivalued maps,” Electronic Journal of Differential Equations, vol. 2002, no. 4, 21pages, 2002.


Research ArticleSome Variational Results Using Generalizations ofSequential Lower Semicontinuity

Ada Bottaro Aruffo and Gianfranco Bottaro

Dipartimento di Matematica, Universita di Genova, Via Dodecaneso 35, 16146 Genova, Italy

Correspondence should be addressed to Ada Bottaro Aruffo, [email protected]

Received 1 October 2009; Accepted 14 February 2010


Copyright q 2010 A. Bottaro Aruffo and G. Bottaro. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Kirk and Saliga and then Chen et al. introduced lower semicontinuity from above, a generalizationof sequential lower semicontinuity, and they showed that well-known results, such as Ekeland’svariational principle and Caristi’s fixed point theorem, remain still true under lower semicon-tinuity from above. In a previous paper we introduced a new concept that generalizes lowersemicontinuity from above. In the present one we continue such study, also introducing othertwo new generalizations of lower semicontinuity from above; we study such extensions, compareeach other five concepts (sequential lower semicontinuity, lower semicontinuity from above, theone by us previously introduced, and the two here defined) and, in particular, we show that theabove quoted well-known results remain still true under one of our such generalizations.

1. Introduction

In [1] Chen et al. proposed the following generalization ([1, Definitions 1.2 and 1.5]).

Definition 1.1. Let (X, τ) be a topological space. Let x ∈ X. A function f : X → [−∞,+∞] issaid to be sequentially lower semicontinuous from above at x (“d-slsc at x”) if (xn)n∈N sequenceof elements of X for which xn → x and (f(xn))n∈N weakly decreasing sequence, impliesf(x) ≤ limn→+∞f(xn). Moreover f is said to be sequentially lower semicontinuous from above(“d-slsc”) if it is sequentially lower semicontinuous from above at x for every x ∈ X.

Actually the same definition was previously considered by Kirk and Saliga in [2,Section 2, definition above Theorem 2.1]. Both in [1, 2] this concept is called lowersemicontinuity from above; furthermore also Borwein and Zhu in [3, Exercise 2.1.4] used thesame concept, naming it partial lower semicontinuity; here we are calling it sequential lowersemicontinuity from above, as it is a generalization of sequential lower semicontinuity.


Moreover the authors of [1] conjectured that, for convex functions on normedspaces, sequential lower semicontinuity from above is equivalent to weak sequential lowersemicontinuity from above (see [1, some rows below Definition 1.5]). We exhibited someexamples showing that such conjecture is false (see [4, Example 3.1 and Examples sketchedin Remark 3.1]).

In [4] we defined the following new concept, that generalizes sequential lowersemicontinuity from above.

Definition 1.2 (see [4, Definition 4.1]). Let (X, τ) be a topological space. Let f be a function,f : X → [−∞,+∞]. Then f is said to be

(i) inf-sequentially lower semicontinuous at x ∈ X (“i-slsc at x”) if (xn)n∈N sequence ofelements of X for which xn → x and limn→+∞f(xn) = inf f, implies f(x) = inf f(equivalently, in the above condition the part limn→+∞f(xn) = inf f can be replacedby f(xn)↘ inf f);

(ii) inf-sequentially lower semicontinuous (“i-slsc”) if it is i-slsc at x for every x ∈ X.

In particular we showed that for convex functions on normed spaces, such concept isequivalent to its weak counterpart ([4, Theorem 4.1]).

Here, with the purpose to continue the study already begun in [4], we define othertwo new concepts (Definitions 3.1), called by us below sequential lower semicontinuity fromabove (bd-slsc) and uniform below sequential lower semicontinuity from above (ubd-slsc), thatgeneralize sequential lower semicontinuity from above and we show the following.

(a) As it already happened for i-slsc, for convex functions on normed spaces, one ofsuch new concepts (bd-slsc) is equivalent to its weak counterpart (Theorem 4.1 andpart (e) of Remarks 3.2); also by means of such result, it can be seen that, for convexfunctions on normed spaces and indifferently with respect to the topology inducedby norm or to the weak topology, i-slsc and bd-slsc are each other equivalent (part(e) of Remarks 3.2, part (b) of Theorem 3.4 and Corollary 4.2).

(b) Some results listed in [1, 2], such as Ekeland’s variational principle and Caristi’sfixed point theorem, remain still true under an hypothesis of ubd-slsc (Section 5).

Moreover we study the five concepts of sequential lower semicontinuity, lowersemicontinuity from above, inf-sequential lower semicontinuity, below sequential lowersemicontinuity from above and uniform below sequential lower semicontinuity from above,supplying further results and examples, with the purpose of getting a comparison betweensuch five concepts, both in the general case (Section 3) and in the case of convex functions(Section 4). In particular, in Theorem 4.10 we prove that every convex ubd-slsc function on aBanach space is continuous in the points of the interior of its effective domain.

Finally, in Section 5, we also give some examples to show that, in the generalization ofEkeland’s variational principle by us proved, some hypotheses cannot be weakened.

2. Notations and Preliminaries

Notations 1. In the sequel, unless otherwise specified, all linear spaces will be consideredon the field F, where F = R or F = C. As convention, in [−∞,+∞], inf ∅ = +∞ and theproduct 0 · (+∞) is considered equal to 0. By N we denote the set of natural numbers (0included), while Z+ := {n ∈ Z : n > 0} and R+ := {x ∈ R : x > 0}; δn,m is the Kronecker


symbol. If Z is a linear space on R or on C, let dim Z denote the algebraic dimensionof Z, and, if A ⊆ Z, let spA and co A denote, respectively, the linear subspace of Z that isgenerated by A and the convex hull of A; if x, y ∈ Z let [x, y] := {λx + (1 − λ)y : λ ∈ [0, 1]}and, if x /=y, let ]x, y] := {λx+(1−λ)y : λ ∈ [0, 1[}; if 0 ∈ A, then the Minkowski functional (orgauge) of A is the function gA : Z → [0,+∞] defined by gA(x) := inf{α ∈ R+ : x ∈ αA} forevery x ∈ Z. If Z is a topological linear space, let Z′ denote the continuous dual of Z; ifA ⊆ Z, let coA be the closure of co A. If Z is a normed space, then |z|Z indicates the normin Z of an element z ∈ Z and SZ(a, r) := {z ∈ Z : |z − a|Z < r}(a ∈ Z, r ∈ R+). Let �2 andc0, respectively, denote the real, or complex, Banach spaces of the sequences whose squares ofmoduli of coordinates are summable, and of the infinitesimal sequences. If A and B are sets,if C ⊆ A and f : A → B is a function, then #A denotes the cardinality of A, f|C means therestriction of f to C; if g : A → [−∞,+∞] is a function, then dom g := {x ∈ A : g(x) < +∞}denotes the effective domain of g. If Z is a topological space and if A ⊆ Z, let ∂A be theboundary of A. Let E denote the integer part function. If (τn)n∈N is a sequence of elementsof [−∞,+∞] and if � ∈ [−∞,+∞], then τn ↘ � means that (τn)n∈N is a weakly decreasingsequence with limn→+∞τn = �. Henceforth we will shorten both lower semicontinuous andlower semicontinuity in “lsc′′, both sequentially lower semicontinuous and sequential lowersemicontinuity in “slsc”.

Definitions 2.1. Let X be a linear space on F, A ⊆ X, y ∈ A. Then (in accordance with [5])

(a) the point y is said to be an internal point of A if for every x ∈ X there exists anαx ∈ R+ such that y + λx ∈ A for all λ ∈ [0, αx];

(b) (see [5, page 8, above Exercise 1.1.20, and page 9, between the two Examples]) Ais said to be absorbing if 0 is an internal point of A, that is, if for every x ∈ X thereexists an αx ∈ R+ such that λx ∈ A for all λ ∈ [0, αx];

(c) the set A is said to be balanced if λx ∈ A for every x ∈ A and λ ∈ F such that |λ| ≤ 1;

(d) the point y is said to be an extreme point of A if x, z ∈ A, λ ∈ ]0, 1[ for whichy = λx + (1 − λ)z implies x = z = y.

Example 2.2. For every infinite dimensional X normed space on F there exists C an absorbingbalanced convex subset of SX(0, 1), C without interior points.

Let T :X → X be a linear bijective not continuous operator (e.g., let en ∈ X be such that|en|X = 1(n ∈ N), en /= em if n,m ∈ N, n /=m, {en : n ∈ N} linearly independent set of vectors,B a Hamel basis of X such that {en : n ∈ N} ⊆ B, T(en) := nen for every n ∈ N, T(b) := b

for every b ∈ B \ {en : n ∈ N}, T extended for linearity to all X). Let C := T−1(SX(0, 1)) ∩SX(0, 1). Then C is a balanced convex as intersection of two balanced convex sets, is boundedbecause contained in SX(0, 1), is absorbing as intersection of two absorbing sets (T−1(SX(0, 1))is absorbing because if x ∈ X \ {0} then T(x) ∈ X \ {0} and so λx ∈ T−1(SX(0, 1)) for everyλ ∈ F with |λ| ≤ 1/|T(x)|X). Moreover, if by absurd there existed an interior point of C, then,being C balanced and convex, 0 should be an interior point of C and therefore T should be acontinuous operator, that is not possible.

Here we are providing also another example, which will be useful in the constructionof the subsequent Example 4.8; to this purpose we describe expressly a set C, that can beobtained using Theorem 1 of [6], provided in the proof of such theorem a bounded Hamelbasis (in case constituted by elements having norm less or equal to 1) is considered, and with


a little change in the complex case for showing that “0 is not an interior point of C” (and sothere are no interior points of C, as above noted).

Let en ∈ X be such that |en|X ≤ 1(n ∈ N), en /= em if n, m ∈ N, n /=m, {en : n ∈ N}linearly independent set of vectors, B a Hamel basis of X such that {en : n ∈ N} ⊆ B, |b|X ≤ 1for every b ∈ B,D := {(1/(n + 1))en : n ∈ N} ∪ (B \ {en : n ∈ N}). Then it is enough to define

C := co{αd : α ∈ F, |α| = 1, d ∈ D}. (2.1)

Such a C is obviously a balanced set; for obtaining the remaining properties the same proof ofTheorem 1 of [6] still works, with the unique following little change if F = C for showing thatcd /∈C when c > 1 and d ∈ D (i.e., one of the points of the demonstration of [6]): if by absurdsuch a cd ∈ C then, using that D is a Hamel basis, there should exist m ∈ Z+, λ1, . . . , λm ∈[0, 1], with

∑mj=1 λj = 1, α1, . . . , αm ∈ C, with |αj | = 1 for every j ∈ {1, . . . , m} such that cd =∑m

j=1 λjαjd, therefore c =∑m

j=1 λjαj , that is impossible because |∑mj=1 λjαj | ≤

∑mj=1 λj |αj | = 1

while c > 1.

Theorem 2.3. Let X be a real normed space with algebraic dimension greater or equal to 2; then∂SX(0, 1) is an arcwise connected set.

Proof. Let x, y ∈ ∂SX(0, 1), x /=y. We will distinguish two cases:

(a) x /= − y,(b) x = −y.

In the case (a) the arc γ : t ∈ [0, 1] �→ ((1 − t)x + ty)/|(1 − t)x + ty|X connects x to y:in fact γ(0) = x, γ(1) = y; moreover γ is defined and continuous, as, if by absurd there existst ∈ [0, 1] such that (1 − t)x = −ty, then 1 − t = (1 − t)|x|X = t|y|X = t, consequently t = 1/2 andso (1/2)x = −(1/2)y, that is in contradiction with the assumption of (a).

In the case (b), being dim X ≥ 2, there exists z ∈ ∂SX(0, 1) linearly independent fromx and y, hence z/= − x and z/= − y; then an arc connecting x to y can be found joiningtogether two arcs, one connecting x to z and another connecting z to y, both of them existingin consequence of (a).

Lemma 2.4. Let I ⊆ R be an interval, x ∈ [−∞,+∞] an extreme of I, � ∈ [−∞,+∞[, f : I →[−∞,+∞] a convex function, xn ∈ I, xn /=x(n ∈ N) such that xn → x and f(xn) ↘ �. Then� = inf {f(y) : y ∈ I \ {x}}.

Proof. If � = −∞, then the conclusion is obvious; therefore we can suppose � ∈ R. Let z ∈I\{x}. Then there exist nz ∈ N such that f(xnz) ∈ R, xnz ∈ ]x, z[ andmz ∈ N,mz > nz such thatxmz ∈ ]x, xnz[; therefore, being � ≤ f(xmz) ≤ f(xnz), we deduce that f(xmz) ∈ R and, for theconvexity of f,we get f(xnz) ≤ ((xmz−xnz)/(xmz−z))f(z)+((xnz−z)/(xmz−z))f(xmz),whence,using in the second inequality that f(xnz) ≥ f(xmz), we obtain f(z) ≥ ((xmz − z)/(xmz −xnz))f(xnz) − ((xnz − z)/(xmz − xnz))f(xmz) ≥ f(xmz) ≥ �.

Theorem 2.5. Let X be an infinite dimensional normed space. Then there exists A ⊆ X, A infinite,countable and linearly independent set such that, for every M, N ⊆ A,M ∩ N = ∅, it is spM ∩sp N = {0}.


Proof. It is enough to use, with respect to an infinite dimensional closed separable subspace Yof X, metrizability and compactness of SY ′(0, 1) with respect to the weak ∗ topology (see, e.g.,[7, proof of Theorem V.5.1], [8, Theorem III.10.2]), and [9, proof of Proposition 1.f.3].

3. New Weaker Concepts of Sequential Lower Semicontinuity

Definitions 3.1. Let (X, τ) be a topological space. Let f be a function, f : X → [−∞,+∞].Then, if f is not +∞ identically, f is said to be

(i) below sequentially lower semicontinuous from above at x ∈ X (“ bd-slsc at x”) if thereexists ax ∈ ]infXf,+∞] such that: (xn)n∈N sequence of elements ofX for which xn →x and f(xn)↘ limn→+∞f(xn) ≤ ax, implies f(x) ≤ limn→+∞f(xn);

(ii) below sequentially lower semicontinuous from above (“bd-slsc”) if it is bd-slsc at x forevery x ∈ X;

(iii) uniformly below sequentially lower semicontinuous from above (“ubd-slsc′′) if thereexists a ∈ ]infXf,+∞] such that: x ∈ X, (xn)n∈N sequence of elements of X for whichxn → x and f(xn)↘ limn→+∞f(xn) ≤ a, imply f(x) ≤ limn→+∞f(xn).

When f has value +∞ constantly, all these properties are assumed to hold in a vacuousway.

In the above definitions, equivalently, we can replace the part “f(xn) ↘limn→+∞f(xn) ≤ ax (resp., ≤ a)” with the following: “(f(xn))n∈N weakly decreasing sequenceand f(xn) ≤ ax (resp., ≤ a) for every n ∈ N.” Indeed one of the two implications is obvious(in both cases) and the other can be proved, for example in the case of (i) (being the othercase similar), in this way: if the above variant of definition (i) is verified relatively to a certainvalue of ax and if bx is such that inf f < bx < ax, then, for every sequence (xn)n∈N of elementsof X for which xn → x and f(xn) ↘ limn→+∞f(xn) ≤ bx, there exists k ∈ N such thatf(xn) ≤ ax for every n > k and so we conclude, applying such variant to the sequence (xn)n>k.

Remarks 3.2. Here we will note some easy comparison between Definitions 3.1 and otherpreviously considered generalizations of sequential lower semicontinuity (Definitions 1.1and 1.2).

Let (X, τ) be a topological space. Let x ∈ X. Let f be a function, f : X → [−∞,+∞].We note that

(a) if f is d-slsc at x, then f is bd-slsc at x;

(b) if f is bd-slsc at x, then f is i-slsc at x, because if f is not constantly +∞, if xn → xand f(xn)↘ inf f, then limn→+∞f(xn) ≤ ax and f(x) ≤ inf f ;

(c) if f is d-slsc, then f is ubd-slsc, because if f is not constantly +∞ it follows that (iii)of Definitions 3.1 is true with respect to an arbitrary value of a > inf f ;

(d) if f is ubd-slsc, then f is bd-slsc;

(e) if f is bd-slsc, then f is i-slsc, for (b);

(f) “f is ubd-slsc and (iii) of Definitions 3.1 is verified with respect to a = +∞” if andonly if f is d-slsc.


Remarks 3.3. Let(X, τ) be a topological space. Let x ∈ X. Let f be a function, f : X →[−∞,+∞]. We observe that:

(a) if lim infy→xf(y) > infXf, then f is bd-slsc at x, because it suffices to considerax ∈]infXf, lim infy→xf(y)[;

(b) for verifying that f satisfies ubd-slsc with respect to a certain a > infXf (a as in (iii)of Definitions 3.1), it is enough to prove that f|{z∈X:lim infy→ zf(y)≤a} is ubd-slsc.

Theorem 3.4. Let (X, τ) be a topological space satisfying the first axiom of countability. Let x ∈ X.Let f be a function, f : X → [−∞,+∞]. The following implications are true:

(a) if f is i-slsc at x, then f is bd-slsc at x;

(b) if f is i-slsc, then f is bd-slsc (so, under hypothesis of fulfillment of the first axiom ofcountability, a vice-versa of part (e) of Remarks 3.2 is true).

Proof. It is sufficient to show the part (a) of the theorem.By absurd, we suppose that f is not bd-slsc at x; then f is not +∞ constantly and,

choosing a sequence (ak)k∈N, with ak > inf f for every k ∈ N, such that ak → inf f, for everyk ∈ N there exists a sequence (xn,k)n∈N of elements of X for which xn,k → x, (f(xn,k))n∈Nis a weakly decreasing sequence with limn→+∞f(xn,k) ≤ ak and f(x) > limn→+∞f(xn,k);moreover, from the hypothesis, we deduce that inf f < limn→+∞f(xn,k) for every k ∈ N.On the other hand, since τ verifies the first axiom of countability, there exists {Uh : h ∈ N}base for the neighbourhood system of x, with Uh+1 ⊆ Uh for every h ∈ N.

Now we will define a sequence (xn)n∈N by means of which we will produce acontradiction. Let x0 = xn0,0, where n0 = min{n ∈ N : xn,0 ∈ U0, inf f < f(xn,0) < a0 + 1} and,for m ∈ N, chosen x0, . . . , xm with inf f < f(xh) (h ∈ {0, . . . , m}), let xm+1 = xnm+1,k(m+1), wherek(m+1) = min{k ∈ N : k ≥ m+1, ak < f(xm)} and nm+1 = min{n ∈ N : xn,k(m+1) ∈ Um+1, inf f <f(xn,k(m+1)) < ak(m+1) + 1/(m + 2), f(xn,k(m+1)) ≤ f(xm)} (these choices are possible, becausef(xn,k) ≥ limp→+∞f(xp,k) > inf f for every n, k ∈ N). For construction xn → x, f(xn)↘ inf f ;furthermore f(x) > inf f, because f(x) > limn→+∞f(xn,k) > inf f (k ∈ N); but these facts arein contradiction with the i-slsc of f at x.

Theorem 3.5. Let X be a topological linear space and let f : X → [0,+∞] be a function such that

f(αx) = αf(x) for every α ∈ [0,+∞[ and for every x ∈ X. (3.1)

Suppose that f is ubd-slsc. Then f is slsc too. Therefore, for such a function f, ubd-slsc, d-slsc,and slsc are each other equivalent conditions (see part (c) of Remarks 3.2).

Proof. Let a be relative to the ubd-slsc of f as in (iii) of Definitions 3.1. Then, since infXf =f(0) = 0, we get that a > 0.

Let x ∈ X and let (xn)n∈N be a sequence of elements of X such that xn → x; let α :=lim infn→+∞f(xn). We will conclude if we will prove that f(x) ≤ α.

Now, since the desired conclusion is obvious if α = +∞, it is enough that we distinguishtwo cases:

(i) α = 0,

(ii) α ∈ R+.


In the case (i), being α = 0 = inf f, then there exists a subsequence (f(xnk))k∈N of (f(xn))n∈Nconstantly 0 or strictly decreasing to 0: anyhow such subsequence is weakly decreasing andconverging to 0 < a; so, applying ubd-slsc of f, the desired result follows.

In the second case, (ii), let (xnk)k∈N be a subsequence of (xn)n∈N such thatlimk→+∞f(xnk) = α; let k0 ∈ N be such that f(xnk) ∈ R+ for every k > k0 and letyk := axnk/f(xnk) for every k ∈ N, k > k0; then, being X a topological linear space, it holdsthat yk → ax/α, moreover f(yk) = (a/f(xnk))f(xnk) = a for every k > k0; so, using ubd-slscof f , we get f(ax/α) ≤ a whence f(x) = (α/a)f(ax/α) ≤ α.

Remark 3.6. Note that, if X is a topological linear space, if A ⊆ X is an absorbing set andif f = gA, then (3.1) is true (see [5, Theorem 1.2.4 (i) and definition foregoing]).

Examples 3.7. (a)There exist a bounded function g: R → R and a point z ∈ R such that g isubd-slsc, but g is not d-slsc at z.

(b)There exists a bounded function h : R → R such that h is bd-slsc, but h is notubd-slsc.

(c) Let W = �2 (on the field F) endowed with its weak topology. Then there exist afunction k : W → [0, 1] and a point w ∈ W such that k is i-slsc, but k is not bd-slscat w (note that an analogue example on a topological space satisfying the first axiom ofcountability does not exist, in consequence of Theorem 3.4).

(a) Let

g(x) =

⎧⎨⎩

arctgx, if x ∈ ]−∞, 0],−1, if x ∈ ]0,+∞[,

(3.2)

and z = 0. Then g is lsc in every point of R\{z}; therefore g is ubd-slsc: indeed it suffices toconsider a ∈ ] − π/2,−1[ and to use (b) of Remarks 3.3. Besides g is not d-slsc at z, becauseif zn := 1/(n + 1) = z + 1/(n + 1) for every n ∈ N we get that zn → z, (g(zn))n∈N is a weaklydecreasing sequence, but g(z) = 0 > −1 = limn→+∞g(zn).

(b) Let

h(x) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

arctgx, if − 2m − 1 < x ≤ −2m (m ∈ N),

0, if − 2m − 2 < x ≤ −2m − 1 (m ∈ N),

0, if x > 0.

(3.3)

Then h is lsc in every point of R\{−2m−1 : m ∈ N}; moreover it is bd-slsc, as for everym ∈ N

it is sufficient to note that lim infx→−2m−1h(x) = arctg (−2m − 1) > −π/2 = infRh and to use(a) of Remarks 3.3. On the other hand h is not ubd-slsc, because for every a > infRh = −π/2there existma ∈ N and a sequence (yn,a)n∈N of real numbers for which limn→+∞yn,a = −2ma−1and h(yn,a) ↘ �a ≤ a, but h(−2ma − 1) = 0 > �a (in fact it is enough to consider ma such thatarctg(−2ma − 1) < a and yn,a = −2ma − 1 + 1/(n + 1) for every n ∈ N).


(c) Let

k(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1E(|λ|) + 1

if x = λem for some m ∈ N and λ ∈ F \ Z,

1|λ| if x = λem for some m ∈ N and λ ∈ Z \ {0},

1 if x ∈ {0}⋃(

W \⋃m∈N

sp{em}),

(3.4)

where en := (δn,m)m∈N for every n ∈ N, and let w = 0. We get that k is i-slsc, as there does notexist a converging sequence (wn)n∈N of elements of W which satisfies

limn→+∞

k(wn) = 0 = infWk : (3.5)

in fact, if a sequence (wn)n∈N verifies (3.5), then it follows that there exists n0 ∈ N such thatfor every n > n0 there are λn ∈ F and mn ∈ N for which wn = λnemn and |λn| → +∞,hence |wn|�2 → +∞, from here the sequence is unbounded and therefore cannot converge.On the other hand k is not bd-slsc at w, because for every a > infWk = 0 there exists asequence (zn,a)n∈N of elements of W for which limn→+∞zn,a = 0 = w and k(zn,a) ↘ �a ≤ a,but k(0) = 1 > �a (indeed it is sufficient to consider sa ∈ N such that sa > 1, 1/sa < a andzn,a = saen for every n ∈ N).

Example 3.8. It is possible to find an example of a Hilbert space Y and of a function that islsc with respect to the topology induced on Y by its norm, but that, with respect to the weaktopology on Y, is not bd-slsc (and so it is neither ubd-slsc nor d-slsc; indeed neither i-slsc):see [4, Example 4.1] and Remarks 3.2.

4. Behavior of Some Weak Concepts of Sequential LowerSemicontinuity with Respect to the Convexity

With respect to [4, Theorem 4.1], a slightly more laborious demonstration allows to get astronger result.

Theorem 4.1. Let X be a normed space, let f : X → [−∞,+∞] be a convex, i-slsc function withrespect to the topology induced on X by its norm. Then f is bd-slsc with respect to the weak topologyon X.

Proof. By absurd, we suppose that f is not bd-slsc with respect to the weak topology onX. Then f is not +∞ constantly and there exists at least a point x ∈ X such that for everya > inf f there exists a sequence (yn,a)n∈N of elements of X for which yn,a ⇀ x, (f(yn,a))n∈N isa weakly decreasing sequence, f(yn,a) ≤ a for every n ∈ N and f(x) > limn→+∞f(yn,a).Hencef(x) > inf f and we will reach an absurd if we will show that

there exists a sequence (xn)n∈N of elements of X

for which xn −→ x and f(xn)↘ inf f,(4.1)


because, using the i-slsc of f at x with respect to the topology induced on X by its norm,relatively to such a sequence, it should be f(x) = inf f. At first, in order to prove (4.1), wewill define a sequence (bk)k∈N of real numbers and by induction we will define other foursequences, (ah)h∈N of real numbers, (kh)h∈N of natural numbers, (jh)h∈N of integer numbersand (zh)h∈N of elements of X such that jh ∈ {0, . . . , h} and f(zjh) > inf f if f(zm) > inf f forsome m ∈ {0, . . . , h}(h ∈ N), by means of the followings:

bk :=

⎧⎨⎩

inf f +1

k + 1if inf f ∈ R

−k if inf f = −∞for every k ∈ N,

k0 := 0, a0 := bk0 , z0 ∈ co{yn,a0 : n ∈ N

}such that |z0 − x|X < 1,

j0 :=

⎧⎨⎩−1 if f(z0) = inf f

0 if f(z0) > inf f

(4.2)

and, if h ∈ N, defined kp, ap, zp and jp for every p ∈ {0, . . . , h}, we define kh+1, ah+1, zh+1 andjh+1 in the following way:

kh+1 :=

⎧⎨⎩kh + 1 if f(z0) = · · · = f(zh) = inf f,

min{k ∈ N : k > kh, bk ≤ f

(zjh)}

if f(zm) > inf f for some m ∈ {0, . . . , h},

ah+1 := bkh+1 , zh+1 ∈ co{yn,ah+1 : n ∈ N

}such that |zh+1 − x|X <

1h + 2

,

jh+1 :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−1, if f(z0) = · · · = f(zh+1) = inf f,

max{j ∈ {0, . . . , h + 1} : f

(zj)> inf f

}, if f(zm) > inf f

for some m ∈ {0, . . . , h + 1}.(4.3)

(such definitions are possible, being x ∈ co{yn,a : n ∈ N} for every a > inf f (as a convexclosed subset of X is weakly closed too) and thanks to choice’s axiom). By definition, (bk)k∈Nis strictly decreasing, with limk→+∞bk = inf f, (kh)h∈N is strictly increasing and so (ah)h∈N isa subsequence of (bk)k∈N and therefore is itself strictly decreasing, with limh→+∞ah = inf f ,limh→+∞zh = x with respect to the topology induced on X by its norm; moreover (jh)h∈N isweakly increasing, j(N) \ {−1} = {n ∈ N : f(zn) > inf f} and hence, if j(N) is a finite set, thereexists a subsequence of (zh)h∈N such that the values of f in the elements of such subsequenceare constantly inf f.

Now we will distinguish two cases:

(i) there exists a subsequence of (zh)h∈N such that the values of f in the elements ofsuch subsequence are constantly inf f,

(ii) there does not exist a subsequence as in (i).

In the case (i) we at once conclude, because, if (xn)n∈N is a subsequence of (zh)h∈N asin (i), then it verifies (4.1).


If we are in the case (ii), then j(N) is an infinite set; therefore by induction we candefine a new sequence (mn)n∈N in this way: let m0 := min(j(N) \ {−1}) and, if n ∈ N, definedmp for every p ∈ {0, . . . , n}, let mn+1 := min(j(N) \ ({−1} ∪ {m0, . . . , mn})); from here, definingxn := zmn for every n ∈ N and taking into account the definition of j, it follows that (xn)n∈N isthe subsequence of (zh)h∈N whose elements are exactly all the “zh” such that f(zh) > inf f.

It will be enough to prove that f(xn)↘ inf f, because in such way we will have proved(4.1).

For every n ∈ N it holds:

f(xn+1) = f(zmn+1) ≤ amn+1 = bkmn+1≤ f(zjmn+1−1

)= f(zmn) = f(xn) (4.4)

(where for obtaining the inequality between second and third terms we used that zmn+1 ∈co{yp,amn+1

: p ∈ N} by definition and for deducing the equality between fifth and sixth termswe used that

jmn+1−1 = max{j ∈ {0, . . . , mn+1 − 1} : f

(zj)> inf f

}= mn (4.5)

by definition).At last and analogously as seen above, it is verified:

f(xn) = f(zmn) ≤ amn for every n ∈ N; (4.6)

besides limn→+∞amn = inf f, as (amn)n∈N is a subsequence of (bk)k∈N and so we conclude.

Corollary 4.2. Let X be a normed space, let f : X → [−∞,+∞] be a convex function, i-slsc withrespect to the weak topology onX; then f is bd-slsc with respect to the same topology (so, in the presentcase, by the help of hypothesis of convexity, the conclusion of Theorem 3.4 is true, although the firstaxiom of countability may be not fulfilled).

Proof. From the easy observation that if τ and σ are two topologies on a set Y, with σ ⊆ τ , andif a function verifies Definition 1.2 with respect to σ then it verifies the same condition withrespect to τ too, we get that f is i-slsc with respect to the topology induced on X by its normand so it is sufficient to use Theorem 4.1.

Examples 4.3. As it will be seen below, in examples (a) and (b), there exist examples of convexfunctions, satisfying “the same semicontinuity conditions” of Examples 3.7 (a) and (b), butthat are not upperly bounded and whose values are in ] − ∞,+∞] instead of in R; on thecontrary, owing to Corollary 4.2, if W is as in (c) of Examples 3.7, an example of convexfunction k : W → [−∞,+∞] i-slsc but not bd-slsc cannot exist.

(a) For every X normed space having, as real space, algebraic dimension greater orequal to 2 and such that

the closed unitary sphere of X admits at least one extreme point, (4.7)

there exist a convex function g : X → [0,+∞] and a point z ∈ X such that g isubd-slsc, but g is not d-slsc at z.


(b) For every X normed space having, as real space, algebraic dimension greater orequal to 2 there exists a convex function h : X → [0,+∞] such that h is bd-slsc, buth is not ubd-slsc.

(a) Let z ∈ ∂SX(0, 1) be an extreme point of SX(0, 1) (such a point exists for (4.7)) andlet

g(x) =

⎧⎨⎩|x|X, if x ∈ SX (0, 1) \ {z},+∞, if x ∈ {z}

⋃(X \ SX(0, 1)

) (4.8)

Then g is convex, because SX(0, 1) \ {z} is a convex set (being z an extreme point ofSX(0, 1)); moreover g is lsc in every point of X \ {z}; therefore g is ubd-slsc: indeedit suffices to consider a ∈ ]0, 1[ and to use (b) of Remarks 3.3. Besides g is notd-slsc at z, because if zn ∈ ∂SX(0, 1) \ {z}(n ∈ N) is chosen in such a way as toconverge to z (this choice is possible for hypothesis, using Theorem 2.3) we get that(g(zn))n∈N is a weakly decreasing sequence (it is constantly 1), but g(z) = +∞ > 1 =limn→+∞g(zn).

(b) Let y ∈ ∂SX(0, 1) and let

h(x) =

⎧⎨⎩|x|X, if x ∈ {0}

⋃SX(y, 1),

+∞, if x ∈ X \({0}⋃SX(y, 1)).

(4.9)

Then h is convex, because {0} ∪ SX(y, 1) is a convex set; furthermore h is lsc inevery point of {0} ∪ (X \ ∂SX(y, 1)); moreover it is bd-slsc, because for everyx ∈ ∂SX(y, 1) \ {0} it is enough to note that lim infz→xh(z) = |x|X > 0 = infXhand to use (a) of Remarks 3.3. On the other hand h is not ubd-slsc, because forevery a > infXh = 0 there exist xa ∈ X and a sequence (yn,a)n∈N of elements ofX for which limn→+∞yn,a = xa, h(yn,a) ↘ �a ≤ a, but h(xa) > �a: in fact, being∂SX(y, 1) a connected set owing to Theorem 2.3 and, chosen 0 < ba < min{a, 2},being ∂SX(y, 1) ∩ (X \ SX(0, ba)) a nonempty open of ∂SX(y, 1) (it contains 2y),the nonempty subset ∂SX(y, 1) ∩ SX(0, ba) of ∂SX(y, 1) (it contains 0) must have atleast a not interior point xa (with respect to ∂SX(y, 1)); so there exists a sequence(zn,a)n∈N of elements of ∂SX(y, 1) with limn→+∞zn,a = xa, |zn,a|X > ba ≥ |xa|X forevery n ∈ N, and it is enough to consider y0,a ∈ SX(y, 1) such that |y0,a − z0,a|X < 1,|y0,a|X > |xa|X and, if n ∈ N, given yn,a (with |yn,a|X > |xa|X) to select hn > n such that|zhn,a|X < |yn,a|X and yn+1,a ∈ SX(y, 1) such that |yn+1,a − zhn,a|X < 1/(n + 2), |xa|X <|yn+1,a|X < |yn,a|X ; with these choices, it results that limn→+∞yn,a = xa, (h(yn,a))n∈N =(|yn,a|X)n∈N is a strictly decreasing sequence with limn→+∞h(yn,a) = |xa|X ≤ ba < a,but h(xa) = +∞ > |xa|X = limn→+∞h(yn,a).

Remark 4.4. Note that hypothesis (4.7) done in (a) of Examples 4.3 is verified for example byevery reflexive Banach space (see [5, Theorem 2.4.5] and use the weak topology), but thereexist Banach spaces (e.g., c0 and L1([a, b])(a, b ∈ R, a /= b)) which do not satisfy it (see [10,Examples II.8]).


Remark 4.5. ForX = R and also if functions with values in [−∞,+∞] are considered, examplesverifying conditions of (a) or (b) of Examples 4.3 do not exist, because the following fact istrue.

If f : R → [−∞,+∞] is a convex, i-slsc function, then f is d-slsc (and so, in this case,taking into account parts (c), (d) and (e) of Remarks 3.2, under hypothesis of convexity, thefour conditions of d-slsc, ubd-slsc, bd-slsc and i-slsc are each other equivalent).

Let x, xn ∈ R (n ∈ N), xn → x, f(xn) ↘ �; then we will conclude, if we will provethat f(x) ≤ �. It is not restrictive to suppose � < +∞, xn /=x, xn ∈ dom f(n ∈ N) and x anextreme of dom f (otherwise, if x ∈ (dom f)◦, the function f should be continuous in x);applying Lemma 2.4 to f|dom f it follows that � = inf{f(y) : y ∈ dom f \ {x}}; then, if byabsurd f(x) > �, it should be � = inf f , that contradicts i-slsc of f.

Example 4.6. For every infinite dimensional X normed space there exists a convex bd-slscfunction g : X → R that is discontinuous in every point of X (and therefore, if X is complete,is neither a lsc function on X (see, e.g., [5, Theorem 3.1.9])).

Indeed, we will show that such a function g can be chosen as whatever a Minkowskifunctional of

an absorbing balanced convex subset C of X such that

0 is not an interior point for C, C ⊆ SX(0, 1)(4.10)

(for the existence of such a set, see Example 2.2). Consequently, if X is a Banach space, usingTheorem 3.5, Remark 3.6 and part (c) of Remarks 3.2, such a function g neither is ubd-slscnor is d-slsc.

Let C be as in (4.10) and let g := gC. Then g is a real valued convex not continuousfunction (see, e.g., [8, Theorems II.12.1 and II.12.3, and foregoing Definition]); hence, using aclassical result of convex analysis (see, e.g., [5, Theorem 3.1.8]) g is discontinuous in everypoint of X.

Owing to part (b) of Theorem 3.4, for showing the remaining condition on g, namelythe bd-slsc of g, it suffices to prove that g is i-slsc.

At first, note that

|x|X ≤ g(x) for every x ∈ X. (4.11)

In fact, being C ⊆ SX(0, 1), for every x ∈ X it holds

|x|X = inf{α ∈ R+ :

∣∣∣xα

∣∣∣X≤ 1}≤ inf

{α ∈ R+ :

x

α∈ C}= g(x). (4.12)

Now let x ∈ X, let (xn)n∈N be a sequence of elements of X for which xn → x andlimn→+∞g(xn) = 0 (= g(0) = inf g); therefore, using (4.11), we get that x = 0, whenceg(x) = g(0) = 0.

Remarks 4.7. (a) Example 4.6 shows that the classical result “every f convex lsc functiondefined on a Banach space with values in [−∞,+∞] is continuous on the interior of dom f”


(see, e.g., [11, Theorem 2.2.20 (b) and sentences afterward Proposition 1.1.11]) fails if lsc isreplaced by bd-slsc (or equivalently (see (e) of Remarks 3.2 and Theorem 3.4) by i-slsc).

(b) On the contrary, the above-mentioned classical result is still true if lsc is replaced byd-slsc (it is enough to use [4, Theorem 3.2], and a classical result of convex analysis (see, e.g.,[5, Theorem 3.1.9]) applied to f|(dom f)◦ , also considering that, if there exists a point x0 ∈ Xsuch that f(x0) = −∞, then f(x) = −∞ for every x ∈ (dom f)◦ (see, e.g., [11, Proposition2.1.4])), or by ubd-slsc (as we will show in the part (c) of Theorem 4.10).

Among other things, by means of such facts there is a different way to prove that, if Xis a Banach space, the function g of Example 4.6 cannot be either ubd-slsc or d-slsc (i.e., whatalready claimed in the last rows of the statement of number 4.6).

(c) The above (a) and (b) show a big difference between convex, bd-slsc (or convex,i-slsc) functions on Banach spaces on the one hand and convex, d-slsc (or convex, ubd-slsc)functions on Banach spaces on the other hand; such situation in a certain way renders, inthe case in which convex functions on Banach spaces are considered, more meaningful thoseresults, as for example [4, Theorems 5.1, 5.3 and Corollary 5.1] in which the classic hypothesisof “lower semicontinuity” can be replaced by an hypothesis of “bd-slsc” (or of “i-slsc”).

Working with some properties of infinite dimensional Banach spaces and with thechoice of the convex set C, we will able to exhibit an example as the following one (that maybe regarded in a certain way as a refinement of Example 4.6, because, also if it does not givea stronger conclusion than the one of Example 4.6, it lets to define in a constructive way asequence of points, by means of which ubd-slsc is showed not to be true).

Moreover, with respect to Examples 4.3 (b), observe that in Example 4.8 we get anexample of a function defined in a less general space, but having real values; hence the pointsby means of which we could prove that such a function is not ubd-slsc, unlike that in part (b)of Examples 4.3, necessarily are all at the interior of its effective domain.

Example 4.8. For every infinite dimensionalX Banach space on F, there exists a convex bd-slscfunction g : X → R that is not ubd-slsc (and therefore, for part (c) of Remarks 3.2, is not evena d-slsc function).

Indeed, we will show that such a function g can be chosen as the Minkowski functionalof a suitable set C satisfying (4.10); then infXg = 0 and we will prove that g is not ubd-slsc,exhibiting a sequence (cq)q∈Z+

of elements of X such that for every q ∈ Z+ there existsa sequence (cq,k)k∈N of points of X for which limk→+∞cq,k = cq, (g(cq,k))k∈N is a weaklydecreasing sequence and g(cq) > 1/q = limk→+∞g(cq,k) for every q ∈ Z+.

We will define the set C by means of the construction of Thorp ([6, Theorem 1]),already cited in the final part of Example 2.2, but choosing a suitable Hamel basis, formedby elements having norm less or equal to 1 and verifying other suitable conditions that weare going to introduce.

For Theorem 2.5 there exists A ⊆ X,A infinite, countable and linearly independent setsuch that

M,N ⊆ A, M ∩N = ∅ =⇒ spM ∩ spN = {0}; (4.13)

moreover let |a|X = 1 for every a ∈ A (this is not a restriction).Consequently if

∑n∈N αnan = 0 with αn ∈ F and an ∈ A (n ∈ N), then αn = 0 for every

n ∈ N.


LetNq ⊆ A (q ∈ Z+) be such thatA =⋃q∈Z+

Nq, eachNq is an infinite set andNq∩Np =∅ (q, p ∈ Z+, q /= p). Further on, let λn ∈ R+(n ∈ N) be such that

∑n∈N λn = 1.

Let bq :=∑

n∈N λneq,n for every q ∈ Z+, where Nq = {eq,n : n ∈ N} with eq,n /= eq,m ifn, m ∈ N, n /=m (q ∈ Z+) (such elements bq (q ∈ Z+) are existing as X is a Banach space).Then |bq|X ≤ 1 for every q ∈ Z+.

Hence bq ∈ spNq \ spNq, spNq ∩ spNt = {0} if q, t ∈ Z+, q /= t and therefore bq /= bt ifq, t ∈ Z+, q /= t, A ∩ {bq : q ∈ Z+} = ∅ and A ∪ {bq : q ∈ Z+} is a linearly independent set ofvectors.

Therefore, we can consider B a Hamel basis of X such that B ⊇ A ∪ D, where D :={bq/q : q ∈ Z+}, with |b|X = 1 for every b ∈ B \D, and C := co{αb : α ∈ F, |α| = 1, b ∈ B}.

Then C satisfies (4.10), because C is a particular case of the convex set defined,following Theorem 1 of [6], in the final part of Example 2.2.

Let g := gC. Since C verifies (4.10) and g is its Minkowski functional, for the proofalready seen in Example 4.6 we get that g is a real valued convex bd-slsc function.

At last we will prove that g is not ubd-slsc, exhibiting a sequence as described in thestatement.

Let

cq :=bq

q, p

(q, k) ∈Nq \ {0, . . . , k},

bq,k :=k∑n=0λneq,n +

∑n>k

λneq,p(q,k),

cq,k :=bq,k

q

(q ∈ Z+, k ∈ N

);

(4.14)

then for each q ∈ Z+, k ∈ N it is bq,k ∈ co{eq,n : n ∈ N} ⊆ C, but there does not exist anα > 1 such that αbq,k ∈ C because, being B a linearly independent set of vectors, if α > 1the following one is the only way in which αbq,k can be written as a linear combination ofelements of B: αbq,k =

∑kn=0 αλneq,n +

∑n>k αλneq,p(q,k) and, on the other hand,

∑kn=0 αλn +∑

n>k αλn = α > 1; therefore g(bq,k) = 1 and, being g positively homogeneous, g(cq,k) =1/q; whence, for each q ∈ Z+, the sequence (g(cq,k))k∈N is constant and therefore weaklydecreasing; furthermore g(cq) = 1 (for a demonstration quite similar to the above proof thatg(bq,k) = 1) and so g(cq) = 1 > 1/q = limk→+∞g(cq,k), besides limk→+∞cq,k = cq and weconclude.

Lemma 4.9. Let Y be a topological space and let X be a topological linear space. The following factshold

(a) if A is a subset of Y , F is a closed subset of Y and G is an open subset of Y such thatA ∩G = F ∩G, then A ∩G ⊆ F (and therefore A ∩G = F ∩G);

(b) if U is an open subset of X and C is a convex subset of X such that U ∩ C/= ∅ and◦C/= ∅,

thenU ∩◦C/= ∅;

(c) if A is a subset of X, C is a convex subset of X and F is a closed subset of X such that◦A ∩ C/= ∅, A ∩

◦C = F∩ ◦

C and◦C /= ∅, then

◦A /= ∅.


Proof. (a) If by absurdA∩G∩(Y \F)/= ∅, then, beingG∩(Y \F) an open, it is ∅/=A∩G∩(Y \F) =F ∩G ∩ (Y \ F) = ∅, that is a contradiction.

(b) Let x ∈ ◦C and y ∈ U ∩ C. If x = y we have the desired result; otherwise, if x /=y,

from [7, demonstration inside of the proof of Theorem V.2.1] it follows that [x, y] \ {y} ⊆ ◦C,

besides from the topological linear structure of X it is ([x, y] \ {y}) ∩ U/= ∅ and so we canconclude.

(c) Applying (b) to U :=◦A, we get

◦A ∩ ◦

C /= ∅. Besides, from (a) applied with G :=◦C,

we deduce◦A ∩ ◦

C ⊆ A ∩◦C ⊆ F and so

◦A ∩ ◦

C ⊆ F ∩◦C ⊆ A, namely,

◦A ∩ ◦

C is a not emptyopen set contained in A and we conclude.

Theorem 4.10. LetX be a topological linear space and let f : X → [−∞,+∞] be a convex, ubd-slscfunction. Suppose that f is not +∞ identically, let a ∈ R be such that a > infXf is relative to f as incondition (iii) of Definitions 3.1 and let A := {x ∈ X : f(x) ≤ a}. Then

(a) the set A ∩ (dom f)◦ is sequentially closed with respect to the relative topology of(dom f)◦;

henceforth suppose that (dom f)◦ /= ∅, then also the following facts hold:

(b) there exists a point x0 ∈ {x ∈ X : f(x) < a}∩ (dom f)◦ and thereforeA−x0 is absorbing;

(c) if X is a Banach space, it results that◦A /= ∅ and f is continuous in the points of (dom f)◦.

Proof. If there exists a point z0 ∈ X such that f(z0) = −∞, then f(x) = −∞ for every x ∈(dom f)◦ (see, e.g., [11, Proposition 2.1.4]) and all the parts of the desired result follow alsoif X is simply a topological linear space (in fact in such case (dom f)◦ ⊆ {x ∈ X : f(x) < a}).

Henceforward we can suppose that f(z) > −∞ for every z ∈ X; with such a furtherhypothesis, we will prove all the three parts of the desired result.

(a) Let (xn)n∈N be a sequence of elements of A∩ (dom f)◦ and let x ∈ (dom f)◦ be suchthat xn → x; let α = lim infn→+∞f(xn). Then α ≤ a; moreover there exists a subsequence(xnk)k∈N of (xn)n∈N such that f(xnk) → α and there exists a further subsequence (xnkh )h∈N of(xnk)k∈N such that (f(xnkh ))h∈N is a weakly monotone sequence. Now, if by absurd a < f(x),we get α < f(x) and hence, using Lemma 3.1 of [4], it is not restrictive to suppose that(f(xnkh ))h∈N is a weakly decreasing sequence; so it is enough to use the ubd-slsc of f to obtaina contradiction.

(b) Let y0 ∈ (dom f)◦ and let z ∈ X be such that f(z) < a; then z ∈ dom f and hence,being dom f a convex set, it is [y0, z] ⊆ dom f ; therefore f|[y0,z] is an upper semicontinuousfunction (see [12, Theorem 10.2]) and so there exists a point x0 ∈ [y0, z] \ {z} such thatf(x0) < a; besides, from [7, demonstration inside of the proof of Theorem V.2.1] it followsthat [y0, z] \ {z} ⊆ (dom f)◦; consequently x0 ∈ {x ∈ X : f(x) < a} ∩ (dom f)◦.

Now, if w ∈ {x ∈ X : f(x) < a} ∩ (dom f)◦, then w is an internal point of A, because,being an interior point of dom f , for every x ∈ X there exists a βx ∈ R+ such that [w,w +βxx] ⊆ (dom f)◦; therefore f|[w,w+βxx] is a continuous function and so, being f(w) < a, thereexists an αx ∈ R+ such that f(w + λx) < a for all λ ∈ [0, αx]. Then A −w is absorbing.

(c) Now let X be a Banach space. First of all we will prove that◦A /= ∅.

From (b) the set A − x0 is absorbing; therefore X =⋃n∈Z+

n(A − x0) and from Baire’s

lemma there exists a n ∈ Z+ such that (n(A − x0))◦/= ∅, hence (A − x0)

◦/= ∅, wherefore

◦A /= ∅.


Now we will prove that part (c) of Lemma 4.9 can be applied with C := dom f . Since

A ⊆ dom f and◦A /= ∅, we get ∅ /=

◦A ⊆

◦A ∩dom f . On the other hand, from (a), the set

A ∩ (dom f)◦ is sequentially closed (and hence closed, satisfying the topology of X the firstaxiom of countability) with respect to the relative topology of (dom f)◦ and therefore thereexists F closed subset of X such that A ∩ (dom f)◦ = F ∩ (dom f)◦. Consequently, from part

(c) of Lemma 4.9, we have that◦A /= ∅.

Then◦A is a not empty open subset of dom f on which f is bounded above (from the

real element a) and so we can conclude, applying Theorem 3.1.8 of [5].

Remark 4.11. The Example 3.1 of [4] and those cited in Remarks 3.1 of [4] give also examplesof Banach spaces Y and of convex, ubd-slsc, with respect to the weak topology, functionsdefined on Y with values in [0,+∞], that are not d-slsc with respect to the weak topology.

In fact it suffices to use (b) of Remarks 3.3 and, using the notations of the above citedexamples, to note that inf f = 0 and that, relatively to whatever value of a ∈ ]0, 1[, it is truethat D ∩ {y ∈ Y : g(y) ≤ a} = C ∩ {y ∈ Y : g(y) ≤ a} is convex and closed, f|D∩{y∈Y :g(y)≤a} =g|D∩{y∈Y :g(y)≤a} and g is continuous (for Example 3.1 of [4]), D ∩ SY (0, a) = C ∩ SY (0, a) isconvex and closed, f|D∩SY (0,a) =| |Y |D∩SY (0,a) (for Example cited in (a) of Remarks 3.1 of [4]),f|SY (0,a) =| |Y |SY (0,a) (for Example cited in (b) of Remarks 3.1 of [4]).

5. Ekeland’s and Caristi’s Theorems

Remark 5.1. In the following theorem we will show an extension of Ekeland’s variationalprinciple (see [13, Theorem 1.1]) to the case in which the hypothesis of lsc is replaced byubd-slsc. The proof is inspired by the demonstration of Theorem 2.1 of [1]: since we will addthe result (b) (that was considered already in [13, Theorem 1.1]) and for reading convenience,here we are writing the whole proof, removing some trivial mistakes of [1].

The authors thank the referee for having pointed out to them that Theorem 2.1 of [1]can be deduced also from [14, Corollary 4 of Theorem 1] that subsequently was generalizedby [15–17]: in fact it is easy to prove that the hypotheses of such corollary are verified if weassume d-slsc (but the same we cannot do if we assume ubd-slsc).

Moreover we wish here to point out another recent paper ([18]) in which othergeneralizations of Caristi-Kirk’s Fixed Point Theorem and Ekeland’s Variational Principle aregiven, in a different environment with respect to the one of the present paper.

Theorem 5.2. Let (X, d) be a complete metric space and let f : X → ] −∞,+∞] be a bounded frombelow and not +∞ identically function. Let ε > 0, λ > 0 and u ∈ X be such that f(u) ≤ infx∈Xf(x)+ε.Moreover,

suppose that f is ubd-slsc and that f(u) ≤ a,where a is relative to f as in (iii) of Definitions 3.1.

(5.1)

Then there exists v ∈ X such that(a)f(v) ≤ f(u),(b)d(u, v) ≤ λ,(c)f(w) > f(v) − (ε/λ)d(w,v) for every w ∈ X \ {v}.


Proof. Let

ε0 := a − infXf. (5.2)

Since f(u) ≤ a = infXf + ε0 and observing that, if ε > ε0, then a point v ∈ X verifying (a), (b)and (c) with respect to ε0, λ and u also satisfies (a), (b) and (c) with respect to ε, λ and u, it isnot restrictive to suppose ε ≤ ε0.

For every z ∈ X let Tz = {x ∈ X : f(x) ≤ f(z) − (ε/λ)d(x, z)}; then Tz /= ∅ and Tz = {z}if and only if f(x) > f(z) − (ε/λ)d(x, z) for every x ∈ X \ {z}.

If z ∈ X and s ∈ Tz, then Ts ⊆ Tz, because if x ∈ Ts we have f(x) ≤ f(s) − (ε/λ)d(x, s);besides f(s) ≤ f(z)−(ε/λ)d(s, z) as s ∈ Tz; so f(x) ≤ f(s)−(ε/λ)d(x, s) ≤ f(z)−(ε/λ)d(s, z)−(ε/λ)d(x, s) ≤ f(z) − (ε/λ)d(x, z) and hence x ∈ Tz.

Let now U := {X} ∪ {Tz : z ∈ X}. Then as a consequence of what was above noted,

Ts ⊆ U for every U ∈ U, s ∈ U. (5.3)

There exists a function h : {(U, s) : U ∈ U, s ∈ U} → X such that h(U, s) ∈ Ts andf(h(U, s)) − infx∈Tsf(x) ≤ (1/2)(f(s) − infx∈Uf(x)) for every U ∈ U and s ∈ U: for showingsuch a fact, if U ∈ U, s ∈ U and if f(s) = infx∈Uf(x) we choose h(U, s) = s (in such a case wemust choose h(U, s) = s, because f(s) ≤ f(x) ≤ f(s) − (ε/λ)d(x, s) for every x ∈ Ts, beingTs ⊆ U on account of (5.3), and so Ts = {s}), otherwise we use the characterization of greatestlower bound and the choice’s axiom.

Then, for definition of Ts, we have that

f(h(U, s)) ≤ f(s) for every U ∈ U, s ∈ U. (5.4)

Now we consider two recursive sequences (xn)n∈N and (Sn)n∈N defined by x0 = u, S0 =X, xn+1 = h(Sn, xn), Sn+1 = Txn for every n ∈ N; then, using (5.4), (5.3) and definition of h, weget

f(xn+1) ≤ f(xn) ≤ f(u) < +∞, Sn+1 ⊆ Sn for every n ∈ N, (5.5)

f(xn+1) − infx∈Sn+1

f(x) ≤ 12

(f(xn) − inf

x∈Snf(x)

)for every n ∈ N; (5.6)

moreover, being xn+1 ∈ Sn+1 = Txn , it holds

ε

λd(xn, xn+1) ≤ f(xn) − f(xn+1) for every n ∈ N, (5.7)

whence (xn)n∈N is a Cauchy sequence, since there exists limn→+∞f(xn) ∈ R because of (5.5)and being f lower bounded for hypothesis.


Let v = limn→+∞xn; since f is a ubd-slsc function, f(u) ≤ infXf + ε0 = a and using(5.5), we obtain that

f(v) ≤ limn→+∞

f(xn) ≤ f(u) (5.8)

and so (a) is verified.From (5.7), using triangular inequality of distance, we deduce

ε

λd(x0, xn) ≤

n∑k=1

ε

λd(xk−1, xk) ≤

n∑k=1

(f(xk−1) − f(xk)

)= f(x0) − f(xn) (5.9)

for every n ∈ Z+; hence, considering that x0 = u, v = limn→+∞xn, f(xn) ≥ infx∈Xf(x) for everyn ∈ N and by means of a limit passage for n → +∞ in the first and the last terms, we get

ε

λd(u, v) ≤ f(u) − lim

n→+∞f(xn) ≤ inf

x∈Xf(x) + ε − lim

n→+∞f(xn) ≤ ε, (5.10)

from whence (b) follows.As an alternative, for showing (b), we can observe that Tz ⊆ {x ∈ X : d(x, z) ≤ λ} for

every z ∈ X such that f(z) ≤ infx∈Xf(x) + ε: indeed, in such hypothesis on z, if y ∈ Tz andif by absurd d(y, z) > λ then f(y) ≤ f(z) − (ε/λ)d(y, z) < f(z) − ε ≤ infx∈Xf(x), that givesa contradiction. Besides, using (5.5), we have that xn ∈ Tu ⊆ {x ∈ X : d(x, u) ≤ λ} for everyn ∈ N. Consequently d(v, u) = limn→+∞d(xn, u) ≤ λ, that is (b).

If, by absurd, (c) is not true, then

there exists x ∈ X \ {v} such that f(x) ≤ f(v) − ελd(x, v); (5.11)

owing to (5.5) and (5.8), it is

f(v) ≤ limk→+∞

f(xk) ≤ f(xm) ≤ f(xn) − ελd(xm, xn) for every n, m ∈ N, m ≥ n (5.12)

and hence, passing to the limit for m → +∞ in the first and the fourth terms, we get f(v) ≤f(xn) − (ε/λ)d(v, xn) for every n ∈ N; therefore, using (5.11), it holds:

f(x) ≤ f(xn) − ελd(v, xn) − ε

λd(x, v) ≤ f(xn) − ε

λd(xn, x) for every n ∈ N; (5.13)

hence x ∈ Sn+1 for every n ∈ N, from whence

f(x) ≥ infSnf for every n ∈ N; (5.14)

besides, since from (5.5) it follows that (infSnf)n∈N is a weakly increasing sequence, we can doa limit passage for n → +∞ in (5.6), obtaining that 0 ≤ α := limn→+∞(f(xn) − infy∈Snf(y)) ≤


(1/2)α, from whence α = 0; then, using (5.11) and (5.8), we get that f(x) < f(v) ≤limn→+∞f(xn) = limn→+∞infSnf , that is in contradiction with (5.14).

Example 5.3. In the hypotheses of Theorem 5.2, but without the additional hypothesis f(u) ≤a, it is easy to verify that conclusions (a) and (c) are still true (in fact, if ε0 = a − infXf and iff(u) > infXf + ε0, then it is enough to consider t ∈ X such that f(t) ≤ infXf + ε0 and in suchcase a point v relative to ε0, λ and t as in Theorem 5.2 solves the question) but it can happenthat there exist ε, λ and u satisfying the remaining hypotheses, for which there is not a pointv that verifies conclusions (b) and (c).

On R we consider the equivalence relation ∼ defined by x ∼ y if x − y ∈ Q(x, y ∈ R)and let R := {A ∩ [0, 2] : A equivalence class with respect to ∼}; then #B = ℵ0 for everyB ∈ R and ∪R = [0, 2], hence #R = 2ℵ0 , so there exists a bijective function ϕ : R → ]0, 1];moreover B = [0, 2] for every B ∈ R. Let now f : R → R be defined by

f(x) =

⎧⎨⎩

arctg x if x ∈ ]−∞, 0[⋃

]2,+∞[,

ϕ(B) if x ∈ B (B ∈ R).(5.15)

Because f is real-valued and continuous in the points y where f(y) < 0, then f is ubd-slsc,considering a = 0 > infRf ; moreover f is bounded from below. Let ε = (π/2) + 1, λ = 1 andu = 1; then f(u) ∈ ϕ(R) ≤ 1 = infRf + ε. If by absurd there exists v ∈ R verifying conclusions(b) and (c), then using (b) we get that |v − 1| ≤ 1 and therefore v ∈ [0, 2]; let B ∈ R such thatv ∈ B. Consequently 1 ≥ f(v) > 0 and, if α ∈ ]0, f(v)[, we obtain that ϕ−1(α) ∈ R, whenceϕ−1(α) = [0, 2] and so there exists a sequence (xn)n∈N of elements of ϕ−1(α) converging to v;from this, using (c), we obtain that α = ϕ(ϕ−1(α)) = f(xn) > f(v) − ε|xn − v| for every n ∈ N

and, by means of a limit passage for n → +∞, we have α ≥ f(v), that is a contradiction.

Example 5.4. If in Theorem 5.2 the hypothesis (5.1) is replaced by a hypothesis of bd-slsc on f,then, in spite of what we noted at the beginning of Example 5.3, conclusion (c) can be not true.Indeed here we will show an example of a bd-slsc function f : [1/2,+∞[ → ]0, 1] for whichthere does not exist v ∈ [1/2,+∞[ verifying conclusion (c) of Theorem 5.2 with respect to ε =λ = 1 (with such a choice the hypothesis f(u) ≤ infx∈[1/2,+∞[f(x)+ε of Theorem 5.2 is satisfiedby every u ∈ [1/2,+∞[), that is for every v ∈ [1/2,+∞[ there exists w ∈ [1/2,+∞[ \{v} suchthat f(w) ≤ f(v) − |w − v|.

Let B :=⋃n∈Z+

(](3n2 + 5n − 1)/(3(n + 2)), n[∪]n, (3n2 + 7n + 1)/(3(n + 2))[) and let

f(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1n + 2

+ 3(n − x), if x ∈]

3n2 + 5n − 13(n + 2)

, n

[(n ∈ Z+),

1n + 2

+ 3(x − n), if x ∈]n,

3n2 + 7n + 13(n + 2)

[(n ∈ Z+),

1, if x ∈[

12,+∞

[\ B

(5.16)


(i.e., if g : [1/2,+∞[ → ]0, 1] is the continuous piecewise affine function, with slopealternatively equal to −3 and 3 in the connected components of B and with value 1 in[1/2,+∞[ \(B ∪ Z+), f coincides with g on [1/2,+∞[ \Z+ and has value 1 in the points ofZ+).

Then f is a bd-slsc function, by the help of (a) of Remarks 3.3.On the other hand, now we will show that for every v ∈ [1/2,+∞[ there exists w ∈

[1/2,+∞[ \{v} such that f(w) ≤ f(v) − |w − v|:

(i) if v ∈ B, it is enough to choose w ∈ B in the same connected component of v andsuch that f(w) < f(v), because with such a choice it holds

f(w) = f(v) − 3|w − v| < f(v) − |w − v|; (5.17)

(ii) if v ∈ Z+, then a point w sufficiently close to v solves what is requested, because

limt→v

f(t) < 1 = limt→v

(f(v) − |t − v|); (5.18)

(iii) if v ∈ [1/2,+∞[ \(B ∪ Z+), then it is sufficient to consider an element n ∈ Z+

such that |n − v| ≤ 1/2 and to choose w in the interval with extremes n and v,w sufficiently close to n, because

limt→n

f(t) =1

n + 2≤ 1

3<

12= 1 − 1

2≤ lim

t→n

(f(v) − |t − v|). (5.19)

Remark 5.5. In the following results we note that Caristi’s fixed point theorem (see [19,Theorem (2.1)’]) and Caristi’s infinite fixed points theorem can be extended to the case inwhich the hypothesis of lsc is replaced by ubd-slsc (see also the extensions given in the cased-slsc by [2, Theorem 2.1] and by [1]).

Theorem 5.6 (Caristi’s fixed point theorem; see [19, Theorem (2.1)’]). Let (X, d) be a completemetric space and let ϕ : X → R be a ubd-slsc and bounded from below function. Let T : X → X be afunction such that d(x, T(x)) ≤ ϕ(x) − ϕ(T(x)) for every x ∈ X. Then there exists x0 ∈ X such thatT(x0) = x0.

Proof. It is enough to repeat the same demonstration of Theorem 2.2 in [1], whereTheorem 5.2 has to be used instead of Theorem 2.1 in [1].

Theorem 5.7 (Caristi’s infinite fixed points theorem). Let (X, d) be a complete metric space andlet ϕ : X → R be a ubd-slsc and bounded from below function, that does not obtain its infimum onX. Let T : X → X be a function such that d(x, T(x)) ≤ ϕ(x) − ϕ(T(x)) for every x ∈ X. Then Tadmits infinite fixed points in X.

Proof. It is enough to repeat the same demonstration of Theorem 2.3 in [1], where Theorems5.2 and 5.6 have to be used instead of Theorems 2.1 and 2.2 of [1].


References

[1] Y. Chen, Y. J. Cho, and L. Yang, “Note on the results with lower semi-continuity,” Bulletin of the KoreanMathematical Society, vol. 39, no. 4, pp. 535–541, 2002.

[2] W. A. Kirk and L. M. Saliga, “The Brezis-Browder order principle and extensions of Caristi’stheorem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 4, pp. 2765–2778, 2001.

[3] J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, vol. 20 of CMS Books in Mathematics,Springer, New York, NY, USA, 2005.

[4] A. Aruffo and G. Bottaro, “Generalizations of sequential lower semicontinuity,” Bollettino della UnioneMatematica Italiana. Serie 9, vol. 1, no. 2, pp. 293–318, 2008.

[5] J. R. Giles, Convex Analysis with Application in the Differentiation of Convex Functions, vol. 58 of ResearchNotes in Mathematics, Pitman, Boston, Mass, USA, 1982.

[6] E. O. Thorp, “Internal points of convex sets,” Journal of the London Mathematical Society, vol. 39, pp.159–160, 1964.

[7] N. Dunford and J. T. Schwartz, Linear Operators, Part I, John Wiley & Sons, New York, NY, USA, 1957.[8] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York, NY, USA,

2nd edition, 1980.[9] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, vol. 92 of Ergebnisse der

Mathematik und ihrer Grenzgebiete, Springer, Berlin, Germany, 1977.[10] S. M. Khaleelulla, Counterexamples in Topological Vector Spaces, vol. 936 of Lecture Notes in Mathematics,

Springer, Berlin, Germany, 1982.[11] C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, USA, 2002.[12] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1970.[13] I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, pp.

324–353, 1974.[14] H. Brezis and F. E. Browder, “A general principle on ordered sets in nonlinear functional analysis,”

Advances in Mathematics, vol. 21, no. 3, pp. 355–364, 1976.[15] M. Altman, “A generalization of the Brezis-Browder principle on ordered sets,” Nonlinear Analysis:

Theory, Methods & Applications, vol. 6, no. 2, pp. 157–165, 1982.[16] M. Turinici, “A generalization of Altman’s ordering principle,” Proceedings of the American

Mathematical Society, vol. 90, no. 1, pp. 128–132, 1984.[17] W.-S. Du, “On some nonlinear problems induced by an abstract maximal element principle,” Journal

of Mathematical Analysis and Applications, vol. 347, no. 2, pp. 391–399, 2008.[18] Z. Wu, “Equivalent extensions to Caristi-Kirk’s fixed point theorem, Ekeland’s variational principle,

and Takahashi’s minimization theorem,” Fixed Point Theory and Applications, vol. 2010, Article ID970579, 20 pages, 2010.



Research ArticleModuli and Characteristics of Monotonicity inSome Banach Lattices

Paweł Foralewski,1 Henryk Hudzik,1 Radosław Kaczmarek,1and Miroslav Krbec2

1 Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87,61-614 Poznan, Poland

2 Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25,115 67 Prague 1, Czech Republic

Correspondence should be addressed to Henryk Hudzik, [email protected]

Received 7 December 2009; Accepted 10 February 2010


Copyright q 2010 Paweł Foralewski et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

First the characteristic of monotonicity of any Banach lattice X is expressed in terms of theleft limit of the modulus of monotonicity of X at the point 1. It is also shown that for Kothespaces the classical characteristic of monotonicity is the same as the characteristic of monotonicitycorresponding to another modulus of monotonicity δm,E. The characteristic of monotonicity ofOrlicz function spaces and Orlicz sequence spaces equipped with the Luxemburg norm arecalculated. In the first case the characteristic is expressed in terms of the generating Orlicz functiononly, but in the sequence case the formula is not so direct. Three examples show why in thesequence case so direct formula is rather impossible. Some other auxiliary and complementedresults are also presented. By the results of Betiuk-Pilarska and Prus (2008) which establish thatBanach lattices X with ε0,m(X) < 1 and weak orthogonality property have the weak fixed pointproperty, our results are related to the fixed point theory (Kirk and Sims (2001)).

1. Introduction

Let us denote S+(X) = S(X) ∩X+, where S(X) is the unit sphere of a Banach lattice X (for itsdefinition, see [1–3]) and X+ is the positive cone of X.

A Banach lattice X is said to be strictly monotone (X ∈ (SM)) if for all x, y ∈ X+ suchthat y ≤ x and y /=x we have ‖y‖ < ‖x‖. A Banach lattice X is said to be uniformly monotone(X ∈ (UM)) if for any ε ∈ (0, 1) there is δ(ε) ∈ (0, 1) such that ‖x − y‖ ≤ 1 − δ(ε) whenever0 ≤ y ≤ x, ‖x‖ = 1, and ‖y‖ ≥ ε (see [1]).

For a given Banach lattice X, the function δm,X : [0, 1] → [0, 1] defined by

δm,X(ε) = inf{

1 − ∥∥x − y∥∥ : 0 ≤ y ≤ x, ‖x‖ = 1,∥∥y∥∥ ≥ ε} (1.1)


is said to be the lower modulus of monotonicity of X. It is easy to show that (see [4])

δm,X(ε) = inf{

1 − ∥∥x − y∥∥ : 0 ≤ y ≤ x, ‖x‖ = 1,∥∥y∥∥ = ε

}= 1 − sup

{∥∥x − y∥∥ : 0 ≤ y ≤ x, ‖x‖ = 1,∥∥y∥∥ ≥ ε}

= 1 − sup{∥∥x − y∥∥ : 0 ≤ y ≤ x, ‖x‖ = 1,

∥∥y∥∥ = ε}.

(1.2)

The lower modulus of monotonicity δm,X is a convex function on the interval [0, 1] (see [5])(so δm,X is continuous on the interval [0, 1) and nondecreasing on [0, 1] as well). It is alsoclear that δm,X(ε) ≤ ε for any ε ∈ [0, 1]. Obviously, X is uniformly monotone if and only ifδm,X(ε) > 0 for every ε ∈ (0, 1]. It is easy to see that a Banach lattice X is strictly monotone ifand only if δm,X(1) = 1.

The number ε0,m(X) ∈ [0, 1] defined by

ε0,m(X) = sup{ε ∈ [0, 1] : δm,X(ε) = 0} = inf{ε ∈ [0, 1] : δm,X(ε) > 0} (1.3)

is said to be the characteristic of monotonicity ofX. Obviously, a Banach latticeX is uniformlymonotone if and only if ε0,m(X) = 0.

We can also define another characteristic of monotonicity of X, namely,

ε0,m(X) = sup{ε ≥ 0 : ηm,X(ε) = 0

}= inf

{ε ≥ 0 : ηm,X(ε) > 0

}, (1.4)

where ηm,X is the upper modulus of monotonicity defined for all ε > 0 by the formula

ηm,X(ε) = inf{∥∥x + y

∥∥ − 1 : x, y ∈ X+, ‖x‖ = 1,∥∥y∥∥ ≥ ε}

= inf{∥∥x + y

∥∥ − 1 : x, y ∈ X+, ‖x‖ = 1,∥∥y∥∥ = ε

} (1.5)

(see [6, 7]). It is clear by the triangle inequality for the norm that ηm,X(ε) ≤ ε for all ε > 0.Obviously, a Banach lattice X is uniformly monotone if and only if ηm,X(ε) > 0 for all ε > 0 orequivalently if ε0,m(X) = 0.

Let us also recall relationships between two moduli of monotonicity δm,X and ηm,X aswell as relationships between the characteristic of monotonicity ε0,m(X) and ε0,m(X).

For arbitrary ε ∈ (0, 1) the following inequalities hold true (see [6]):

δm,X(ε/(1 + ε))1 − δm,X(ε/(1 + ε))

≤ ηm,X(ε) ≤δm,X(ε)

1 − δm,X(ε) . (1.6)

Notice that inequalities (1.6) are equivalent to the following ones:

ηm,X(ε)1 + ηm,X(ε)

≤ δm,X(ε) ≤ηm,X(ε/(1 − ε))

1 + ηm,X(ε/(1 − ε)) (1.7)

for any ε ∈ (0, 1). In [4, Theorem 1], it has been shown that

ε0,m(X) ≤ ε0,m(X) ≤ 2ε0,m(X). (1.8)


It is easy to show that the upper estimate of the characteristic of monotonicity ε0,m(X) of aBanach lattice X given above can be improved. Namely, since ‖x + y‖ ≥ max(‖x‖, ‖y‖) forany couple x, y ≥ 0, we have ηm,X(ε) > 0 for all ε > 1, whence we get ε0,m(X) ≤ 1. Therefore

ε0,m(X) ≤ ε0,m(X) ≤ min{1, 2ε0,m(X)} (1.9)

for any Banach lattice X.For more information on the monotonicity properties and coefficient of monotonicity

in some Kothe spaces, we refer to [4–14].

2. Some General Results

In this part of the paper we will give a few general results. First we will present a new formulafor the characteristic of monotonicity ε0,m(X) and we will introduce another modulus ofmonotonicity and characteristic of monotonicity for Kothe spaces. Obtained results will beuseful in the last part of the paper in order to calculate the characteristic of monotonicity inOrlicz spaces. Finally we will investigate ε0,m(X).

2.1. A New Formula for the Characteristic of Monotonicity ε0,m(X)

Theorem 2.1. For any normed lattice X the following equality is true:

ε0,m(X) = 1 − δm,X(1−), (2.1)

where δm,X(1−) = limε→ 1−δm,X(ε). Moreover,

δm,X(1 − δm,X(ε)) = 1 − ε (2.2)

for arbitrary ε ∈ (ε0,m(X), 1] if ε0,m(X) < 1 as well as in the case when ε = ε0,m(X) = 1.

Proof. If ε0,m(X) = 1, then by the definition of ε0,m(X), we have δm,X(ε) = 0 for any ε ∈ (0, 1),whence we get 1 − δm,X(1−) = 1.

Let now ε0,m(X) < 1, ε ∈ (ε0,m(X), 1), and η ∈ (0, 1 − δm,X(ε)). Then for any x ∈ S(X)and y ∈ X satisfying 0 ≤ y ≤ x, ‖y‖ = ε, and ‖x − y‖ ≥ 1 − δm,X(ε) − η, we have

ε =∥∥y∥∥ =

∥∥x − (x − y)∥∥ ≤ 1 − δm,X(∥∥x − y∥∥) ≤ 1 − δm,X

(1 − δm,X(ε) − η

). (2.3)

Since δm,X is a continuous function on the interval [0, 1), by δm,X(ε) > 0 and arbitrariness ofη ∈ (0, 1 − δm,X(ε)), we get

ε ≤ 1 − δm,X(1 − δm,X(ε)). (2.4)

Letting ε → 1−, we have

1 ≤ 1 − δm,X(1 − δm,X

(1−)), (2.5)


that is, δm,X(1 − δm,X(1−)) ≤ 0, whence

δm,X(1 − δm,X

(1−))

= 0. (2.6)

Therefore, ε0,m(X) ≥ 1 − δm,X(1−). Letting ε ↘ ε0,m(X) in (2.4), we get the opposite inequality,which ends the proof of equality (2.1).

Now we will show that equality (2.2) holds true. Suppose first that ε ∈ (ε0,m(X), 1).Since δm,X is a nondecreasing function on the interval [0, 1], by inequality (2.4), definingt = 1 − δm,X(ε), we get

1 − δm,X(ε) ≥ 1 − δm,X(1 − δm,X(1 − δm,X(ε))) = 1 − δm,X(1 − δm,X(t)). (2.7)

Simultaneously, since δm,X is strictly increasing on the interval (ε0,m(X), 1], by equality (2.1),we have

ε0,m(X) = 1 − δm,X(1−)< 1 − δm,X(ε) = t < 1 (2.8)

for any ε ∈ (ε0,m(X), 1). In consequence, inequality (2.4) holds also for t in place of ε, whichmeans that

1 − δm,X(1 − δm,X(t)) ≥ t = 1 − δm,X(ε). (2.9)

Combining inequalities (2.7) and (2.9), we get the equality

1 − δm,X(1 − δm,X(t)) = 1 − δm,X(ε). (2.10)

Since ε, t ∈ (ε0,m(X), 1) and δm,X is strictly increasing on this interval, we get the equalityδm,X(t) = 1 − ε, which is just equality (2.2) for ε ∈ (ε0,m(X), 1).

Let now ε = 1. Since δm,X(1−) ≤ δm,X(1), by inequality (2.1), we get 1 − δm,X(1) ≤1 − δm,X(1−) = ε0,m(X), whence δm,X(1 − δm,X(1)) = 0. Indeed, if δm,X is continuous at 1, thenδm,X(1−) = δm,X(1) and so 1 − δm,X(1) = 1 − δm,X(1−) = ε0,m(X), whence δm,X(1 − δm,X(1)) =δm,X(ε0,m(X)) = 0. If δm,X is not continuous at 1, then δm,X(1−) < δm,X(1) and so 1 − δm,X(1) <1 − δm,X(1−) = ε0,m(X), whence, by the definition of ε0,m(X), we have δm,X(1 − δm,X(1)) = 0.Therefore equality (2.2) holds also in this case.

Remark 2.2. In equality (2.1), δm,X(1−) cannot be replaced by δm,X(1). In Examples 2.3 and 2.4we will present Banach lattices X for which δm,X(ε) = 0 for any ε ∈ [0, 1) and δm,X(1) = 1.


Example 2.3. Let us first consider the space Lp = Lp([0, 1],Σ, m) with 1 ≤ p < ∞ over theLebesgue measure space ([0, 1],Σ, m). If x ∈ S+(Lp) andA ∈ Σ is such that ‖xχA‖p = ε ∈ [0, 1],then we have

1 = ‖x‖pp =∥∥xχ[0,1]\A

∥∥pp +∥∥xχA∥∥pp. (2.11)

Hence for y = xχA, we get ‖y‖p = ε and ‖x − y‖p = ‖xχ[0,1]\A‖p = (1 − εp)1/p, whence 1 −‖x − y‖p = 1 − (1 − εp)1/p. In consequence δm,Lp(ε) ≤ 1 − (1 − εp)1/p. In order to show theopposite inequality, let us take arbitrary 0 ≤ y ≤ x ∈ Lp, ‖x‖p = 1, ‖y‖p ≥ ε. Then

1 = ‖x‖pp =∫1

0xp(t)dμ(t) =

∫1

0

[(x − y) + y]p(t)dμ(t)

≥∫1

0

(x − y)p(t)dμ(t) +

∫1

0yp(t)dμ(t) =

∥∥x − y∥∥pp +∥∥y∥∥pp,

(2.12)

whence

∥∥x − y∥∥p ≤(

1 − ∥∥y∥∥pp)1/p ≤ (1 − εp)1/p. (2.13)

This means that 1 − ‖x − y‖p ≥ 1 − (1 − εp)1/p, whence, by arbitrariness of x and y, we get

δm,Lp(ε) ≥ 1 − (1 − εp)1/p. Therefore we have δm,Lp(ε) = 1 − (1 − εp)1/p for every ε ∈ [0, 1].Let us define X = ⊕Lpn , the �1-direct sum of the spaces Lpn , where pn ≥ 1 for any n ∈ N,

and pn ↗ ∞ as n → ∞, equipped with the norm ‖x‖ = ∑∞n=1 ‖xn‖pn for any x = (xn)∞n=1 ∈ X

with xn ∈ Lpn for any n ∈ N. Since any space Lpn is order linearly isometrically embedded intoX, where the embedding operator is defined by

Lpn � xn −→ (0, 0, . . . , 0, xn, 0, 0, . . .) (2.14)

with xn on the nth place, for any ε ∈ [0, 1), we have

0 ≤ δm,X(ε) ≤ δm,Lpn (ε) = 1 − (1 − εpn)1pn ↘ 0 (2.15)

as n → ∞, and consequently, δm,X(ε) = 0 for any ε ∈ [0, 1). Simultaneously, the space X isstrictly monotone as the �1-direct sum of uniformly monotone spaces Lpn with 1 ≤ pn <∞ forany n ∈ N. Therefore δm,X(1) = 1 and δm,X(1−) = 0.

Example 2.4. Let now L0 = L0([0,∞)) be the space of all (equivalence classes of) Lebesguemeasurable real-valued functions defined on the interval [0,∞). For any x ∈ L0 we define itsdistribution function μ by

μx(λ) = m{t ∈ [0, γ) : |x(t)| > λ} (2.16)


(see [3, 15, 16]) and the nonincreasing rearrangement x∗ of x as

x∗(t) = inf{λ ≥ 0 : μx(λ) ≤ t

}(2.17)

(under the convention inf ∅ =∞).Let ω : [0,∞) → R+ be a nonincreasing locally integrable function, called a weight

function. We say that the weight function is regular if there exists η > 0 such that∫2t

0 ω(t)dt ≥(1 + η)

∫ t0ω(t)dt for any t ∈ [0,∞) (see [9, 17]).

For any weight function ω, we define the Lorentz space by the formula

Λω ={x ∈ L0 : ‖x‖ =

∫∞0x∗(t)ω(t)dt <∞

}. (2.18)

Now we will show that for any Lorentz space Λω such that the weight function is not regularbut

∫∞0 ω(t)dt = ∞ (e.g., ω(t) = min(1, 1/t) for t ∈ [0,∞)), we have δm,Λω(1

−) = 0 < 1 =δm,Λω(1).

In fact, since Λω is strictly monotone (see [18, Proposition 4.1]), we have the equalityδm,Λω(1) = 1. Simultaneously, sinceω is not regular, there exists an increasing sequence (tn)

∞n=1

in the interval [0,∞) such that

∫2tn

0ω(t)dt ≤

(1 +

1n

)∫ tn0ω(t)dt. (2.19)

We can find a decreasing sequence of positive numbers (un)∞n=1 such that

∫2tn

0unω(t)dt = 1 (2.20)

for any n ∈ N. For xn := unχ[0,2tn) and yn := unχ[0,tn) (n ∈ N), we get 0 ≤ yn ≤ xn, ‖xn‖ = 1and, by inequality (2.19), n/(n + 1) ≤ ‖yn‖ ≤ 1. Since (xn − yn)∗ = y∗n, we also have thatn/(n + 1) ≤ ‖xn − yn‖ ≤ 1 for any n ∈ N. Therefore δm,Λω(ε) = 0 for any ε ∈ [0, 1).

Problem 1. In the above examples it has been shown that there are Banach lattices for whichδm,X(1−) < δm,X(1) and ε0,m(X) = 1, that is, δm,X(1−) = 0. It is natural to ask whether thereexist Banach lattices X, for which 0 < δm,X(1−) < δm,X(1).

From Theorem 2.1 and the definition of the modulus δm,X (see the preliminaries), wehave the following.

Corollary 2.5. For arbitrary Banach lattice X the following formulas hold true:

ε0,m(X) = limε→ 1−

(sup{∥∥x − y∥∥ : 0 ≤ y ≤ x, ‖x‖ = 1,

∥∥y∥∥ ≥ ε})

= limε→ 1−

(sup{∥∥x − y∥∥ : 0 ≤ y ≤ x, ‖x‖ = 1,

∥∥y∥∥ = ε}).

(2.21)


2.2. Modulus and Characteristic of Monotonicity in Kothe Spaces

Denote by (T,Σ, μ) a positive, complete, and σ-finite measure space and by L0 = L0(T,Σ, μ)the space of all (equivalence classes of) real-valued and Σ-measurable functions defined onT. For two functions x, y ∈ L0 we write x ≤ y if x(t) ≤ y(t) μ—a.e. in T. By E = (E,≤, ‖ · ‖E)we denote a Kothe space over the measure space (T,Σ, μ), that is, E is a Banach subspace ofL0 which satisfies the following conditions (see [2, 3]).

(i) If |x| ≤ |y|, y ∈ E, and x ∈ L0, then x ∈ E and ‖x‖E ≤ ‖y‖E.(ii) There exists a function x ∈ E which is strictly positive μ—a.e. in T.

In Kothe spaces the definition of the characteristic of monotonicity can be simplified by usinganother modulus. Using the new formula for the characteristic of monotonicity of Kothespaces, it should be easier to calculate this coefficient in concrete classes of Kothe spaces. Wewill see this advantage of the new formula in the class of Orlicz sequence spaces endowedwith the Luxemburg norm. Let us define for E the modulus δm,E : [0, 1] → [0, 1] by theformula

δm,E(ε) = inf{

1 − ∥∥x − xχA∥∥E : x ≥ 0, ‖x‖E = 1, A ∈ Σ,∥∥xχA∥∥E ≥ ε}. (2.22)

Obviously, the modulus δm,E is nondecreasing with respect to ε ∈ [0, 1] and δm,X(ε) ≤δm,E(ε) ≤ ε for any ε ∈ [0, 1]. It is also possible to prove similarly as for the modulus δm,Xin [4] that

δm,E(ε) = inf{

1 − ∥∥x − xχA∥∥E : x ≥ 0, ‖x‖E = 1, A ∈ Σ,∥∥xχA∥∥E = ε

}= 1 − sup

{∥∥x − xχA∥∥E : x ≥ 0, ‖x‖E = 1, A ∈ Σ,∥∥xχA∥∥E ≥ ε}

= 1 − sup{∥∥x − xχA∥∥E : x ≥ 0, ‖x‖E = 1, A ∈ Σ,

∥∥xχA∥∥E = ε}.

(2.23)

The characteristic of monotonicity ε0,m(E) corresponding to the modulus δm,E is defined by

ε0,m(E) = sup{ε ∈ [0, 1] : δm,E(ε) = 0

}= inf

{ε ∈ [0, 1] : δm,E(ε) > 0

}. (2.24)

We have the following:

Proposition 2.6. For arbitrary Kothe space E the following formula holds true:

ε0,m(E) = sup{

lim supn→∞

∥∥xnχA′n∥∥E : (xn) ⊂ S+(E), (An) ⊂ Σ,∥∥xnχAn

∥∥E −→ 1

}. (2.25)

Proof. Let us denote

α(E) = sup{

lim supn→∞


∥∥E −→ 1

}. (2.26)


First, we will show that ε0,m(E) ≤ α(E). In order to do it, assume that ε0,m(E) > 0 and ε ∈[0, ε0,m(E)). Then δm,E(ε) = 0 and so

sup{∥∥xχA∥∥E : x ≥ 0, ‖x‖E = 1, A ∈ Σ,

∥∥xχA′∥∥E = ε}= 1. (2.27)

Next there exist a sequence (xn) in S+(E) and a sequence (An) in Σ such that ‖xnχA′n‖E = εand ‖xnχAn‖E → 1. Therefore ε ≤ α(E), whence ε0,m(E) ≤ α(E).

In order to prove the opposite inequality assume that ε0,m(E) < 1 and ε ∈ (ε0,m(E), 1],that is,

sup{∥∥xχA∥∥E : x ≥ 0, ‖x‖E = 1, A ∈ Σ,

∥∥xχA′∥∥E ≥ ε} < 1 (2.28)

because of δm,E(ε) > 0. We will show that α(E) ≤ ε. Otherwise we would have α(E) > εand then there were a sequence (xn) in S+(E) and a sequence of sets (An) in Σ such that‖xnχAn‖E → 1 and ‖xnχA′n‖E > ε for n large enough. Hence we have

sup{∥∥xχA∥∥E : x ≥ 0, ‖x‖E = 1, A ∈ Σ,

∥∥xχA′∥∥E ≥ ε} = 1, (2.29)

which contradicts inequality (2.28). Therefore, α(E) ≤ ε and in consequence, by thearbitrariness of ε ∈ (ε0,m(E), 1], we conclude that α(E) ≤ ε0,m(E).

Now we will show that both characteristics of monotonicity ε0,m(E) and ε0,m(E) areequal in Kothe spaces. In order to prove this fact we will prove first a result that will behelpful to prove this equality.

Lemma 2.7. If E is a Kothe space, then for any positive ε and δ satisfying the condition ε + δ < 1 theinequality δm,E(ε + δ) ≥ δδm,E(ε) holds true.

Proof. Let ε, δ ∈ (0, 1) be such that ε + δ < 1 and δm,E(ε) > 0. Assume that 0 ≤ y ≤ x, ‖x‖E = 1,and ‖y‖E ≥ ε + δ. Let us define

A ={t ∈ T : y(t) < δx(t)

}. (2.30)

Then ‖yχA‖E ≤ ‖δx‖E = δ. Since ε + δ ≤ ‖y‖E ≤ ‖yχA‖E + ‖yχA′ ‖E, we get that ‖yχA′ ‖E ≥ ε.Therefore

∥∥x − y∥∥E ≤∥∥x − yχA′∥∥E ≤

∥∥x − δxχA′∥∥E =∥∥(1 − δ)x + δx − δxχA′

∥∥E

≤ (1 − δ)‖x‖E + δ∥∥x − xχA′∥∥E ≤ (1 − δ) + δ

(1 − δm,E(ε)

)

= 1 − δδm,E(ε).

(2.31)

Hence for all 0 ≤ y ≤ x such that ‖x‖E = 1, ‖y‖E ≥ ε + δ, we have that 1 − ‖x − y‖E ≥ δδm,E(ε),whence δm,E(ε + δ) ≥ δδm,E(ε).


Theorem 2.8. For arbitrary Kothe space E one has the equality

ε0,m(E) = ε0,m(E). (2.32)

Proof. Since δm,E(ε) ≤ δm,E(ε) for all ε ∈ [0, 1], we have

ε0,m(E) ≤ ε0,m(E). (2.33)

In order to get the inequality ε0,m(E) ≥ ε0,m(E), we need to consider separately two cases,namely, the case when ε0,m(E) < 1 and the case when ε0,m(E) = 1.

Case 1. Assume that ε0,m(E) < 1. By virtue of inequality (2.33), we have ε0,m(E) < 1 andδm,E(ε) > 0 for all ε ∈ (ε0,m(E), 1). By Lemma 2.7, we have

δm,E(ε1) ≥ (ε1 − ε)δm,E(ε) > 0 (2.34)

for all ε and ε1 such that ε0,m(E) < ε < ε1 < 1. Therefore, we obtained that δm,E(ε1) > 0 for anyε1 ∈ (ε0,m(E), 1). Hence

ε0,m(E) := inf{ε1 : δm,E(ε1) > 0} ≤ ε0,m(E). (2.35)

Case 2. Assume now that ε0,m(E) = 1. We will prove that ε0,m(E) = 1. Assume for the contrarythat ε0,m(E) < 1. Then, similarly as in Case 1, we get that δm,E(ε1) > 0 for all ε1 ∈ (ε0,m(E), 1),whence ε0,m(E) ≤ ε0,m(E) < 1, a contradiction. Therefore ε0,m(E) = 1 implies that ε0,m(E) = 1.

Now, we will prove the following:

Corollary 2.9. For arbitrary Kothe space E the following formulas are true:

ε0,m(E) = ε0,m(E) = limε→ 1−

sup{∥∥xχA′∥∥E : x ∈ S+(E), A ∈ Σ,

∥∥xχA∥∥E ≥ ε}

= limε→ 1−

sup{∥∥xχA′∥∥E : x ∈ S+(E), A ∈ Σ,

∥∥xχA∥∥E = ε}.

(2.36)

Proof. Note that, for any ε ∈ (0, 1),

sup{

lim supn→∞


∥∥E −→ 1

}

≤ sup{∥∥xχA′∥∥E : x ∈ S+(E), A ∈ Σ,

∥∥xχA∥∥E ≥ ε}.(2.37)

Hence, by Proposition 2.6 and the arbitrariness of ε ∈ (0, 1), we get

ε0,m(E) ≤ limε→ 1−

sup{∥∥xχA′∥∥E : x ∈ S+(E), A ∈ Σ,

∥∥xχA∥∥E ≥ ε}. (2.38)


Simultaneously, by Corollary 2.5 and Theorem 2.8,

limε→ 1−

sup{∥∥xχA′∥∥E : x ∈ S+(E), A ∈ Σ,

∥∥xχA∥∥E ≥ ε}

≤ limε→ 1−

(sup{∥∥x − y∥∥ : 0 ≤ y ≤ x, ‖x‖ = 1,

∥∥y∥∥ ≥ ε}) = ε0,m(E) = ε0,m(E).(2.39)

Combining (2.38) and (2.39), we get inequality (2.36).

Problem 2. We have δm,X(ε) ≤ δm,E(ε) ≤ ε, ε0,m(E) = ε0,m(E) (i.e., δm,X(ε) = δm,E(ε) = 0 forany ε ∈ [0, ε0,m(E)), and limε→ 1−δm,X(ε) = limε→ 1− δm,E(ε). It follows from Example 2.3 thatδm,X(ε) = δm,E(ε) for any ε ∈ [0, 1] for the space E = Lp([0, 1],Σ, m). So, it is natural to askwhether these two moduli are equal in arbitrary Kothe spaces.

2.3. Characteristic of Monotonicity ε0,m(X) of a Banach Lattice X

Analogously as for ε0,m(X) (see [4, Theorem 5]) we get the following:

Proposition 2.10. For arbitrary Banach lattice X the following formula holds true:

ε0,m(X) = sup{

lim supn→∞

‖zn − xn‖ : 0 ≤ xn ≤ zn, ‖xn‖ = 1, ‖zn‖ −→ 1}. (2.40)

Proof. Let us denote

α(X) = sup{

lim supn→∞

‖zn − xn‖ : 0 ≤ xn ≤ zn, ‖xn‖ = 1, ‖zn‖ −→ 1}. (2.41)

First, we will show that ε0,m(X) ≤ α(X). In order to do it, assume that ε > 0 and let ηm,X(ε) = 0,that is,

inf{‖z‖ : 0 ≤ x ≤ z, ‖x‖ = 1, ‖z − x‖ = ε} = 1. (2.42)

Therefore there exist sequences (xn)∞n=1 ⊂ S+(X) and (zn)

∞n=1 ⊂ X+ such that 0 ≤ xn ≤ zn and

‖zn − xn‖ = ε for any n ∈ N and ‖zn‖ → 1. Hence, for arbitrary ε > 0 such that ηm,X(ε) = 0,we have

ε ≤ sup{

lim supn→∞

‖zn − xn‖ : 0 ≤ xn ≤ zn, ‖xn‖ = 1, ‖zn‖ −→ 1}

= α(X). (2.43)

Therefore

ε0,m(X) ≤ α(X). (2.44)


Now, we will show the opposite inequality. In order to do this, assume that ε > 0 andηm,X(ε) > 0, that is,

inf{‖z‖ : 0 ≤ x ≤ z, ‖x‖ = 1, ‖z − x‖ ≥ ε} > 1. (2.45)

Then α(X) ≤ ε. Indeed, in the opposite case it would be α(X) > ε, and then there would existsequences (xn)

∞n=1 ⊂ S+(X) and (zn)

∞n=1 ⊂ X+ such that 0 ≤ xn ≤ zn for all n ∈ N, ‖zn‖ → 1 and

‖zn − xn‖ > ε for n ∈ N large enough. Hence we get

inf{‖z‖ : 0 ≤ x ≤ z, ‖x‖ = 1, ‖z − x‖ ≥ ε} = 1, (2.46)

which contradicts inequality (2.45). Therefore, α(X) ≤ ε whenever ε > 0 and ηm,X(ε) > 0.Consequently, α(X) ≤ ε0,m(X), which together with (2.44) ends the proof.

3. Characteristics of Monotonicity in Orlicz Spaces

In the last part of our paper we will present formulas for the characteristic of monotonicity inOrlicz function spaces and Orlicz sequence spaces. Let us start with some basic notions.

A map Φ : R → [0,∞] is said to be an Orlicz function if Φ is a nonzero function that isconvex, even, vanishing and continuous at zero and left continuous on R+, which means thatlimu→ b(Φ)−Φ(u) = Φ(b(Φ)) (for the definition of b(Φ), see below).

Given any Orlicz function Φ, we define on L0 = L0(T,Σ, μ), where μ is nonatomic, aconvex modular by the formula

IΦ(x) =∫T

Φ(x(t))dμ (3.1)

(see [19–24]). The Orlicz function space LΦ = LΦ(T,Σ, μ) generated by an Orlicz function Φis defined as

LΦ ={x ∈ L0 : IΦ(λx) < +∞ for some λ > 0

}. (3.2)

We equip this space with the Luxemburg norm

‖x‖Φ = inf{λ > 0 : IΦ

(xλ

)≤ 1}. (3.3)

In the sequence case, that is, when T = N, Σ = 2N, and μ(A) = card(A) for any A ⊂ N,we define on �0 = �0(N, 2N, μ) a convex modular IΦ by

IΦ(x) =∞∑n=1

Φ(x(n)). (3.4)

We define the Orlicz sequence space �Φ analogously as LΦ and also consider it with theLuxemburg norm.


We say that an Orlicz function Φ satisfies condition Δ2 for all u ∈ R+(at infinity) [atzero] if there isK > 0 such that the inequality Φ(2u) ≤ KΦ(u) holds for all u ∈ R (for all u ∈ R

satisfying |u| ≥ u0 with some u0 > 0 such that Φ(u0) < ∞) [for all u ∈ R satisfying |u| ≤ u0

with some u0 > 0 such that Φ(u0) > 0]. We write then Φ ∈ Δ2(R+) (Φ ∈ Δ2(∞))[Φ ∈ Δ2(0)],respectively. Let us note that Φ ∈ Δ2(0) implies that Φ vanishes only at zero and Φ ∈ Δ2(∞)implies that Φ(u) <∞ for all u ∈ R.

We will use two well-known parameters for the Orlicz function Φ: a(Φ) := sup{u > 0 :Φ(u) = 0} and b(Φ) := sup{u > 0 : Φ(u) <∞}.

3.1. The Characteristic of Monotonicity ε0,m(LΦ) of Orlicz Function Spaces

We start with the following:

Lemma 3.1. Assume thatΦ is an Orlicz function with a(Φ) > 0 and satisfying the conditionΔ2(∞)and let c ∈ (a(Φ),+∞). Then for any ε ∈ (0, 1) there exists δ(ε) ∈ (0, 1) such that if x ∈ LΦ,|x(t)| ≥ c for μ—a.e. t ∈ T, and IΦ(x) ≤ δ(ε), then ‖x‖Φ ≤ ε.

Proof. Since Φ ∈ Δ2(∞), so there are u0 > a(Φ) and K ≥ 2 such that Φ(2u) ≤ KΦ(u)for any u ≥ u0. We can assume that c < u0. Since the interval [c, u0] is compact and thefunction Φ(2u)/Φ(u) is continuous on this interval, we have that L := sup{(Φ(2u)/Φ(u)) :u ∈ [c, u0]} < ∞. In consequence, Φ(2u) ≤ max(K,L)Φ(u) for all u ≥ c. Let us denote byϕ the right-hand side derivative of Φ. Since for any t ≥ c, tϕ(t) ≤ Φ(2t) ≤ γΦ(t), whereγ := max(K,L), we have

M := supt≥c

tϕ(t)Φ(t)

<∞. (3.5)

Therefore, taking any u ≥ c and α ≥ 1, we have

∫αuu

ϕ(t)Φ(t)

dt ≤∫αuu

M

tdt, (3.6)

whence

Φ(αu) ≤ αMΦ(u). (3.7)

In consequence, if 0 < β ≤ 1 and u ≥ 0 are such that βu ≥ c, then we have

Φ(u) = Φ(

1β

(βu)) ≤ 1

βMΦ(βu), (3.8)

whence

Φ(βu) ≥ βMΦ(u). (3.9)


Therefore, if x ∈ LΦ and ε are as in the formulation of the Lemma, then, assuming that IΦ(x) ≤εM < 1, we have

εM ≥ IΦ(x) = IΦ(‖x‖Φ

x

‖x‖Φ

)≥ ‖x‖MΦ IΦ

(x

‖x‖Φ

)= ‖x‖MΦ , (3.10)

whence ‖x‖Φ ≤ ε. In such a way we proved our lemma with δ(ε) := εM.

Lemma 3.2 (see [4, Lemma 4]). Let μ(T) < ∞ and Φ ∈ Δ2(∞). Then for any ε ∈ (0, 1) there isp(ε) ∈ (0, 1) such that if 1 ≥ ‖xn‖Φ ≥ 1 − p(ε), then IΦ(x) ≥ 1 − ε.

Theorem 3.3 (see [4, Theorem 6]). If μ(T) <∞, Φ ∈ Δ2(∞), and a(Φ) > 0, then

δm,LΦ(1) = 1 − a(Φ)c(Φ)

, (3.11)

where c(Φ) is the nonnegative constant satisfying the equality Φ(c(Φ))μ(T) = 1.

Theorem 3.4. Let LΦ be an Orlicz function space. If μ(T) < ∞, then the following statements holdtrue.

(i) If Φ ∈ Δ2(∞) and a(Φ) = 0, then ε0,m(LΦ) = 0.

(ii) If Φ ∈ Δ2(∞) and a(Φ) > 0, then ε0,m(LΦ) = a(Φ)/c(Φ), where c(Φ) is the nonnegativeconstant satisfying the equality Φ(c(Φ))μ(T) = 1.

(iii) If Φ/∈Δ2(∞), then ε0,m(LΦ) = 1.

Proof. (i) If Φ ∈ Δ2(∞) and a(Φ) = 0, then the Orlicz space LΦ is uniformly monotone (see[6]), so ε0,m(LΦ) = 0.

(ii) By Theorems 2.1 and 3.3, we have

ε0,m

(LΦ)≥ a(Φ)c(Φ)

. (3.12)

Now, we will show that for any θ ∈ (0, 1) there exists σ(θ) ∈ (0, 1) (close enough to 1) suchthat if 0 ≤ y ≤ x ∈ S+(LΦ) and ‖y‖Φ ≥ σ(θ), then

∥∥x − y∥∥Φ ≤ (1 + θ)a(Φ)c(Φ)

+ θ. (3.13)

Then, by Corollary 2.5 and inequality (3.12), we will get ε0,m(LΦ) = a(Φ)/c(Φ).For any fixed θ ∈ (0, 1), by Lemma 3.1, we can find δ(θ) ∈ (0, 1) such that ‖z‖Φ ≤ θ for

any z satisfying IΦ(z) ≤ δ(θ) and |z(t)| ≥ (1 + θ)a(Φ) for μ—a.e. t ∈ T . Next, by Lemma 3.2,we can find that p(δ(θ)) ∈ (0, 1) such that IΦ(z) ≥ 1 − δ(θ) whenever ‖z‖Φ ≥ 1 − p(δ(θ)).Denote σ(θ) = 1 − p(δ(θ)).

Now for any fixed x and y such that 0 ≤ y ≤ x ∈ S+(LΦ) and ‖y‖Φ ≥ σ(θ), we definethe set

Ax,y ={t ∈ T : x(t) − y(t) > (1 + θ)a(Φ)

}. (3.14)


Since Φ is superadditive on R+, we have

1 = IΦ(x) = IΦ((x − y) + y) ≥ IΦ(x − y) + IΦ(y), (3.15)

whence, by ‖y‖Φ ≥ σ(θ), we get

IΦ(x − y) ≤ 1 − IΦ

(y) ≤ 1 − (1 − δ(θ)) = δ(θ). (3.16)

In consequence

IΦ((x − y)χAx,y

)≤ δ(θ), (3.17)

and, by virtue of Lemma 3.1,

∥∥∥(x − y)χAx,y

∥∥∥Φ≤ θ. (3.18)

Simultaneously, 0 ≤ (x − y)χA′x,y ≤ (1 + θ)a(Φ)χA′x,y ≤ (1 + θ)a(Φ)χT , whence

∥∥∥(x − y)χA′x,y∥∥∥Φ≤ (1 + θ)a(Φ)

∥∥χT∥∥Φ = (1 + θ)a(Φ)c(Φ)

. (3.19)

Combining (3.18) and (3.19), we get (3.13), and the proof is finished.(iii) Recall also that if Φ/∈Δ2(∞), then the Orlicz space LΦ contains an order

isomorphically isometric copy of l∞ (see [25, 26]), whence δm,LΦ(1) = 0 and consequentlyε0,m(LΦ) = 1.

Proceeding analogously as in proof of Theorem 3.4(i) and (iii), we get the following:

Theorem 3.5. Let LΦ be an Orlicz function space. If μ(T) = ∞, then ε0,m(LΦ) = 0 whenever Φ ∈Δ2(R) and ε0,m(LΦ) = 1 otherwise.

3.2. Characteristic of Monotonicity of Orlicz Sequence Spaces

We start with a result that will be important for proving the main result of this section.

Theorem 3.6. If the Orlicz function Φ satisfies the condition Δ2(0) and Φ(b(Φ)) ∈ (1/2, 1), then

δm,�Φ(1) = 1 − sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))}. (3.20)

Proof. Let us take arbitrary x such that IΦ(x) = 1 −Φ(b(Φ)) and define

y = (b(Φ), |x(1)|, |x(2)|, . . .), z = (b(Φ), 0, . . .). (3.21)


Then 0 ≤ z ≤ y, ‖z‖Φ = ‖y‖Φ = 1, and ‖y − z‖Φ = ‖x‖Φ. Therefore,

δm,�Φ(1) ≤ 1 − ∥∥y − z∥∥Φ = 1 − ‖x‖Φ (3.22)

and, by the arbitrariness of x, we have

δm,�Φ(1) ≤ 1 − sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))}. (3.23)

In order to prove the opposite inequality it is enough to show that the inequality

∥∥y − z∥∥Φ ≤ sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))} (3.24)

holds for any couple of elements y and z such that 0 ≤ z ≤ y and ‖z‖Φ = ‖y‖Φ = 1.First assume that y(i) < b(Φ) for every i ∈ N. Since there is at most one coordinate i0

satisfying Φ−1(1/2) < y(i0) < b(Φ), we can find λ > 1 such that λy(i) ≤ b(Φ) for any i ∈ N.Hence applying the assumption Φ ∈ Δ2(0), we get that IΦ(λy) < ∞, whence IΦ(y) = 1. Since0 ≤ z ≤ y, then in a similar way as for y we obtain that IΦ(z) = 1. Since Φ ∈ Δ2(0), we havea(Φ) = 0, whence we get that z(i) = y(i) for any i ∈ N. Therefore ‖y − z‖Φ = 0, and inequality(3.24) is true.

Let now there exist n ∈ N, for which y(n) = b(Φ). Since ‖z‖Φ = 1 and 0 ≤ z ≤ y, we getz(n) = b(Φ). Let us denote by y the element y if IΦ(y) = 1 or the element (y(1), y(2), . . . , y(n−1), b(Φ), y(n+1), y(n+2), . . .), where y(n+1) is chosen in such a way that IΦ(y) = 1 if IΦ(y) < 1.Then

∥∥y − z∥∥Φ =∥∥(y − z)χ

N\{n}

∥∥Φ ≤∥∥yχ

N\{n}

∥∥Φ ≤∥∥yχ

N\{n}

∥∥Φ

≤ sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))},(3.25)

which finishes the proof.

For the sake of completeness we will give proofs of Lemmas 3.7 and 3.8 because we donot know the papers in which they were also proved for degenerated Orlicz functions, thatis, for Orlicz functions Φ with Φ(b(Φ)) < 1.

Lemma 3.7. Let Φ ∈ Δ2(0), Φ(b(Φ)) < 1, and 0 < a < b(Φ). Then IΦ(xm) → 1 provided that‖xm‖Φ → 1 for any sequence (xm) such that xm ∈ B(�Φ) and |xm(n)| ≤ a for allm,n ∈ N.

Proof. Assume that there exists a sequence (xm) in B(�Φ) such that ‖xm‖Φ → 1, |xm(n)| ≤ afor any m,n ∈ N, and IΦ(xm) does not tend to 1 as n → ∞. Passing to a subsequence ifnecessary, we can assume that there exists δ > 0 such that IΦ(xm) ≤ 1 − δ for all m ∈ N. SinceΦ ∈ Δ2(0), we can find that η > 1 such that η ≤ b(Φ)/a and Φ(ηu) ≤ (1/(1 − δ))Φ(u) foru ∈ [0, a]. Therefore IΦ(ηxm) ≤ (1/(1 − δ))IΦ(xm) = 1, whence we have ‖xm‖Φ ≤ 1/η < 1,which is a contradiction.

Lemma 3.8. Assume that Φ ∈ Δ2(0) and b(Φ) < ∞. Then for any sequence (xm) such thatIΦ(xm) → 0 there holds ‖xm‖Φ → 0.


Proof. Let us take an arbitrary but fixed sequence (xm) such that IΦ(xm) → 0. We will showthat IΦ(λxm) → 0 for arbitrary λ > 0, whence we obtain that ‖xm‖Φ → 0 (see [23]).

Take an arbitrary but fixed λ > 0 and ε ∈ (0, 1) and let n be the smallest natural numbersuch that λ ≤ 2n. Since Φ ∈ Δ2(0), there existsK > 0 such that Φ(2u) ≤ KΦ(u) for u ≤ b(Φ)/22.By IΦ(xm) → 0 we can find m0 ∈ N such that

IΦ(xm) ≤ min{Φ(b(Φ)2n+1

),ε

Kn

}(3.26)

for m ≥ m0. Hence |xm(n)| ≤ b(Φ)/2n+1 for any n ∈ N and m ≥ m0, and finally

IΦ(λxm) ≤ IΦ(2nxm) ≤ KnIΦ(xm) ≤ Kn ε

Kn= ε (3.27)

for m ≥ m0, which ends the proof.

Theorem 3.9. Let �Φ be an Orlicz sequence space. Then the following statements are true:

(i) If Φ/∈Δ2(0) or Φ(b(Φ)) ≤ 1/2, then ε0,m(�Φ) = 1.

(ii) If Φ ∈ Δ2(0) and 1/2 < Φ(b(Φ)) < 1, then

ε0,m

(�Φ)= sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))}. (3.28)

(iii) If Φ ∈ Δ2(0) and Φ(b(Φ)) ≥ 1, then ε0,m(�Φ) = 0.

Proof. (i) If Φ/∈Δ2(0), then the Orlicz sequence space �Φ contains an order isomorphicallyisometric copy of �∞ (see [25, 26]), whence δm,�Φ(1) = 0 and consequently ε0,m(�Φ) = 1.Assume now that Φ(b(Φ)) ≤ 1/2. Defining

x = (b(Φ), b(Φ), 0, 0, . . .), y = (b(Φ), 0, 0, 0, . . .), (3.29)

we have that 0 ≤ y ≤ x and x, y ∈ S+(�Φ). Moreover x − y = (0, b(Φ), 0, 0, . . .), so ‖x − y‖ = 1.Consequently δm,�Φ(1) = 0, so ε0,m(�Φ) = 1.

(ii) In the first part of the proof we will show that there exists ε0 ∈ (0, 1) such that theinequality

sup{∥∥xχ

A′∥∥Φ : x ∈ S+

(�Φ), A ⊂ N,

∥∥xχA

∥∥Φ ≥ ε

}≤ sup{‖x‖Φ : IΦ(x) = 1 −Φ(ε · b(Φ))}

(3.30)

is true for every ε ∈ [ε0, 1). In order to do this, let a = Φ−1(max(1/2, (5/4)Φ(b(Φ)) − 1/4)).Then obviously Φ−1(1/2) ≤ a < b(Φ). By virtue of Lemma 3.7, we can find ε1 ∈ (0, 1) suchthat IΦ(x) ≥ Φ(b(Φ)) if ‖x‖Φ ≥ ε1, for every x ∈ B(�Φ) satisfying |x(i)| ≤ a for any i ∈ N. Let


us define the constant ε2 ∈ (0, 1) by the equality ε2 · b(Φ) = a. Since Φ ∈ Δ2(0), we can alsofind ε3 from the interval (0, 1) such that the inequality

Φ(u

ε

)≤(

1 +12

)Φ(u) (3.31)

holds for ε ∈ [ε3, 1) and u ∈ [0, a]. Finally, we put ε0 = max(ε1, ε2, ε3).Take now arbitrary ε ∈ [ε0, 1). We will show that IΦ(xχA′ ) ≤ 1 − Φ(ε · b(Φ)) for any

x ∈ S+(�Φ) and any set A ⊂ N such that ‖xχA‖Φ ≥ ε, whence we will obtain inequality (3.30).

We need to consider two cases.Let |x(i)| ≤ a for every i ∈ N. Then the definition of ε0 (ε0 ≥ ε1) yields that IΦ(xχA) ≥

Φ(b(Φ)). Hence IΦ(xχA′ ) ≤ 1 −Φ(b(Φ)) < 1 −Φ(ε · b(Φ)).Assume now that there exists exactly one n ∈ N such that x(n) ∈ (a, b(Φ)]. Since

IΦ(xχ

N\{n}) ≤ 1 −Φ(x(n)) < 1 −Φ(a) ≤ 1

2< Φ(b(Φ)), (3.32)

by the definition of ε0 (ε0 ≥ ε1), we get ‖xχN\{n}‖Φ < ε, whence n ∈ A. We have to consider

two different subcases.First, if x(n) ∈ (a, ε · b(Φ)), then (x(n)/ε) < b(Φ). Hence, by ‖xχA‖ ≥ ε, we get

Φ(x(n)ε

)+∑

i∈A\{n}Φ(x(i)ε

)≥ 1, (3.33)

and consequently

∑i∈A\{n}

Φ(x(i)ε

)≥ 1 −Φ

(x(n)ε

)> 1 −Φ(b(Φ)). (3.34)

Since x(i) ≤ a for any i ∈ N \ {n}, by the definition of ε0 (ε0 ≥ ε3) and inequality (3.31), weobtain

∑i∈A\{n}

Φ(x(i)) ≥ 23(1 −Φ(b(Φ))). (3.35)

Therefore

IΦ(xχ

A

)= Φ(xn) +

∑i∈A\{n}

Φ(x(i)) ≥ Φ(a) +23(1 −Φ(b(Φ)))

≥(

54Φ(b(Φ)) − 1

4

)+(

23− 2

3Φ(b(Φ))

)≥ Φ(b(Φ)),

(3.36)

whence IΦ(xχA′ ) ≤ 1 −Φ(b(Φ)) < 1 −Φ(εb(Φ)).


Let now x(n) ∈ [ε · b(Φ), b(Φ)). Then

IΦ(xχ

A′)= 1 − IΦ

(xχA

) ≤ 1 −Φ(x(n)) ≤ 1 −Φ(εb(Φ)). (3.37)

It is worth noticing that in the above inequality we can obtain the equality for A = {n} andx(n) = ε · b(Φ).

In the second part of the proof we will show that

limε→ 1−

sup{‖x‖Φ : IΦ(x) = 1 −Φ(ε · b(Φ))} = sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))}, (3.38)

whence, by virtue of inequality (3.30) and Corollary 2.9, we will get

ε0,m

(�Φ)= ε0,m

(�Φ)≤ sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))}. (3.39)

Since, by Theorem 3.6, we have the inequality opposite to (3.39), the proof will be finished.Let ε ∈ [ε0, 1). Then for arbitrary x satisfying IΦ(x) = 1−Φ(ε ·b(Φ)) we can find y such

that 0 ≤ y ≤ x and IΦ(y) = 1−Φ(b(Φ)). By superadditivity of the Orlicz function Φ on [0,∞),we can write

Φ(x(n)) = Φ(x(n) − y(n) + y(n)) ≥ Φ

(x(n) − y(n)) + Φ

(y(n)

)(3.40)

for all n ∈ N, whence

IΦ(x − y) ≤ IΦ(x) − IΦ(y) = Φ(b(Φ)) −Φ(ε · b(Φ)) <

12. (3.41)

By virtue of Lemma 3.8, there is σ(ε) > 0 such that ‖x − y‖Φ ≤ σ(ε), whence ‖x‖Φ ≤ ‖y‖Φ +σ(ε). Consequently

sup{‖x‖Φ : IΦ(x) = 1 −Φ(ε · b(Φ))} ≤ sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))} + σ(ε). (3.42)

Assuming now that ε → 1− and applying again Lemma 3.8, we have that σ(ε) → 0, whichgives (3.38).

(iii) It is well known that the condition Φ ∈ Δ2(0) implies that a(Φ) = 0, whichtogether with the condition Φ(b(Φ)) ≥ 1 gives that �Φ is uniformly monotone (see [27]),that is, ε0,m(�Φ) = 0.

Remark 3.10. The formulas given in Theorems 3.6 and 3.9(ii), respectively, are not completelyconstructive because they are not expressed in terms of the generating Orlicz functions only.However, finding better, that is, “more evident” formulas will be probably very difficultbecause these formulas can have different forms depending on the generating Orlicz functionΦ. We will illustrate this phenomena in some examples below.


In Example 3.11 we will show that for some Orlicz functions Φ,

sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))} = Φ−1(1 −Φ(b(Φ)))b(Φ)

. (3.43)

Example 3.11. Assume that Φ(u) = un for u ∈ [0, b(Φ)] and Φ(u) = ∞ for u ∈ (b(Φ),∞),where n ∈ N and b(Φ) ∈ ( n

√1/2, 1). Let us take an arbitrary x such that IΦ(x) = 1−Φ(b(Φ)) =

1 − (b(Φ))n. We will consider two cases separately.First assume that μ(supp x) = 1, that is, |x| = Φ−1(1 − Φ(b(Φ)))ei = n

√(1 − (b(Φ))n)ei

for some i ∈ N. Then

IΦ

(x

Φ−1(1 −Φ(b(Φ)))/b(Φ)

)=

(n√

1 − (b(Φ))n)n

(n√

1 − (b(Φ))n)n/(b(Φ))n

= (b(Φ))n < 1. (3.44)

Simultaneously, for λ < Φ−1(1 − Φ(b(Φ)))/b(Φ), we have that |x(i)|/λ > b(Φ), whenceIΦ(x/λ) =∞ and consequently ‖x‖Φ = Φ−1(1 −Φ(b(Φ)))/b(Φ).

Assume now that μ(supp x) ≥ 2. Then there exists δx > 0 such that |x(i)| ≤ Φ−1(1 −Φ(b(Φ))) − δx = n

√1 − (b(Φ))n − δx for any i ∈ supp x. Then

IΦ

(x

Φ−1(1 −Φ(b(Φ)))/b(Φ)

)=

∑i∈supp x |x(i)|n(

n√

1 − (b(Φ))n)n

(b(Φ))n=

(1 − (b(Φ))n

) · (b(Φ))n

1 − (b(Φ))n< 1.

(3.45)

Since Φ ∈ Δ2(0), there exists λ < Φ−1(1 − Φ(b(Φ)))/b(Φ) such that IΦ(x/λ) ≤ 1, so ‖x‖Φ <Φ−1(1 −Φ(b(Φ)))/b(Φ).

In the next example we will find an Orlicz function Φ, for which

sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))} = Φ−1((1 −Φ(b(Φ)))/2)Φ−1(1/2)

. (3.46)

Example 3.12. Let

Φ(u) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

u for u ∈[

0,5

50

),

5u − 25

for u ∈[

550,

1250

],

∞ for u >1250.

(3.47)


Then b(Φ) = 12/50, Φ(12/50) = 8/10, Φ−1(1 − Φ(12/50)) = Φ−1(2/10) = 6/50, Φ−1((1 −Φ(12/50))/2) = 1/10, and Φ−1(1/2) = 9/50, so

Φ−1((1 −Φ(12/50))/2)Φ−1(1/2)

=59. (3.48)

For x such that μ(supp(x)) = 2 and |x(i)| = 1/10 for i ∈ supp(x), we have that IΦ(x) =1 −Φ(12/50) = 2/10 and IΦ(x/(5/9)) = 2 ·Φ((9/5) · (1/10)) = 1, whence ‖x‖Φ = 1.

Notice also that if |x| = (6/50)ei for some i ∈ N, then IΦ(x) = 2/10 and ‖x‖Φ = 1/2 <5/9. Finally, let us take arbitrary x such that IΦ(x) = 2/10, μ(supp(x)) ≥ 2, and |x(i)|/= 1/10for some i ∈ supp(x). It is easy to see that we can find j ∈ supp(x) for which |x(j)| < 1/10.Moreover, Φ(u) ≥ u for any u ≥ 0. Hence, denoting by ϕ the right-hand-side derivative of theOrlicz function Φ, we have

IΦ

(x

5/9

)=

∑i∈supp(x)

Φ(

95x(i))

=∑

i∈supp(x)

(Φ(

95x(i))−Φ(x(i))

)+∑

i∈supp(x)

Φ(x(i))

=∑

i∈supp(x)

∫ (9/5)x(i)

x(i)ϕ(t)dt + 0, 2 <

∑i∈supp(x)

∫ (9/5)x(i)

x(i)5dt + 0, 2

= 5(

95− 1)∑

xi + 0, 2 ≤ 1.

(3.49)

This inequality and Φ ∈ Δ2(0) imply that ‖x‖Φ < 5/9.

In the last example we will show that for some Orlicz functions Φ,

sup{‖x‖Φ : IΦ(x) = 1 −Φ(b(Φ))} > max

{Φ−1(1 −Φ(b(Φ)))

b(Φ), supn≥2

Φ−1((1 −Φ(b(Φ)))/n)Φ−1(1/n)

}.

(3.50)

Example 3.13. Let

Φ(u) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

u for u ∈[

0,11

100

],

5u − 44100

for u ∈(

11100

,20

100

],

6u − 64100

for u ∈(

20100

,24

100

],

∞ for u >24

100.

(3.51)


Then b(Φ) = 24/100, Φ(b(Φ)) = 8/10, and Φ−1(1 −Φ(24/100)) = Φ−1(2/10) = 16/125, so

Φ−1(1 −Φ(24/100))24/100

=8

15. (3.52)

Since (1 −Φ(24/100))/n ≤ 1/10 for n ≥ 2, we have

Φ−1(

1 −Φ(24/100)n

)=

1 −Φ(24/100)n

=2

10n(3.53)

for all n ≥ 2. Obviously Φ−1(1/n) = 1/n for n ≥ 10, whence

Φ−1((1 −Φ(24/100))/n)Φ−1(1/n)

=2

10(3.54)

for the same n. By Φ−1(1/n) ∈ (11/100, 2/10) for n = 2, . . . , 9, we get Φ−1(1/n) = (100 +44n)/500n for the same n. Thus for those n (i.e., for n = 2, . . . , 9), we get

Φ−1((1 −Φ(24/100))/n)Φ−1(1/n)

=(2/10n)

(100 + 44n)/500n=

2525 + 11n

≤ 2547. (3.55)

Finally,

max

{Φ−1(1 −Φ(24/100))

24/100, supn≥2

Φ−1((1 −Φ(24/100))/n)Φ−1(1/n)

}=

815. (3.56)

Simultaneously, for x such that |x| = (11/100)ei + (9/100)ej for some i, j ∈ N, we obtain thatIΦ(x) = 2/10 and ‖x‖Φ = 111/208, so sup{‖x‖Φ : IΦ(x) = 1 −Φ(24/100)} > 8/15.

Acknowledgments

The first, second, and the third authors gratefully acknowledge the support of the StateCommittee for Scientific Research, Poland, Grant no. N N201 362236. The fourth authorgratefully acknowledges the support of the Academy of Sciences of the Czech Republic,Institutional Research Plan no. AV0Z10190503, and of the Grant no. IAA100190804 of GAof CAS and by the Necas Center, LC 06052.


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Research ArticleOrdered Non-Archimedean Fuzzy Metric Spacesand Some Fixed Point Results

Ishak Altun1 and Dorel Mihet2

1 Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan,Kirikkale, Turkey

2 Departament of Mathematics, Faculty of Mathematics and Computer Science,West University of Timisoara, Bv. V. Parvan 4, 300223 Timisoara, Romania

Correspondence should be addressed to Ishak Altun, [email protected]

Received 2 July 2009; Accepted 9 February 2010

Academic Editor: Mohamed A. Khamsi

Copyright q 2010 I. Altun and D. Mihet. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

In the present paper we provide two different kinds of fixed point theorems on orderednonArchimedean fuzzy metric spaces. First, two fixed point theorems are proved for fuzzy orderψ-contractive type mappings. Then a common fixed point theorem is given for noncontractive typemappings. Kirk’s problem on an extension of Caristi’s theorem is also discussed.

1. Introduction and Preliminaries

After the definition of the concept of fuzzy metric space by some authors [1–3], the fixedpoint theory on these spaces has been developing (see, e.g., [4–9]). Generally, this theoryon fuzzy metric space is done for contractive or contractive-type mappings (see [2, 10–13]and references therein). In this paper we introduce the concept of fuzzy order ψ-contractivemappings and give two fixed point theorems on ordered non-Archimedean fuzzy metricspaces for fuzzy order ψ-contractive type mappings. Then, using an idea in [14], we willprovide a common fixed point theorem for weakly increasing single-valued mappings ina complete fuzzy metric space endowed with a partial order induced by an appropriatefunction. Some fixed point results on ordered probabilistic metric spaces can be found in [15].

For the sake of completeness, we briefly recall some notions from the theory of fuzzymetric spaces used in this paper.

Definition 1.1 (see [16]). A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is called a continuoust-norm if ([0, 1], ∗) is an Abelian topological monoid with the unit 1 such that a ∗ b ≤ c ∗ dwhenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1].


A continuous t-norm ∗ is of Hadzic-type if there exists a strictly increasing sequence{bn} ⊂ (0, 1) such that bn ∗ bn = bn for all n ∈ N.

Definition 1.2 (see [3]). A fuzzy metric space (in the sense of Kramosil and Michalek) is atriple (X,M, ∗), where X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set onX2 × [0,∞), satisfying the following properties:

(KM-1) M(x, y, 0) = 0, for all x, y ∈ X,(KM-2) M(x, y, t) = 1, for all t > 0 if and only if x = y,

(KM-3) M(x, y, t) =M(y, x, t), for all x, y ∈ X and t > 0,

(KM-4) M(x, y, ·) : [0,∞) → [0, 1] is left continuous, for all x, y ∈ X,(KM-5) M(x, z, t + s) ≥M(x, y, t) ∗M(y, z, s), for all x, y, z ∈ X, for all t, s > 0.

If, in the above definition, the triangular inequality (KM-5) is replaced by

M(x, z,max{t, s}) ≥M(x, y, t) ∗M(y, z, s), ∀x, y, z ∈ X, ∀t, s > 0, (NA)

then the triple (X,M, ∗) is called a non-Archimedean fuzzy metric space. It is easy to check thatthe triangular inequality (NA) implies (KM-5), that is, every non-Archimedean fuzzy metricspace is itself a fuzzy metric space.

Example 1.3. Let (X, d) be an ordinary metric space and let θ be a nondecreasing andcontinuous function from (0,∞) into (0, 1) such that limt→∞θ(t) = 1. Some examples of thesefunctions are θ(t) = t/(t + 1), θ(t) = 1 − e−t and θ(t) = e−1/t. Let a ∗ b ≤ ab for all a, b ∈ [0, 1].For each t ∈ (0,∞), define

M(x, y, t

)= [θ(t)]d(x,y) (1.1)

for all x, y ∈ X. It is easy to see that (X,M, ∗) is a non-Archimedean fuzzy metric space.

Definition 1.4 (see [1, 16]). Let (X,M, ∗) be a fuzzy metric space. A sequence {xn} in X iscalled an M-Cauchy sequence, if for each ε ∈ (0, 1) and t > 0 there exists n0 ∈ N such thatM(xn, xm, t) > 1 − ε for all m,n ≥ n0. A sequence {xn} in a fuzzy metric space (X,M, ∗) issaid to be convergent to x ∈ X if limn→∞M(xn, x, t) = 1 for all t > 0. A fuzzy metric space(X,M, ∗) is called M-complete if every M-Cauchy sequence is convergent.

Definition 1.5 (see [7]). Let (X,M, ∗) be a fuzzy metric space. A sequence {xn} in X is calledG-Cauchy if

limn→∞

M(xn, xn+1, t) = 1 (1.2)

for all t > 0. The space (X,M, ∗) is called G-complete if every G-Cauchy sequence isconvergent.

Lemma 1.6 (see [11]). Each M -complete non-Archimedean fuzzy metric space (X,M, T) with T ofHadzic-type is G-complete.


Theorem 2.10 in the next section is related to a partial order on a fuzzy metric spaceunder the Łukasiewicz t-norm. We will refer to [14].

Lemma 1.7 (see [14]). Let (X,M, ∗) be a non-Archimedean fuzzy metric space with a∗b ≥ max{a+b − 1, 0} and φ : X × [0,∞) → R. Define the relation “” on X as follows:

x y ⇐⇒M(x, y, t

) ≥ 1 + φ(x, t) − φ(y, t), ∀t > 0. (1.3)

Then is a (partial) order on X, named the partial order induced by φ.

2. Main Results

The first two theorems in this section are related to Theorem 2.1 in [17]. We begin by givingthe following definitions.

Definition 2.1. Let be an order relation on X. A mapping f : X → X is called nondecreasingw.r.t if x y implies fx fy.

Definition 2.2. Let (X,) be a partially ordered set, let (X,M, ∗) be a fuzzy metric space, andlet ψ be a function from [0, 1] to [0, 1]. A mapping f : X → X is called a fuzzy order ψ-contractive mapping if the following implication holds:

x, y ∈ X, x y =⇒ [M(fx, fy, t) ≥ ψ(M(x, y, t)) ∀t > 0]. (2.1)

Theorem 2.3. Let (X,) be a partially ordered set and (X,M, ∗) be anM-complete non-Archimedeanfuzzy metric space with ∗ of Hadzic-type. Let ψ : [0, 1] → [0, 1] be a continuous, nondecreasingfunction and let f : X → X be a fuzzy order ψ-contractive and nondecreasing mapping w.r.t .Suppose that either

f is continuous, (2.2)

or

xn x ∀n, whenever{xn} ⊂ X is nondecreasing sequence with xn −→ x ∈ X

(2.3)

hold. If there exists x0 ∈ X such that

x0 fx0, limn→∞

ψn(M(x0, fx0, t

))= 1 (2.4)

for each t > 0, then f has a fixed point.

Proof. Let xn = fxn−1 for n ∈ {1, 2, . . .}. Since x0 fx0 and f is nondecreasing w.r.t , we have

x0 x1 x2 · · · xn xn+1 · · · . (2.5)


Then, it immediately follows by induction that

M(xn+1, xn+2, t) ≥ ψ(M(xn, xn+1, t)), (n ∈ N, t > 0), (2.6)

hence

M(xn, xn+1, t) ≥ ψn(M(x0, fx0, t

)), (n ∈ N, t > 0). (2.7)

By taking the limit as n → ∞ we obtain

limn→∞

M(xn, xn+1, t) = 1 (2.8)

for all t > 0, that is, {xn} is G-Cauchy. Since X is G-complete (Lemma 1.6), then there existsx ∈ X such that limn→∞xn = x.

Now, if f is continuous then it is clear that fx = x, while if the condition (2.3) holdthen, for all t > 0,

M(xn+1, fx, t

)=M

(fxn, fx, t

) ≥ ψ(M(xn, x, t)) (2.9)

and letting n → ∞ it follows

M(x, fx, t

) ≥ ψ(1) = 1, (2.10)

hence fx = x.

Theorem 2.4. Let (X,) be a partially ordered set, let (X,M, ∗) be anM-complete non-Archimedeanfuzzy metric space, and let ψ : [0, 1] → [0, 1] be a continuous mapping such that ψ(t) > t for allt ∈ (0, 1). Also, let f : X → X be a nondecreasing mapping w.r.t , with the property

M(fx, fy, t

) ≥ ψ(M(x, y, t)) ∀t > 0, whenever x y. (2.11)

Suppose that either (2.2) or (2.3) holds. If there exists x0 ∈ X such that

x0 fx0, M(x0, fx0, t

)> 0 (2.12)

for all t > 0, then f has a fixed point.

Proof. Let xn = fxn−1 for n ∈ {1, 2, . . .}. Then, as in the proof of the preceding theorem we canprove that

M(xn+1, xn+2, t) ≥ ψ(M(xn, xn+1, t)) ≥M(xn, xn+1, t), (n ∈ N, t > 0). (2.13)


Therefore, for every t > 0, {M(xn, xn+1, t)}n∈N is a nondecreasing sequence of numbersin (0, 1]. Let, for fixed t > 0, limn→∞M(xn, xn+1, t) = l. Then we have l ∈ (0, 1], sinceM(x0, x1, t) > 0. Also, since

M(xn+1, xn+2, t) ≥ ψ(M(xn, xn+1, t)) (2.14)

and ψ is continuous, we have l ≥ ψ(l). This implies l = 1 and therefore, for all t > 0,

limn→∞

M(xn, xn+1, t) = 1. (2.15)

Now we show that {xn} is an M-Cauchy sequence. Supposing this is not true, then there areε ∈ (0, 1) and t > 0 such that for each k ∈ N there exist m(k), n(k) ∈ N with m(k) > n(k) ≥ kand

M(xm(k), xn(k), t

) ≤ 1 − ε. (2.16)

Let, for each k, m(k) be the least integer exceeding n(k) satisfying the inequality (2.16), thatis,

M(xm(k)−1, xn(k), t

)> 1 − ε. (2.17)

Then, for each k,

1 − ε ≥M(xm(k), xn(k), t)

≥M(xm(k)−1, xn(k), t) ∗M(xm(k)−1, xm(k), t

)≥ (1 − ε) ∗M(xm(k)−1, xm(k), t

).

(2.18)

Letting k → ∞ and using (2.15), we have, for t > 0,

limk→∞

M(xm(k), xn(k), t

)= 1 − ε. (2.19)

Then, since xn(k) xm(k), we have

M(xm(k), xn(k), t

) ≥M(xm(k), xm(k)+1, t) ∗M(xm(k)+1, xn(k)+1, t

) ∗M(xn(k)+1, xn(k), t)

≥M(xm(k), xm(k)+1, t) ∗ ψ(M(xm(k), xn(k), t

)) ∗M(xn(k)+1, xn(k), t).

(2.20)

Letting k → ∞ and using (2.15) and (2.19), we obtain

1 − ε ≥ 1 ∗ ψ(1 − ε) ∗ 1 = ψ(1 − ε) > 1 − ε, (2.21)


which is a contradiction. Thus {xn} is an M-Cauchy sequence. Since X is M-complete, thenthere exists x ∈ X such that

limn→∞

xn = x. (2.22)

If f is continuous, then from xn = fxn−1 (n ∈ N) it follows that fx = x. Also, if (2.3) holds,then (since xn x) we have

M(xn+1, fx, t

)=M

(fxn, fx, t

) ≥ ψ(M(xn, x, t)), (n ∈ N, t > 0). (2.23)

Letting n → ∞, we obtain that

M(x, fx, t

)= 1 ∀t > 0, (2.24)

hence fx = x.

Example 2.5. Let X = (0,∞). Consider the following relation on X:

x y ⇐⇒ (x = y or x, y ∈ [1, 4], x ≤ y). (2.25)

It is easy to see that is a partial order on X. Let a ∗ b = ab and

M(x, y, t

)=

min{x, y}

max{x, y} , ∀t > 0. (2.26)

Then (X,M, ∗) is an M-complete non-Archimedean fuzzy metric space (see [18]) satisfyingM(x, y, t) > 0 for all t > 0. Define a self map f of X as follows:

fx =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

2x, 0 < x < 1

x + 53

, 1 ≤ x ≤ 4

2x − 5, x > 4.

(2.27)

Now, it is easy to see that f is continuous and nondecreasing w.r.t . Also, for x0 = 1 we have1 = x0 fx0 = 2. Now we can see that f is fuzzy order ψ-contractive with ψ(t) =

√t.

Indeed, let x, y ∈ X with x y. Now if x = y, then

M(fx, fy, t

)= 1 ≥ ψ(1) = ψ(M(x, y, t)). (2.28)


If x, y ∈ [1, 4] with x ≤ y, then

M(fx, fy, t

)=

min{fx, fy

}max

{fx, fy

}

=min{(x + 5)/3,

(y + 5

)/3}

max{(x + 5)/3,

(y + 5

)/3}

=x + 5y + 5

≥√x

y

= ψ(M(x, y, t

)).

(2.29)

Therefore f is fuzzy order ψ-contractive with ψ(t) =√t. Hence all conditions of Theorem 2.4

are satisfied and so f has a fixed point on X.

In order to state our next theorem, we give the concept of weakly comparablemappings on an ordered space.

Definition 2.6. Let (X,) be an ordered space. Two mappings f, g : X → X are said to beweakly comparable if fx gfx and gx fgx for all x ∈ X.

Note that two weakly comparable mappings need not to be nondecreasing.

Example 2.7. Let X = [0,∞) and ≤ be usual ordering. Let f, g : X → X defined by

fx =

⎧⎨⎩x if 0 ≤ x ≤ 1,

0 if 1 < x <∞,gx =

⎧⎨⎩√x if 0 ≤ x ≤ 1,

0 if 1 < x <∞.(2.30)

Then it is obvious that fx ≤ gfx and gx ≤ fgx for all x ∈ X. Thus f and g are weaklycomparable mappings. Note that both f and g are not nondecreasing.

Example 2.8. Let X = [1,∞) × [1,∞) and be coordinate-wise ordering, that is, (x, y) (z,w) ⇔ x ≤ z and y ≤ w. Let f, g : X → X be defined by f(x, y) = (2x, 3y)and g(x, y) = (x2, y2), then f(x, y) = (2x, 3y) gf(x, y) = g(2x, 3y) = (4x2, 9y2) andg(x, y) = (x2, y2) fg(x, y) = f(x2, y2) = (2x2, 3y2). Thus f and g are weakly comparablemappings.

Example 2.9. Let X = R2 and be lexicographical ordering, that is, (x, y) (z,w) ⇔ (x < z)

or (if x = z, then y ≤ w). Let f, g : X → X be defined by

f(x, y)=(max

{x, y},min

{x, y}),

g(x, y)=(

max{x, y},x + y

2

),

(2.31)


then f(x, y) gf(x, y) and g(x, y) fg(x, y) for all (x, y) ∈ X. Thus f and g are weaklycomparable mappings. Note that, (1, 4) (2, 3) but f(1, 4) = (4, 1)(3, 2) = f(2, 3), then f is notnondecreasing. Similarly g is not nondecreasing.

Theorem 2.10. Let (X,M, ∗) be an M -complete non-Archimedean fuzzy metric space with a ∗ b ≥max{a + b − 1, 0}, φ : X × [0,∞) → R be a bounded-from-above function, and let be the partialorder induced by φ. If f, g : X → X are two continuous and weakly comparable mappings, then fand g have a common fixed point in X.

Proof. Let X0 be an arbitrary point of X and let us define a sequence {xn} in X as follows:

x2n+1 = fx2n, x2n+2n = gx2n+1 for n ∈ {0, 1, . . .}. (2.32)

Note that, since f and g are weakly comparable, we have

x1 = fx0 gfx0 = gx1 = x2,

x2 = gx1 fgx1 = fx2 = x3.(2.33)

By continuing this process we get

x1 x2 · · · xn xn+1 · · · , (2.34)

that is, the sequence {xn} is nondecreasing. By the definition of we have φ(x0, t) ≤ φ(x1, t) ≤φ(x2, t) ≤ · · · for all t > 0, that is, for even t > 0, the sequence {φ(xn, t)} is a nondecreasingsequence in R. Since φ is bounded from above, {φ(xn, t)} is convergent and hence it is Cauchy.Then, for all ε > 0 there exists n0 ∈ N such that for all m > n > n0 and t > 0 we have|φ(xm, t) − φ(xn, t)| = φ(xm, t) − φ(xn, t) < ε. Therefore, since xn xm, we have

M(xn, xm, t) ≥ 1 + φ(xn, t) − φ(xm, t)= 1 − [φ(xm, t) − φ(xn, t)]> 1 − ε.

(2.35)

This shows that the sequence {xn} is M-Cauchy. SinceX is M-complete, it converges to a pointz ∈ X. As x2n+1 → z and x2n+2 → z, by the continuity of f and g we get fz = gz = z.

Corollary 2.11 ([Caristi fixed point theorem in non-Archimedean fuzzy metric spaces]). Let(X,M, ∗) be an M -complete non-Archimedean fuzzy metric space with a ∗ b ≥ max{a+ b − 1, 0}, letφ : X × [0,∞) → R be a bounded-from-above function and f : X → X be a continuous mapping,such that

M(x, fx, t

) ≥ 1 + φ(x, t) − φ(fx, t) (2.36)

for all x ∈ X and t > 0. Then f has a fixed point in X.


Proof. We take in the above theorem g = 1X and note that the weak comparability of f and greduces to (2.36).

The generalization suggested by Kirk of Caristi’s fixed point theorem [19] is wellknown. A similar theorem in the setting of non-Archimedean fuzzy metric spaces is stated inthe final part of our paper.

In what follows ν : [0, 1] → [0, 1] is nondecreasing, subadditive mapping (i.e., ν(a +b) ≤ ν(a) + ν(b) for all a, b ∈ [0, 1]), with ν(0) = 0.

Theorem 2.12. Let (X,M, ∗) be a non-Archimedean fuzzy metric space with a∗b ≥ max{a+b−1, 0}and φ : X × [0,∞) → R. Define the relation “” on X through

x y ⇐⇒ φ(y, t) − φ(x, t) ≥ ν(1 −M(x, y, t)), ∀t > 0. (2.37)

Then “” is a (partial) order on X.

Proof. Since ν(0) = 0, then for all x ∈ X and t > 0,

0 = φ(x, t) − φ(x, t) ≥ ν(1 −M(x, x, t)) = 0, (2.38)

that is, “” is reflexive.Let x, y ∈ X be such that x y and y x. Then for all t > 0,

φ(y, t) − φ(x, t) ≥ ν(1 −M(x, y, t)),

φ(x, t) − φ(y, t) ≥ ν(1 −M(x, y, t)), (2.39)

implying that M(x, y, t) = 1 for all t > 0, that is, x = y. Thus “” is antisymmetric.Now for x, y, z ∈ X, let x y and y z. Then, for given t > 0,

φ(y, t) − φ(x, t) ≥ ν(1 −M(x, y, t)), (2.40)

φ(z, t) − φ(y, t) ≥ ν(1 −M(z, y, t)). (2.41)

By using (2.40) and (2.41) we get

φ(z, t) − φ(x, t) ≥ ν(1 −M(x, y, t)) + ν(1 −M(y, z, t))≥ ν(1 −M(x, y, t) + 1 −M(y, z, t)). (2.42)

On the other hand, from the triangular inequality (NA), the inequality

M(x, z, t) ≥M(x, y, t) +M(y, z, t) − 1 (2.43)

holds. This implies

1 −M(x, y, t) + 1 −M(y, z, t) ≥ 1 −M(x, z, t). (2.44)


As ν is nondecreasing, it follows that

ν(1 −M(x, y, t) + 1 −M(y, z, t)) ≥ ν(1 −M(x, z, t)) (2.45)

and therefore

φ(z, t) − φ(x, t) ≥ ν(1 −M(x, z, t)). (2.46)

This shows that x z, that is, “” is transitive.

From the above theorem we can immediately obtain the following generalization ofCorollary 2.11.

Corollary 2.13. Let (X,M, ∗) be an M -complete non-Archimedean fuzzy metric space with a ∗ b ≥max{a + b − 1, 0}, let φ : X × [0,∞) → R be a bounded-from-above function and f : X → X be acontinuous mapping, such that

φ(fx, t

) − φ(x, t) ≥ ν(1 −M(x, fx, t)) (2.47)

for all x ∈ X and t > 0. If ν satisfies the property

∀ε > 0 ∃δ > 0 : ν(x) < δ =⇒ x < ε, (2.48)

then f has a fixed point in X.

The reader is referred to the nice paper [20] for some discussion of Kirk’s problem onan extension of Caristi’s fixed point theorem.

References

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[2] O. Kaleva and S. Seikkala, “On fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 12, no. 3, pp. 215–229,1984.

[3] I. Kramosil and J. Michalek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 11, no. 5,pp. 336–344, 1975.

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[5] Y. J. Cho, “Fixed points in fuzzy metric spaces,” Journal of Fuzzy Mathematics, vol. 5, no. 4, pp. 949–962,1997.

[6] J. X. Fang, “On fixed point theorems in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 46, no. 1, pp.107–113, 1992.

[7] M. Grabiec, “Fixed points in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 27, no. 3, pp. 385–389,1988.

[8] V. Gregori and A. Sapena, “On fixed-point theorems in fuzzy metric spaces,” Fuzzy Sets and Systems,vol. 125, no. 2, pp. 245–252, 2002.

[9] O. Hadzic and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 of Mathematics and ItsApplications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.


[10] D. Mihet, “On fuzzy contractive mappings in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 158,no. 8, pp. 915–921, 2007.

[11] D. Mihet, “Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces,” Fuzzy Sets andSystems, vol. 159, no. 6, pp. 739–744, 2008.

[12] D. Mihet, “Fuzzy quasi-metric versions of a theorem of Gregori and Sapena,” Iranian Journal of FuzzySystems, vol. 7, no. 1, pp. 59–64, 2010.

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[17] R. P. Agarwal, M. A. El-Gebeily, and D. O’Regan, “Generalized contractions in partially orderedmetric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008.

[18] V. Radu, “Some remarks on the probabilistic contractions on fuzzy Menger spaces,” AutomationComputers Applied Mathematics, vol. 11, no. 1, pp. 125–131, 2002.

[19] J. Caristi, “Fixed point theory and inwardness conditions,” in Applied Nonlinear Analysis, pp. 479–483,Academic Press, New York, NY, USA, 1979.

[20] M. A. Khamsi, “Remarks on Caristi’s fixed point theorem,” Nonlinear Analysis: Theory, Methods &Applications, vol. 71, no. 1-2, pp. 227–231, 2009.


Research ArticleA Set of Axioms for the Degree of a Tangent VectorField on Differentiable Manifolds

Massimo Furi, Maria Patrizia Pera, and Marco Spadini

Dipartimento di Matematica Applicata “Giovanni Sansone”, Universita degli Studi di Firenze,Via San Marta 3, 50139 Firenze, Italy

Correspondence should be addressed to Massimo Furi, [email protected]

Received 28 September 2009; Accepted 7 February 2010


Copyright q 2010 Massimo Furi et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Given a tangent vector field on a finite-dimensional real smooth manifold, its degree (also knownas characteristic or rotation) is, in some sense, an algebraic count of its zeros and gives usefulinformation for its associated ordinary differential equation. When, in particular, the ambientmanifold is an open subset U of R

m, a tangent vector field v on U can be identified with a map�v : U → R

m, and its degree, when defined, coincides with the Brouwer degree with respect to zeroof the corresponding map �v. As is well known, the Brouwer degree in R

m is uniquely determinedby three axioms called Normalization, Additivity, and Homotopy Invariance. Here we shall provide asimple proof that in the context of differentiable manifolds the degree of a tangent vector field isuniquely determined by suitably adapted versions of the above three axioms.

1. Introduction

The degree of a tangent vector field on a differentiable manifold is a very well-known toolof nonlinear analysis used, in particular, in the theory of ordinary differential equations anddynamical systems. This notion is more often known by the names of rotation or of (Euler)characteristic of a vector field (see, e.g., [1–6]). Here, we depart from the established traditionby choosing the name “degree” because of the following consideration: in the case that theambient manifold is an open subset U of R

m, there is a natural identification of a vector fieldv on U with a map �v : U → R

m, and the degree deg(v,U) of v on U, when defined, isjust the Brouwer degree degB(�v,U, 0) of �v on U with respect to zero. Thus the degree of avector field can be seen as a generalization to the context of differentiable manifolds of thenotion of Brouwer degree in R

m. As is well-known, this extension of degB does not require theorientability of the underlying manifold, and is therefore different from the classical extensionof degB for maps acting between oriented differentiable manifolds.


A result of Amann and Weiss [7] (see also [8]) asserts that the Brouwer degree in Rm is

uniquely determined by three axioms: Normalization, Additivity, and Homotopy Invariance.A similar statement is true (e.g., as a consequence of a result of Staecker [9]) for the degreeof maps between oriented differentiable manifolds of the same dimension. In this paper,which is closely related in both spirit and demonstrative techniques to [10], we will provethat suitably adapted versions of the above axioms are sufficient to uniquely determine thedegree of a tangent vector field on a (not necessarily orientable) differentiable manifold.We will not deal with the problem of existence of such a degree, for which we refer to[1–5].

2. Preliminaries

Given two sets X and Y , by a local map with source X and target Y we mean a triple g =(X,Y,Γ), where Γ, the graph of g, is a subset of X × Y such that for any x ∈ X there exists atmost one y ∈ Y with (x, y) ∈ Γ. The domain D(g) of g is the set of all x ∈ X for which thereexists y = g(x) ∈ Y such that (x, y) ∈ Γ; that is,D(g) = π1(Γ), where π1 denotes the projectionof X × Y onto the first factor. The restriction of a local map g = (X,Y,Γ) to a subset C of X isthe triple

g∣∣C = (C, Y,Γ ∩ (C × Y )) (2.1)

with domain C ∩ D(g).Incidentally, we point out that sets and local maps (with the obvious composition)

constitute a category. Although the notation g : X → Y would be acceptable in the contextof category theory, it will be reserved for the case when D(g) = X.

Whenever it makes sense (e.g., when source and target spaces are differentiablemanifolds), local maps are tacitly assumed to be continuous.

Throughout the paper all of the differentiable manifolds will be assumed to be finitedimensional, smooth, real, Hausdorff, and second countable. Thus, they can be embedded insome R

k. Moreover, M and N will always denote arbitrary differentiable manifolds. Givenany x ∈M, TxMwill denote the tangent space ofM at x. Furthermore TMwill be the tangentbundle of M; that is,

TM = {(x, v) : x ∈M, v ∈ TxM}. (2.2)

The map π : TM → M given by π(x, v) = x will be the bundle projection of TM. It will alsobe convenient, given any x ∈M, to denote by 0x the zero element of TxM.

Given a smooth map f : M → N, by Tf : TM → TN we will mean the map thatto each (x, v) ∈ TM associates (f(x), dfx(v)) ∈ TN. Here dfx : TxM → Tf(x)N denotes thedifferential of f at x. Notice that if f : M → N is a diffeomorphism, then so is Tf : TM →TN and one has T(f−1) = (Tf)−1.

By a local tangent vector field on M we mean a local map v having M as source andTM as target, with the property that the composition π ◦v is the identity on D(v). Therefore,given a local tangent vector field v on M, for all x ∈ D(v) there exists �v(x) ∈ TxM such thatv(x) = (x, �v(x)).


Let V and W be differentiable manifolds and let ψ : V → W be a diffeomorphism.Recall that two tangent vector fields v : V → TV and w : W → TW correspond under ψ if thefollowing diagram commutes:

TVTψ

TW

V

v

ψW

w

Let V be an open subset of M and suppose that v is a local tangent vector field on Mwith V ⊆ D(v). We say that v is identity-like on V if there exists a diffeomorphism ψ of V ontoRm such that v|V and the identity in R

m correspond under ψ. Notice that any diffeomorphismψ from an open subset V of M onto R

m induces an identity-like vector field on V .Let v be a local tangent vector field onM and let p ∈M be a zero of v; that is, �v(p) = 0p.

Consider a diffeomorphism ϕ of a neighborhood U ⊆M of p onto Rm and let w : R

m → TRm

be the tangent vector field on Rm that corresponds to v under ϕ. Since TR

m = Rm × R

m, thenthe map �w associated to w sends R

m into itself. Assuming that v is smooth in a neighborhoodof p, the function �w is Frechet differentiable at q = ϕ(p). Denote by D �w(q) : R

m → Rm its

Frechet derivative and let v′(p) : TpM → TpM be the endomorphism of TpM which makesthe following diagram commutative:

TpMv′(p)

dϕp

TpM

dϕp

Rm

D �w(q)Rm

(2.3)

Using the fact that p is a zero of v, it is not difficult to prove that v′(p) does not depend onthe choice of ϕ. This endomorphism of TpM is called the linearization of v at p. Observe that,when M = R

m, the linearization v′(p) of a tangent vector field v at a zero p is just the Frechetderivative D�v(p) at p of the map �v associated to v.

The following fact will play an important role in the proof of our main result.

Remark 2.1. Let v, w, p, and q be as above. Then, the commutativity of diagram (2.3) implies

detv′(p)= detD �w

(q). (2.4)

3. Degree of a Tangent Vector Field

Given an open subset U of M and a local tangent vector field v on M, the pair (v,U) is saidto be admissible onU if U ⊆ D(v) and the set

Z(v,U) := {x ∈ U : �v(x) = 0x} (3.1)

of the zeros of v in U is compact. In particular, (v,U) is admissible if the closure U of U is acompact subset of D(v) and �v is nonzero on the boundary ∂U of U.


Given an open subset U of M and a (continuous) local map H with source M × [0, 1]and target TM, we say that H is a homotopy of tangent vector fields on U if U × [0, 1] ⊆ D(H),and if H(·, λ) is a local tangent vector field for all λ ∈ [0, 1]. If, in addition, the set

{(x, λ) ∈ U × [0, 1] : �H(x, λ) = 0x

}(3.2)

is compact, the homotopy H is said to be admissible. Thus, if U is compact and U × [0, 1] ⊆D(H), a sufficient condition for H to be admissible on U is the following:

�H(x, λ)/= 0x, ∀(x, λ) ∈ ∂U × [0, 1], (3.3)

which, by abuse of terminology, will be referred to as “H is nonzero on ∂U”.We will show that there exists at most one function that, to any admissible pair (v,U),

assigns a real number deg(v,U) called the degree (or characteristic or rotation) of the tangentvector field v on U, which satisfies the following three properties that will be regarded asaxioms. Moreover, this function (if it exists) must be integer valued.

Normalization

Let v be identity-like on an open subset U of M. Then,

deg(v,U) = 1. (3.4)

Additivity

Given an admissible pair (v,U), if U1 and U2 are two disjoint open subsets of U such thatZ(v,U) ⊆ U1 ∪U2, then

deg(v,U) = deg(v|U1

, U1)+ deg

(v|U2

, U2). (3.5)

Homotopy Invariance

If H is an admissible homotopy on U, then

deg(H(·, 0), U) = deg(H(·, 1), U). (3.6)

From now on we will assume the existence of a function deg defined on the family ofall admissible pairs and satisfying the above three properties that we will regard as axioms.

Remark 3.1. The pair (v, ∅) is admissible. This includes the case when D(v) is the empty set(D(v) = ∅ is coherent with the notion of local tangent vector field). A simple application ofthe Additivity Property shows that deg(v|∅, ∅) = 0 and deg(v, ∅) = 0.

As a consequence of the Additivity Property and Remark 3.1, one easily gets thefollowing (often neglected) property, which shows that the degree of an admissible pair(v,U) does not depend on the behavior of v outside U. To prove it, take U1 = U and U2 = ∅in the Additivity Property.


Localization

If (v,U) is admissible, then deg(v,U) = deg(v|U,U).

A further important property of the degree of a tangent vector field is the following.

Excision

Given an admissible pair (v,U) and an open subset U1 of U containing Z(v,U), one hasdeg(v,U) = deg(v,U1).

To prove this property, observe that by Additivity, Remark 3.1, and Localization onegets

deg(v,U) = deg(v|U1

, U1)+ deg

(v|∅, ∅

)= deg(v,U1). (3.7)

As a consequence, we have the following property.

Solution

If deg(v,U)/= 0, then Z(v,U)/= ∅.To obtain it, observe that if Z(v,U) = ∅, taking U1 = ∅, we get

deg(v,U) = deg(v, ∅) = 0. (3.8)

4. The Degree for Linear Vector Fields

By L(Rm) we will mean the normed space of linear endomorphisms of Rm, and by GL(Rm)

we will denote the group of invertible ones. In this section we will consider linear vector fieldson R

m, namely, vector fields L : Rm → TR

m with the property that �L ∈ L(Rm). Notice that(L,Rm), with L a linear vector field, is an admissible pair if and only if �L ∈ GL(Rm).

The following consequence of the axioms asserts that the degree of an admissible pair(L,Rm), with �L ∈ GL(Rm), is either 1 or −1.

Lemma 4.1. Let �L be a nonsingular linear operator in Rm. Then

deg(L,Rm) = sign det �L. (4.1)

Proof. It is well-known (see, e.g., [11]) that GL(Rm) has exactly two connected components.Equivalently, the following two subsets of L(Rm) are connected:

GL+(Rm) = {A ∈ L(Rm) : detA > 0},GL−(Rm) = {A ∈ L(Rm) : detA < 0}.

(4.2)

Since the connected sets GL+(Rm) and GL−(Rm) are open in L(Rm), they are actuallypath connected. Consequently, given a linear tangent vector field L on R

m with �L ∈ GL(Rm),


Homotopy Invariance implies that deg(L,Rm) depends only on the component of GL(Rm)containing �L. Therefore, if �L ∈ GL+(Rm), one has deg(L,Rm) = deg(I,Rm), where �I is theidentity on R

m. Thus, by Normalization, we get

deg(L,Rm) = 1. (4.3)

It remains to prove that deg(L,Rm) = −1 when �L ∈ GL−(Rm). For this purpose considerthe vector field f : R

m → TRm determined by

�f(ξ1, . . . , ξm−1, ξm) = (ξ1, . . . , ξm−1, |ξm| − 1). (4.4)

Notice that deg(f,Rm) is well defined because �f−1(0) is compact. Observe also that deg(f,Rm)is zero, because f is admissibly homotopic in R

m to the never-vanishing vector field g : Rm →

TRm given by �g(ξ1, . . . , ξm) = (ξ1, . . . , |ξm| + 1).

Let U− and U+ denote, respectively, the open half-spaces of the points in Rm with

negative and positive last coordinate. Consider the two solutions

x− = (0, . . . , 0,−1), x+ = (0, . . . , 0, 1) (4.5)

of the equation �f(x) = 0 and observe that x− ∈ U−, x+ ∈ U+.By Additivity (and taking into account the Localization property), we get

0 = deg(f,Rm) = deg

(f,U−

)+ deg

(f,U+

). (4.6)

Now, observe that f in U+ coincides with the vector field f+ : Rm → TR

m determined by

�f+(ξ1, . . . , ξm−1, ξm) = (ξ1, . . . , ξm−1, ξm − 1), (4.7)

which is admissibly homotopic (in Rm) to the tangent vector field I : R

m → TRm, given

by I(x) = (x, x). Therefore, because of the properties of Localization, Excision, HomotopyInvariance, and Normalization, one has

deg(f,U+

)= deg

(f+, U+

)= deg

(f+,R

m) = deg(I,Rm) = 1, (4.8)

which, by (4.6), implies that

deg(f,U−

)= −1. (4.9)

Notice that f in U− coincides with the vector field f− : Rm → TR

m defined by

�f−(ξ1, . . . , ξm−1, ξm) = (ξ1, . . . , ξm−1,−ξm − 1), (4.10)


which is admissibly homotopic (in Rm) to the linear vector field L− defined by �L− ∈ GL−(Rm)

with

�L−(ξ1, . . . , ξm−1, ξm) = (ξ1, . . . , ξm−1,−ξm). (4.11)

Thus, by Homotopy Invariance, Excision, Localization, and formula (4.9)

deg(L−,Rm) = deg(f−,Rm) = deg

(f−, U−

)= deg

(f,U−

)= −1. (4.12)

Hence, GL−(Rm) being path connected, we finally get deg(L,Rm) = −1 for all linear tangentvector fields L on R

m such that �L ∈ GL−(Rm), and the proof is complete.

We conclude this section with a consequence as well as an extension of Lemma 4.1.The Euclidean norm of an element x ∈ R

m will be denoted by |x|.

Lemma 4.2. Let v be a local vector field on Rm and letU ⊆ D(v) be open and such that the equation

�v(x) = 0 has a unique solution x0 ∈ U. If �v is smooth in a neighborhood of x0 and the linearizationv′(x0) of v at x0 is invertible, then deg(v,U) = sign detv′(x0).

Proof. Since �v is Frechet differentiable at x0 and D�v(x0) = v′(x0), we have

�v(x0 + h) = v′(x0)h + |h|ε(h), ∀h ∈ −x0 +U, (4.13)

where ε(h) is a continuous function such that ε(0) = 0. Consider the vector field g : Rm →

TRm determined by �g(x) = v′(x0)(x − x0), and let H be the homotopy on U, joining g with v,

defined by

�H(x, λ) = v′(x0)(x − x0) + λ|x − x0|ε(x − x0). (4.14)

For all x in U we have

∣∣∣ �H(x, λ)∣∣∣ ≥ (m − |ε(x − x0)|)|x − x0|, (4.15)

where m = inf{|v′(x0)y| : |y| = 1} is positive because v′(x0) is invertible. This shows thatthere exists a neighborhood V of x0 such that (V × [0, 1])∩ �H−1(0) coincides with the compactset {x0} × [0, 1]. Thus, by Excision and Homotopy Invariance,

deg(v,U) = deg(v, V ) = deg(g, V

). (4.16)

Let L : Rm → TR

m be the linear tangent vector field given by ξ �→ (ξ, v′(x0)ξ). Clearly, L isadmissibly homotopic to g in R

m. By Excision, Homotopy Invariance, and Lemma 4.1, we get

deg(g, V

)= deg

(g,Rm) = deg(L,Rm) = sign det �L. (4.17)

The assertion now follows from (4.16), (4.17), and the fact that �L coincides with v′(x0).


5. The Uniqueness Result

Given a local tangent vector field v on M, a zero p of v is called nondegenerate if v is smooth ina neighborhood of p and its linearization v′(p) at p is an automorphism of TpM. It is knownthat this is equivalent to the assumption that v is transversal at p to the zero section M0 ={(x, 0x) ∈ TM : x ∈M} of TM (for the theory of transversality see, e.g., [3, 4]). We recall thata nondegenerate zero is, in particular, an isolated zero.

Let v be a local tangent vector field on M. A pair (v,U) will be called nondegenerateif U is a relatively compact open subset of M, v is smooth on a neighborhood of the closureU of U, being nonzero on ∂U, and all its zeros in U are nondegenerate. Note that, in thiscase, (v,U) is an admissible pair and Z(v,U) is a discrete set and therefore finite because itis closed in the compact set U.

The following result, which is an easy consequence of transversality theory, showsthat the computation of the degree of any admissible pair can be reduced to that of anondegenerate pair.

Lemma 5.1. Let v be a local tangent vector field on M and let (v,U) be admissible. Let V be arelatively compact open subset of M containing Z(v,U) and such that V ⊆ U. Then, there exists alocal tangent vector field w onM which is admissibly homotopic to v in V and such that (w,V ) is anondegenerate pair. Consequently, deg(v,U) = deg(w,V ).

Proof. Without loss of generality we can assume that M ⊆ Rk. Let

δ = minx∈∂V|�v(x)| > 0. (5.1)

From the Transversality theorem (see, e.g., [3, 4]) it follows that one can find a smooth tangentvector field w : U → TU ⊆ TM that is transversal to the zero section M0 of TM and suchthat

maxx∈∂V|�v(x) − �w(x)| < δ. (5.2)

Since M0 is closed in TM, the set Z(w,V ) = w−1(M0) ∩ V is a compact subset of V .Thus, this inequality shows that (w,V ) is admissible. Moreover, at any zero x ∈ Z(w,U) =w−1(M0) ∩U the endomorphism w′(x) : TxM → TxM is invertible. This implies that (w,V )is nondegenerate.

The conclusion follows by observing that the homotopy H on U of tangent vectorfields given by

�H(x, λ) = λ�v(x) + (1 − λ) �w(x) (5.3)

is nonzero on ∂V × [0, 1] and therefore it is admissible on V . The last assertion follows fromExcision, and Homotopy Invariance.

Theorem 5.2 below provides a formula for the computation of the degree of a tangentvector field that is valid for any nondegenerate pair. This implies the existence of at mostone real function on the family of admissible pairs that satisfies the axioms for the degree


of a tangent vector field. We recall that the property of Localization as well as Lemmas 5.1and 4.2, which are needed in the proof of our result, are consequences of the properties ofNormalization, Additivity and Homotopy Invariance.

Theorem 5.2 (uniqueness of the degree). Let deg be a real function on the family of admissiblepairs satisfying the properties of Normalization, Additivity, and Homotopy Invariance. If (v,U) is anondegenerate pair, then

deg(v,U) =∑

x∈Z(v,U)

sign detv′(x). (5.4)

Consequently, there exists at most one function on the family of admissible pairs satisfying the axiomsfor the degree of a tangent vector field, and this function, if it exists, must be integer valued.

Proof. Consider first the case M = Rm. Let (v,U) be a nondegenerate pair in R

m and, for anyx ∈ Z(v,U), let Vx be an isolating neighborhood of x. We may assume that the neighborhoodsVx are pairwise disjoint. Additivity and Localization together with Lemma 4.2 yield

deg(v,U) =∑

x∈Z(v,U)

deg(v, Vx) =∑

x∈Z(v,U)

sign detv′(x). (5.5)

Now the uniqueness of the degree of a tangent vector field on Rm follows immediately from

Lemma 5.1.Let us now consider the general case and denote by m the dimension of M. Let W

be any open subset of M which is diffeomorphic to Rm and let ψ : W → R

m be anydiffeomorphism onto R

m. Denote by U the set of all pairs (v,U) which are admissible andsuch that U ⊆W . We claim that for any (v,U) ∈ U one necessarily has

deg(v,U) = deg(Tψ ◦ v ◦ ψ−1, ψ(U)

). (5.6)

To show this, denote by V the set of admissible pairs (w,V ) with V ⊆ Rm and consider the

map α : U → V defined by

α(v,U) =(Tψ ◦ v ◦ ψ−1, ψ(U)

). (5.7)

Our claim means that the restriction deg |U of deg to U coincides with deg ◦ α. Observe thatα is invertible and

α−1(w,V ) =(Tψ−1 ◦w ◦ ψ, ψ−1(V )

). (5.8)

Moreover if two pairs (v,U) ∈ U and (w,V ) ∈ V correspond under α, then the sets Z(v,U)and Z(w,V ) correspond under ψ. It is also evident that the function deg ◦ α−1 : V → R

satisfies the axioms. Thus, by the first part of the proof, it coincides with the restriction deg |V,and this implies our claim.


Now let (v,U) be a given nondegenerate pair in M. Let Z(v,U) = {x1, . . . , xn} and letW1, . . . ,Wn be n pairwise disjoint open subsets of U such that xj ∈ Wj , for j = 1, . . . , n. Sinceany point of M has a fundamental system of neighborhoods which are diffeomorphic to R

m,we may assume that each Wj is diffeomorphic to R

m by a diffeomorphism ψj . Additivity andLocalization yield

deg(v,U) =n∑j=1

deg(v,Wj

), (5.9)

and, by the above claim, we get

n∑j=1

deg(v,Wj

)=

n∑j=1

deg(Tψj ◦ v ◦ ψ−1

j , ψj(Wj

)). (5.10)

By Lemma 4.2 and Remark 2.1

deg(Tψj ◦ v ◦ ψ−1

j , ψj(Wj

))= sign det

(Tψj ◦ v ◦ ψ−1

j

)′(ψj(xj))

= sign detv′(xj),

(5.11)

for j = 1, . . . , n. Thus

deg(v,U) =n∑j=1

sign detv′(xj). (5.12)

As in the case M = Rm, the uniqueness of the degree of a tangent vector field is now a

consequence of Lemma 5.1.

Acknowledgment

The author is dedicated to Professor William Art Kirk for his outstanding contributions in thetheory fixed points

References

[1] M. A. Krasnosel’skiı, The Operator of Translation along the Trajectories of Differential Equations,Translations of Mathematical Monographs, vol. 19, American Mathematical Society, Providence, RI,USA, 1968.

[2] M. A. Krasnosel’skiı and P. P. Zabreıko, Geometrical Methods of Nonlinear Analysis, vol. 263 ofGrundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1984.

[3] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, NJ, USA, 1974.[4] M. W. Hirsch, Differential Topology, Springer, New York, NY, USA, 1976, Graduate Texts in

Mathematics, no. 33.[5] J. W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville,

Va, USA, 1965.


[6] A. J. Tromba, “The Euler characteristic of vector fields on Banach manifolds and a globalization ofLeray-Schauder degree,” Advances in Mathematics, vol. 28, no. 2, pp. 148–173, 1978.

[7] H. Amann and S. A. Weiss, “On the uniqueness of the topological degree,” Mathematische Zeitschrift,vol. 130, pp. 39–54, 1973.

[8] L. Fuhrer, “Ein elementarer analytischer beweis zur eindeutigkeit des abbildungsgrades im Rn,”

Mathematische Nachrichten, vol. 54, pp. 259–267, 1972.[9] P. C. Staecker, “On the uniqueness of the coincidence index on orientable differentiable manifolds,”

Topology and Its Applications, vol. 154, no. 9, pp. 1961–1970, 2007.[10] M. Furi, M. P. Pera, and M. Spadini, “On the uniqueness of the fixed point index on differentiable

manifolds,” Fixed Point Theory and Applications, vol. 2004, no. 4, pp. 251–259, 2004.[11] F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, vol. 94 of Graduate Texts in

Mathematics, Springer, New York, NY, USA, 1983.


Research ArticleApproximating Fixed Points of Some Maps inUniformly Convex Metric Spaces

Abdul Rahim Khan,1 Hafiz Fukhar-ud-din,2and Abdul Aziz Domlo3

1 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals,Dahran 31261, Saudi Arabia

2 Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan3 Department of Mathematics, Taibah University, P.O. Box 30002, Madinah Munawarah, Saudi Arabia

Correspondence should be addressed to Abdul Rahim Khan, [email protected]

Received 1 October 2009; Accepted 19 January 2010


Copyright q 2010 Abdul Rahim Khan et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study strong convergence of the Ishikawa iterates of qasi-nonexpansive (generalizednonexpansive) maps and some related results in uniformly convex metric spaces. Our workimproves and generalizes the corresponding results existing in the literature for uniformly convexBanach spaces.


Let C be a nonempty subset of a metric space (X, d) and let T : C → C be a map. Denotethe set of fixed points of T, {x ∈ C : T(x) = x} by F. The map T is said to be (i) quasi-nonexpansive if F /=φ and d(Tx, p) ≤ d(x, p) for all x ∈ C and p ∈ F, (ii) k-Lipschitz if forsome k > 0, we have d(Tx, Ty) ≤ kd(x, y) for all x, y ∈ C; for k = 1, it becomes nonexpansive,and (iii) generalized nonexpansive (cf. [1] and the references therein) if

d(Tx, Ty

) ≤ ad(x, y) + b{d(x, Tx) + d(y, Ty)} + c{d(x, Ty) + d(y, Tx)} (∗)

for all x, y ∈ C where a, b, c ≥ 0 with a + 2b + 2c ≤ 1.The concept of quasi-nonexpansiveness is more general than that of nonexpansive-

ness. A nonexpansive map with at least one fixed point is quasi-nonexpansive but there arequasi-nonexpansive maps which are not nonexpansive [2].


Mann and Ishikawa type iterates for nonexpansive and quasi-nonexpansive mapshave been extensively studied in uniformly convex Banach spaces [1, 3–6]. Senter andDotson [7] established convergence of Mann type iterates of quais-nonexpansive maps undera condition in uniformly convex Banach spaces. In 1973, Goebel et al. [8] proved thatgeneralized nonexpansive self maps have fixed points in uniformly convex Banach spaces.Based on their work, Bose and Mukerjee [1] proved theorems for the convergence of Manntype iterates of generalized nonexpansive maps and obtained a result of Kannan [9] underrelaxed conditions. Maiti and Ghosh [6] generalized the results of Bose and Mukerjee [1] forIshikawa iterates by using modified conditions of Senter and Dotson [7] (see, also [10]). Forthe sake of completeness, we state the result of Kannan [9] and its generalization by Bose andMukerjee [1].

Theorem 1.1 (see [9]). Let C be a nonempty, bounded, closed, and convex subset of a uniformlyconvex Banach space. Let T be a map of C into itself such that

(i) ‖Tx − Ty‖ ≤ (1/2)‖x − Tx‖ + (1/2)‖y − Ty‖ for all x, y ∈ C,(ii) supz∈K‖z−Tz‖ ≤ δ(K)/2, whereK is any nonempty convex subset of C which is mapped

into itself by T and δ(K) is the diameter of K.

Then the sequence {xn} defined by xn+1 = (1/2)xn+(1/2)Txn converges to the fixed point of T,wherex1 is any arbitrary point of C.

Theorem 1.2 (see [1]). Let C be a nonempty, bounded, closed, and convex subset of a uniformlyconvex Banach space. Let T be a map of C into itself such that

∥∥Tx − Ty∥∥ ≤ a∥∥x − y∥∥ + b{‖x − Tx‖ + ∥∥y − Ty∥∥} + c{∥∥x − Ty∥∥ +

∥∥y − Tx∥∥} (1.1)

for all x, y ∈ C where a, b, c ≥ 0 and 3a+2b+4c ≤ 1.Define a sequence {xn} in C for x1 ∈ C, xn+1 =(1 − αn)xn + αnTxn, for all n ≥ 1, where 0 < β ≤ αn ≤ γ < 1. Then {xn} converges to a fixed point ofT .

In Theorem 1.2, taking a = c = 0, b = 1/2, and αn = 1/2 for all n ≥ 1, it becomesTheorem 1.1 without requiring condition (ii).

In 1970, Takahashi [11] introduced a notion of convexity in a metric space (X, d) asfollows: a map W : X ×X × I → X is a convex structure in X if

d(u,W

(x, y, λ

)) ≤ λd(u, x) + (1 − λ)d(u, y) (1.2)

for all x, y ∈ X and λ ∈ I = [0, 1]. A metric space together with a convex structure issaid to be convex metric space. A nonempty subset C of a convex metric space is convexif W(x, y, λ) ∈ C for all x, y ∈ C and λ ∈ I. In fact, every normed space and its convexsubsets are convex metric spaces but the converse is not true, in general (see [11]). Later on,Shimizu and Takahashi [12] obtained some fixed point theorems for nonexpansive maps inconvex metric spaces. This notion of convexity has been used in [13–15] to study Mann andIshikawa iterations in convex metric spaces. For other fixed point results in the closely relatedclasses of spaces, namely, hyperbolic and hyperconvex metric spaces, we refer to [16–19].


In the sequel, we assume that C is a nonempty convex subset of a convex metric spaceX and T is a selfmap onC. For an initial value x1 ∈ C,we define the Ishikawa iteration schemein C as follows:

x1 ∈ C,xn+1 =W

(Tyn, xn, αn

),

yn =W(Txn, xn, βn

) ∀n ≥ 1,

(1.3)

where {αn} and {βn} are control sequences in [0, 1].If we choose βn = 0, then (1.3) reduces to the following Mann iteration scheme:

x1 ∈ C, xn+1 =W(Txn, xn, αn), ∀n ≥ 1, (1.4)

where {αn} is a control sequence in [0, 1].If X is a normed space with C as its convex subset, then W(x, y, λ) = λx + (1 − λ)y is a

convex structure in X; consequently (1.3) and (1.4), respectively, become

x1 ∈ C, xn+1 = (1 − αn)xn + αnTyn,yn =

(1 − βn

)xn + βnTxn ∀n ≥ 1.

x1 ∈ C, xn+1 = (1 − αn)xn + αnTxn, ∀n ≥ 1,

(1.5)

where {αn} and {βn} are control sequences in [0, 1].A convex metric space X is said to be uniformly convex [11] if for arbitrary positive

numbers ε and r, there exists α(ε) > 0 such that

d

(z,W

(x, y,

12

))≤ r(1 − α) (1.6)

whenever x, y, z ∈ X, d(z, x) ≤ r, d(z, y) ≤ r and d(x, y) ≥ rε.In 1989, Maiti and Ghosh [6] generalized the two conditions due to Senter and Dotson

[7]. We state all these conditions in convex metric spaces:Let T be a map with nonempty fixed point set F and d(x, F) = infp∈Fd(x, p). Then T is said

to satisfy the following Condotions.

Condition 1. If there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0 andf(r) > 0 for all r > 0 such that d(x, Tx) ≥ f(d(x, F)) for x ∈ C.

Condition 2. If there exists a real number k > 0 such that d(x, Tx) ≥ kd(x, F) for x ∈ C.

Condition 3. If there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(r) >0 for all r > 0 such that d(x, Ty) ≥ f(d(x, F)) for x ∈ C and all corresponding y =W(Tx, x, t)where 0 ≤ t ≤ β < 1.

Condition 4. If there exists a real number k > 0 such that d(x, Ty) ≥ kd(x, F) for x ∈ C and allcorresponding y =W(Tx, x, t) where 0 ≤ t ≤ β < 1.


Note that if T satisfies Condition 1 (resp., 3), then it satisfies Condition 2 (resp., 4).We also note that Conditions 1 and 2 become Conditions A and B, respectively, of Senterand Dotson [7] while Conditions 3 and 4 become Conditions I and II, respectively, of Maitiand Ghosh [6] in a normed space. Further, Conditions 3 and 4 reduce to Conditions 1 and 2,respectively, when t = 0.

In this note, we present results under relaxed control conditions which generalize thecorresponding results of Kannan [9], Bose and Mukerjee [1], and Maiti and Ghosh [6] fromuniformly convex Banach spaces to uniformly convex metric spaces. We present sufficientconditions for the convergence of Ishikawa iterates of k−Lipschitz maps to their fixed pointsin convex metric spaces and improve [3, Lemma 2]. A necessary and sufficient condition isobtained for the convergence of a sequence to fixed point of a generalized nonexpansive mapin metric spaces.

We need the following fundamental result for the developmant of our results.

Theorem 1.3 (see [20]). Let X be a uniformly convex metric space with a continuous convexstructureW : X ×X × 0, 1] → X. Then for arbitrary positive numbers ε and r, there exists α(ε) > 0such that

d(z,W

(x, y, λ

)) ≤ r(1 − 2 min{λ, 1 − λ}α) (1.7)

for all x, y, z ∈ X, d(z, x) ≤ r, d(z, y) ≤ r, d(x, y) ≥ rε and λ[∈ 0, 1].

2. Convergence Analysis

We prove a lemma which plays key role to establish strong convergence of the iterativeschemes (1.3) and (1.4).

Lemma 2.1. Let X be a uniformly convex metric space. Let C be a nonempty closed convex subsetof X, T : C → C a quasi-nonexpansive map and {xn} as in (1.3). If

∑∞n=0 αn(1 − αn) = ∞ and

0 ≤ βn ≤ β < 1, then lim infn→∞d(xn, Tyn) = 0.

Proof. For p ∈ F, we consider

d(xn+1, p

)= d

(p,W

(Tyn, xn, αn

))≤ αnd

(p, Tyn

)+ (1 − αn)d

(p, xn

)≤ αnd

(p, yn

)+ (1 − αn)d

(p, xn

)= αnd

(p,W

(Txn, xn, βn

))+ (1 − αn)d

(p, xn

)≤ αnβnd

(p, Txn

)+ αn

(1 − βn

)d(p, xn

)+ (1 − αn)d

(p, xn

)≤ αnβnd

(p, xn

)+ αn

(1 − βn

)d(p, xn

)+ (1 − αn)d

(p, xn

)= d

(xn, p

).

(2.1)

This implies that the sequence {d(xn, p)} is nonincreasing and bounded below. Thuslimn→∞d(xn, p) exists. We may assume that c = limn→∞d(xn, p) > 0.


For any p ∈ F, we have that

d(xn, Tyn

) ≤ d(xn, p) + d(Tyn, p)≤ d(xn, p) + d(yn, p)= d

(xn, p

)+ d

(p,W

(Txn, xn, βn

))≤ d(xn, p) + βnd(Txn, p) + (

1 − βn)d(xn, p

)≤ d(xn, p) + βnd(xn, p) + (

1 − βn)d(xn, p

)= 2d

(xn, p

).

(2.2)

Since limn→∞d(xn, p) exists, so d(xn, Tyn) is bounded and hence infn≥1d(xn, Tyn)exists. We show that infn≥1d(xn, Tyn) = 0. Assume that infn≥1d(xn, Tyn) = σ > 0.

Then

d(xn, Tyn

) ≥ d(xn, p) · σ

d(xn, p

)

≥ d(xn, p) · σ

d(x1, p

) .(2.3)

Hence by Theorem 1.3, there exists α(σ/d(x1, p)) > 0 such that

d(xn−1,p

)= d

(W

(Tyn, xn, αn

), p)

≤ d(xn, p)(1 − 2 min{αn, 1 − αn}α)≤ d(xn, p)(1 − 2αn(1 − αn)α).

(2.4)

That is,

2cαn(1 − αn)α ≤ d(xn, p

) − d(xn+1, p). (2.5)

Taking m � 1 and summing up the (m + 1) terms on the both sides in the above inequality,we have

2cαm∑n=1

αn(1 − αn) ≤ d(p, x1

) − d(p, xm) ∀m ≥ 1. (2.6)

Let m → ∞. Then, we have

∞ ≤ d(p, x1)<∞. (2.7)

This is contradiction and hence infn≥1d(xn, Tyn) = 0.


In the light of above result, we can construct subsequences {xni} and {yni} of {xn} and{yn}, respectively, such that limi→∞d(xni , Tyni) = 0 and hence lim infn→∞d(xn, Tyn) = 0.

Now we state and prove Ishikawa type convergence result in uniformly convex metricspaces.

Theorem 2.2. Let X be a uniformly convex complete metric space with continuous convex structureand let C be its nonempty closed convex subset. Let T be a continuous quasi-nonexpansive map of Cinto itself satisfying Condition 3. If {xn} is as in (1.3), where

∑∞n=0 αn(1−αn) =∞ and 0 ≤ βn ≤ β <

1, then {xn} converges to a fixed point of T .

Proof. In Lemma 2.1, we have shown that d(xn+1,p) ≤ d(xn, p). Therefore d(xn+1, F) ≤d(xn, F). This implies that the sequence {d(xn, F)} is nonincreasing and bounded below. Thuslimn→∞d(xn, F) exists. Now by Condition 3, we have

lim infn→∞

f(d(xn, F)) ≤ lim infn→∞

d(Tyn, xn

)= 0. (2.8)

Using the properties of f, we have lim infn→∞d(xn, F) = 0. As limn→∞d(xn, F) exists,therefore limn→∞d(xn, F) = 0.

Now, we show that {xn} is a Cauchy sequence. For ε > 0, there exists a constant n0 suchthat for all n ≥ n0, we have d(xn, F) < ε/4. In particular, d(xn0 , F) < ε/4. That is, inf{d(xn0 , p) :p ∈ F} < ε/4. There must exist p∗ ∈ F such that d(xn0 , p

∗) < ε/2. Now, for m,n ≥ n0, we havethat

d(xn+m, xn) ≤ d(xn+m, p

∗) + d(xn, p∗)≤ 2d

(xn0 , p

∗)< ε.

(2.9)

This proves that {xn} is a Cauchy sequence in C. Since C is a closed subset of acomplete metric space X, therefore it must converge to a point q in C.

Finally, we prove that q is a fixed point of T.Since

d(q, F

) ≤ d(q, xn) + d(xn, F), (2.10)

therefore d(q, F) = 0. As F is closed, so q ∈ F.

Choose βn = 0 for all n ≥ 1, in the above theorem; it reduces to the following Manntype convergence result.

Theorem 2.3. Let X be a uniformly convex complete metric space with continuous convex structureand let C be its nonempty closed convex subset. Let T be a continuous quasi-nonexpansive map ofC into itself satisfying Condition 1. If {xn} is as in (1.4), where

∑∞n=0 αn(1 − αn) = ∞, then {xn}

converges to a fixed point of T .

Next we establish strong convergence of Ishikawa iterates of a generalized nonexpan-sive map.


Theorem 2.4. LetX and C be as in Theorem 2.3. Let T be a continuous generalied nonexpansive mapof C into itself with at least one fixed point. If {xn} is as in (1.3), where

∑∞n=0 αn(1 − αn) = ∞ and

0 ≤ βn ≤ β < 1,then {xn} converges to a fixed point of T .

Proof. Let p be any fixed point of T. Then setting y = p in (∗), we have

d(Tx, p

) ≤ (a + c)d(x, p

)+ bd(x, Tx) + cd

(Tx, p

)≤ (a + b + c)d

(x, p

)+ (b + c)d

(Tx, p

),

(2.11)

which implies

d(Tx, p

) ≤ a + b + c1 − b − c d

(x, p

) ≤ d(x, p). (2.12)

Thus T is quasi-nonexpansive.For any y ∈ C, we also observe that

d(Ty, p

) ≤ (a + c)d(y, p

)+ bd

(y, Ty

)+ cd

(Ty, p

). (2.13)

If y =W(Tx, x, t), where 0 ≤ t ≤ β < 1, then

d(y, p

)= d

(W(Tx, x, t), p

)≤ td(Tx, p) + (1 − t)d(x, p)≤ d(x, p),

(2.14)

d(y, x

)= d(W(Tx, x, t), x)

≤ td(x, Tx) + (1 − t)d(x, x)= td(x, Tx)

≤ t[d(x, p) + d(Tx, p)]≤ 2td

(x, p

).

(2.15)

Using (2.14) in (2.13), we have

d(Ty, p

) ≤ (a + c)d(y, p

)+ bd

(y, Ty

)+ cd

(Ty, p

)≤ (a + c)d

(y, p

)+ c

{d(x, p

)+ d

(x, Ty

)}+ b

{d(x, y

)+ d

(x, Ty

)}≤ (a + 2c)d

(x, p

)+ bd

(x, y

)+ (b + c)d

(x, Ty

).

(2.16)

Also it is obvious that

d(Ty, p

) ≥ d(x, p) − d(x, Ty). (2.17)


Combining (2.16) and (2.17), we get that

bd(x, y

)+ (1 + b + c)d

(x, Ty

) ≥ (1 − a − 2c)d(x, p

)≥ 2bd

(x, p

).

(2.18)

Now inserting (2.15) in (2.18), we derive

(1 + b + c)d(x, Ty

) ≥ 2bd(x, p

) − bd(x, y)≥ 2b(1 − t)d(x, p). (2.19)

That is,

d(x, Ty

) ≥ 2b(1 − t)1 + b + c

d(x, p

) ≥ 2b(1 − β)

1 + b + cd(x, p

), (2.20)

where 2b(1 − t)/(1 + b + c) > 0. Thus T satisfies Condition 4 (and hence Condition 3). Theresult now follows from Theorem 2.2.

Remark 2.5. In the above theorem, we have assumed that the generalied nonexpansive mapT has a fixed point. It remains an open questions: what conditions on a, b, and c in (∗) aresufficient to guarantee the existence of a fixed point of T even in the setting of a metric space.

Choose βn = 0 for all n ≥ 1 in Theorem 2.4 to get the following Mann type convergenceresult.

Theorem 2.6. Let X,C,and T be as in Theorem 2.4. If{xn} is as in (1.4), where∑∞

n=0 αn(1 − αn) =∞,then {xn} converges to a fixed point of T .

Proof. For βn = 0 for all n ≥ 1, y = W(Tx, x, 0) = x, the inequality (2.20) in the proof ofTheorem 2.4 becomes

d(x, Tx) ≥ 2b1 + b + c

d(x, p

). (2.21)

Thus T satisfies Condition 2 (and hence Condition 1) and so the result follows fromTheorem 2.3.

The analogue of Kannan result in uniformly convex metric space can be deduced fromTheorem 2.6 (by taking a = c = 0, b = 1/2, and αn = 1/2 for all n ≥ 1) as follows.

Theorem 2.7. Let X be a uniformly convex complete metric space with continuous convex structureand let C be its nonempty closed convex subset. Let T be a continuous map of C into itself with atleast one fixed point such that d(Tx, Ty) ≤ (1/2)d(x, Tx) + (1/2)d(y, Ty) for all x, y ∈ C. Thenthe sequence {xn} where x1 ∈ C and xn+1 =W(Txn, xn, 1/2) converges to a fixed point of T.

Next we give sufficient conditions for the existence of fixed point of a k-Lipschitz mapin terms of the Ishikawa iterates.


Theorem 2.8. Let (X, d) be a convex metric space and let C be its nonempty convex subset. LetT be a k-Lipschitz selfmap of C. Let {xn} be the sequence as in (1.3), where {αn} and {βn} satisfy(i)0 ≤ αn, βn ≤ 1 for all n ≥ 1(ii) lim infn→∞αn > 0 and (iii) lim infn→∞βn < k−1. If d(xn+1, xn) =αnd(xn, Tyn) and xn → p, then p is a fixed point of T.

Proof. Let p ∈ C. Then

d(p, Tp

) ≤ d(xn, p) + d(xn, Tyn) + d(Tyn, Tp)

= d(xn, p

)+

1and(xn+1, xn) + d

(Tyn, Tp

)

≤ d(xn, p) + 1and(xn+1, xn) + kd

(yn, p

)

= d(xn, p

)+

1and(xn+1, xn) + kd

(W

(Txn, xn, βn

), p)

≤ d(xn, p) + 1and(xn+1, xn) + k

{βnd

(Txn, p

)+(1 − βn

)d(xn, p

)}

≤ d(xn, p) + 1and(xn+1, xn) + kβn

{d(Txn, Tp

)+ d

(p, Tp

)}

+ k(1 − βn

)d(xn, p

).

(2.22)

That is,

(1 − kβn

)d(p, Tp

) ≤ (1 + k2βn + k

(1 − βn

))d(xn, p

)+

1and(xn+1, xn), (2.23)

Since lim inf an > 0, therefore there exists a > 0 such that an > a for all n ≥ 1. This implies that

(1 − kβn

)d(p, Tp

) ≤ (1 + k2βn + k

(1 − βn

))d(xn, p

)+

1ad(xn+1, xn), (2.24)

Taking lim sup on both the sides in the above inequality and using the conditionlim infn→∞βn < k−1, we have d(p, Tp) = 0.

Finally, using a generalized nonexpansive map T on a metric space X, we provide anecessary and sufficient condition for the convergence of an arbitrary sequence {xn} in X toa fixed point of T in terms of the approximating sequence {d(xn, Txn)}.

Theorem 2.9. Suppose that C is a closed subset of a complete metric space (X, d) and T : C → C isa continuous map such that for some a, b, c ≥ 0, a + 2c < 1, the following inequality holds:

d(Tx, Ty

) ≤ ad(x, y) + b{d(x, Tx) + d(y, Ty)} + c{d(x, Ty) + d(y, Tx)} (2.25)

for all x, y ∈ C. Then a sequence {xn} in C converges to a fixed point of T if and only iflimn→∞d(xn, Txn) = 0.


Proof. Suppose that limn→∞d(xn, Txn) = 0. First we show that {xn} is a Cauchy sequence inC. To acheive this goal, consider:

d(Txn, Txm) ≤ ad(xn, xm) + b{d(xn, Txn) + d(xm, Txm)}+ c{d(xn, Txm) + d(xm, Txn)}

≤ ad(xn, xm) + b{d(xn, Txn) + d(xm, Txm)}+ c{d(xn, xm) + d(xm, Txm) + d(xm, xn) + d(xn, Txn)}

= (a + 2c)d(xn, xm)

+ (b + c){d(xn, Txn) + d(xm, Txm)}≤ (a + b + 3c){d(xn, Txn) + d(xm, Txm)}+ (a + 2c)d(Txn, Txm).

(2.26)

That is,

(1 − a − 2c)d(Txn, Txm) ≤ (a + b + 3c){d(xn, Txn) + d(xm, Txm)}. (2.27)

Since limn→∞d(xn, Txn) = 0 and a + 2c < 1, therefore from the above inequality, itfollows that {Txn} is a Cauchy sequence in C. In view of closedness of C, this sequenceconverges to an element p of C. Also limn→∞d(xn, Txn) = 0 gives that limn→∞xn = p. Nowusing the continuity of T, we have T(p) = T(limn→∞xn) = limn→∞T(xn) = p. Hence p is afixed point of T.

Conversely, suppose that {xn} converges to a fixed point p of T. Using the continuityof T, we have that limn→∞Txn = p. Thus limn→∞d(xn, Txn) = 0.

Remark 2.10. Theorem 2.8 improves Lemma 2 in [3] from real line to convex metric spacesetting. Theorem 2.9 is an extension of Theorem 4 in [21] to metric spaces. If we choose c = 0in Theorem 2.9, it is still an improvement of [21, Theorem 4].

Remark 2.11. We have proved our results (2.1)–(2.8) in convex metric space setting. All theseresults, in particular, hold in Banach spaces if we set W(x, y, λ) = λx + (1 − λ)y.

Acknowledgment

The author A. R. Khan is grateful to King Fahd University of Petroleum & Minerals forsupport during this research.

References

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[7] H. F. Senter and W. G. Dotson Jr., “Approximating fixed points of nonexpansive mappings,”Proceedings of the American Mathematical Society, vol. 44, pp. 375–380, 1974.

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[9] R. Kannan, “Some results on fixed points. III,” Fundamenta Mathematicae, vol. 70, no. 2, pp. 169–177,1971.

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Research ArticleDemiclosed Principle for AsymptoticallyNonexpansive Mappings in CAT(0) Spaces

B. Nanjaras and B. Panyanak

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to B. Panyanak, [email protected]

Received 21 August 2009; Accepted 18 January 2010


Copyright q 2010 B. Nanjaras and B. Panyanak. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We prove the demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces.As a consequence, we obtain a Δ-convergence theorem of the Krasnosel’skii-Mann iteration forasymptotically nonexpansive mappings in this setting. Our results extend and improve manyresults in the literature.

1. Introduction

One of the fundamental and celebrated results in the theory of nonexpansive mappings isBrowder’s demiclosed principle [1] which states that if X is a uniformly convex Banachspace, then C is a nonempty closed convex subset of X, and if T : C → X is a nonexpansivemapping, then I−T is demiclosed at each y ∈ X, that is, for any sequence {xn} inC conditionsxn → x weakly and (I−T)(xn) → y strongly imply that (I−T)(x) = y (where I is the identitymapping of X). It is known that the demiclosed principle plays important role in studyingthe asymptotic behavior for nonexpansive mappings (see, e.g., [2–10]). In [11], Xu proved thedemiclosed principle for asymptotically nonexpansive mappings in the setting of a uniformlyconvex Banach space. The purpose of this paper is to extend Xu’s result to a special kind ofmetric spaces, namely, CAT(0) spaces, which will be defined in the next section. We alsoapply our result to obtain a Δ-convergence theorem of the Krasnosel’skii-Mann iteration forasymptotically nonexpansive mappings in the CAT(0) space setting.

2. CAT(0) Spaces

A metric space X is a CAT(0) space if it is geodesically connected and if every geodesictriangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. It is


well known that any complete, simply connected Riemannian manifold having nonpositivesectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces, R-trees (see[12]), Euclidean buildings (see [13]), the complex Hilbert ball with a hyperbolic metric (see[14]), and many others. For a thorough discussion of these spaces and of the fundamental rolethey play in geometry, see Bridson and Haefliger [12]. Burago et al. [15] provide a somewhatmore elementary treatment and Gromov [16] presents a deeper study.

Fixed point theory in a CAT(0) space was first studied by Kirk (see [17, 18]). Heshowed that every nonexpansive (single-valued) mapping defined on a bounded closedconvex subset of a complete CAT(0) space always has a fixed point. Since then the fixedpoint theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidlydeveloped and many papers have appeared (see, e.g., [19–34]). It is worth mentioning thatthe results in CAT(0) spaces can be applied to any CAT(κ) space with κ ≤ 0 since any CAT(κ)space is a CAT(κ′) space for every κ′ ≥ κ (see [12, page 165]).

Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, ageodesic from x to y) is a map c from a closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) =y, and d(c(t), c(t′)) = |t − t′| for all t, t′ ∈ [0, l]. In particular, c is an isometry and d(x, y) = l.The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique,this geodesic is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every twopoints of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly onegeodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convex if Y includesevery geodesic segment joining any two of its points.

A geodesic triangle Δ(x1, x2, x3) in a geodesic space (X, d) consists of three pointsx1, x2, x3 in X (the vertices of Δ) and a geodesic segment between each pair of vertices (theedges of Δ). A comparison triangle for geodesic triangle Δ(x1, x2, x3) in (X, d) is a triangleΔ(x1, x2, x3) := Δ(x1, x2, x3) in the Euclidean plane E

2 such that dE2(xi, xj) = d(xi, xj) fori, j ∈ {1, 2, 3}.

A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy thefollowing comparison axiom.

CAT(0): Let Δ be a geodesic triangle in X and let Δ be a comparison triangle for Δ.Then Δ is said to satisfy the CAT(0) inequality if, for all x, y ∈ Δ and all comparison pointsx, y ∈ Δ,

d(x, y) ≤ dE2

(x, y). (2.1)

Let x, y ∈ X, by [26, Lemma 2.1(iv)] for each t ∈ [0, 1], then there exists a unique pointz ∈ [x, y] such that

d(x, z) = td(x, y), d

(y, z)= (1 − t)d(x, y). (2.2)

From now on, we will use the notation (1 − t)x ⊕ ty for the unique point z satisfying (2.2). Byusing this notation, Dhompongsa and Panyanak [26] obtained the following lemma whichwill be used frequently in the proof of our main results.


Lemma 2.1. Let X be a CAT(0) space. Then

d((1 − t)x ⊕ ty, z) ≤ (1 − t)d(x, z) + td(y, z), (2.3)

for all x, y, z ∈ X and t ∈ [0, 1].

If x, y1, y2 are points in a CAT(0) space and if y0 = (1/2)y1 ⊕ (1/2)y2, then the CAT(0)inequality implies that

d(x, y0

)2 ≤ 12d(x, y1

)2 +12d(x, y2

)2 − 14d(y1, y2

)2. (CN)

This is the (CN) inequality of Bruhat and Tits [35]. In fact (cf. [12, page 163]), a geodesicspace is a CAT(0) space if and only if it satisfies (CN).

The following lemma is a generalization of the (CN) inequality which can be found in[26].

Lemma 2.2. Let (X, d) be a CAT(0) space. Then

d((1 − t)x ⊕ ty, z)2 ≤ (1 − t)d(x, z)2 + td

(y, z)2 − t(1 − t)d(x, y)2

, (2.4)

for all t ∈ [0, 1] and x, y, z ∈ X.

3. Demiclosed Principle

In 1976, Lim [36] introduced a concept of convergence in a general metric space whichhe called “Δ-convergence”. In 2008, Kirk and Panyanak [37] specialized Lim’s concept toCAT(0) spaces and showed that many Banach space results involving weak convergencehave precise analogs in this setting. Since then the notion of Δ-convergence has been widelystudied and a number of papers have appeared (see, e.g., [26, 29, 31, 32, 34]).

We now give the concept of Δ-convergence and collect some of its basic properties.Let {xn} be a bounded sequence in a CAT(0) space X. For x ∈ X, we set

r(x, {xn}) = lim supn→∞

d(x, xn). (3.1)

The asymptotic radius r({xn}) of {xn} is given by

r({xn}) = inf{r(x, {xn}) : x ∈ X}, (3.2)

the asymptotic radius rC({xn}) of {xn}with respect to C ⊂ X is given by

rC({xn}) = inf{r(x, {xn}) : x ∈ C}, (3.3)


the asymptotic center A({xn}) of {xn} is the set

A({xn}) = {x ∈ X : r(x, {xn}) = r({xn})}, (3.4)

and the asymptotic center AC({xn}) of {xn}with respect to C ⊂ X is the set

AC({xn}) = {x ∈ C : r(x, {xn}) = rC({xn})}. (3.5)

Recall that a bounded sequence {xn} in X is said to be regular if r({xn}) = r({un}) forevery subsequence {un} of {xn}.

The following proposition was proved in [22].

Proposition 3.1. If {xn} is a bounded sequence in a complete CAT(0) space X and if C is a closedconvex subset of X, then there exists a unique point u ∈ C such that

r(u, {xn}) = infx∈C

r(x, {xn}). (3.6)

This fact immediately yields the following proposition.

Proposition 3.2. Let {xn}, C, and X be as in Proposition 3.1. Then A({xn}) and AC({xn}) aresingletons.

The following lemma can be found in [25].

Lemma 3.3. If C is a closed convex subset of X and if {xn} is a bounded sequence in C, then theasymptotic center of {xn} is in C.

Definition 3.4 (see [36, 37]). A sequence {xn} in X is said to Δ-converge to x ∈ X if x is theunique asymptotic center of {un} for every subsequence {un} of {xn}. In this case we writeΔ − limnxn = x and call x the Δ-limit of {xn}.

Lemma 3.5 (see [37]). Every bounded sequence in a complete CAT(0) space always has a Δ-convergent subsequence.

There is another concept of convergence in geodesic spaces; it was introduced by Sosov[38] and was specialized to CAT(0) spaces by Espınola and Fernandez-Leon [29] as follows.

Let X be a CAT(0) space and let p be a point in X. Let S be the set of all the geodesicsegments containing the point p. Given l ∈ S and x ∈ X, we define the function φl : X → R

as φl(x) = d(p, Pl(x)) where Pl(x) is the projection of x onto l. The set of all these φl is denotedby Φp(X).

Definition 3.6. A bounded sequence {xn} in X is said to φp-converge to a point x ∈ X if

limn→∞

φ(xn) = φ(x) for any φ ∈ Φp(X). (3.7)

In [29], the authors showed that Δ-convergence and φ-convergence are equivalent asthe following result.


Proposition 3.7. A sequence {xn} in a CAT(0) space Δ-converges to p if and only if it φp-convergesto p.

Recall that a mapping T on a metric space X is said to be nonexpansive if

d(T(x), T

(y)) ≤ d(x, y), ∀x, y ∈ X. (3.8)

T is called asymptotically nonexpansive if there is a sequence {kn} of positive numberswith the property limn→∞kn = 1 such that

d(Tn(x), Tn

(y)) ≤ knd(x, y), ∀n ≥ 1, x, y ∈ X. (3.9)

A point x ∈ X is called a fixed point of T if x = T(x). We will denote with F(T) the setof fixed points of T. The existence of fixed points for asymptotically nonexpansive mappingsin CAT(0) spaces was proved by Kirk [18] as the following result.

Theorem 3.8. Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space Xand let T : C → C be asymptotically nonexpansive. Then T has a fixed point.

Now, we discuss the notion of asymptotic contractions which was introduced by Kirk[39] as the following statement.

Let Ψ denote the class of all mappings ψ : [0,∞) → [0,∞) satisfying what follows:

(i) ψ is continuous,

(ii) ψ(s) < s for all s > 0.

Definition 3.9. Let (X, d) be a metric space. A mapping T : X → X is said to be an asymptoticcontraction (see[39]) if

d(Tn(x), Tn

(y)) ≤ ψn(d(x, y)) ∀x, y ∈ X, (3.10)

where ψn : [0,∞) → [0,∞) and ψn → ψ ∈ Ψ uniformly on the range of d.

T is called a pointwise contraction (see[40]) if there exists a mapping α : X → [0, 1)such that

d(T(x), T

(y)) ≤ α(x)d(x, y) for each y ∈ X. (3.11)

Definition 3.10. Let (X, d) be a metric space. A mapping T : X → X is called an asymptoticpointwise mapping (see[30]) if there exists a sequence of mappings αn : X → [0,∞) such that

d(Tn(x), Tn

(y)) ≤ αn(x)d(x, y) for any y ∈ X. (3.12)

(i) If {αn} converges pointwise to α : X → [0, 1), then T is called an asymptoticpointwise contraction.


(ii) If lim supn→∞αn(x) ≤ 1, then T is called asymptotic pointwise nonexpansive.

(iii) If lim supn→∞αn(x) ≤ k, with 0 < k < 1, then T is called strongly asymptoticpointwise contraction.

It is immediately clear that an asymptotically nonexpansive mapping is asymptoticpointwise nonexpansive. By using the ultrapower technique, Kirk [39] established theexistence of fixed points for asymptotic contractions in complete metric spaces. In [40], Kirkand Xu gave simple and elementary proofs for the existence of fixed points for asymptoticpointwise contractions and asymptotic pointwise nonexpansive mappings in Banach spaceswithout the use of ultrapowers. Very recently, Hussain and Khamsi [30] extended Kirk-Xu’s results to CAT(0) spaces. Moreover, they introduced a notion of convergence in CAT(0)spaces as follows.

Let {xn} be a bounded sequence in a CAT(0) space X and let C be a closed convexsubset of X which contains {xn}. We denote the notation

{xn}⇀ w iff Φ(w) = infx∈C

Φ(x), (3.13)

where Φ(x) = lim supn→∞d(xn, x). By using this notation, they obtained the demiclosedprinciple for asymptotic pointwise nonexpansive mappings as the following result.

Proposition 3.11. Let C be a closed and convex subset of a complete CAT(0) space X, T : C → Cbe an asymptotic pointwise nonexpansive mapping. Let {xn} be a bounded sequence in C such thatlimnd(xn, T(xn)) = 0, and {xn}⇀ w. Then T(w) = w.

We now give a connection between this kind of convergence and Δ-convergence.

Proposition 3.12. Let {xn} be a bounded sequence in a CAT(0) space X and let C be a closed convexsubset of X which contains {xn}. Then

(1) Δ − limnxn = x implies that {xn}⇀ x,

(2) the converse of (2.2) is true if {xn} is regular.

Proof. (1) Suppose that Δ − limnxn = x, then x ∈ C by Lemma 3.3. Since A({xn}) = {x},we have r({xn}) = r(x, {xn}). This implies that Φ(x) = infy∈CΦ(y). Therefore we obtain thedesired result.

(2) Suppose that {xn} is regular and {xn} ⇀ x. We note that {xn} ⇀ x if and only ifAC({xn}) = {x}. Suppose that A({xn}) = {y}, again by Lemma 3.3, we have y ∈ C. Thereforex = y, and hence, A({xn}) = {x}. By the regularity of {xn}, we have A({xn}) ⊂ A({un})for each subsequence {un} of {xn}. Thus Δ − limnxn = x since the asymptotic center of anybounded sequence in X must be a singleton.

The following example shows that the regularity in Proposition 3.12 is necessary.

Example 3.13. Let X = R, d be the usual metric on X, C = [−1, 1], {xn} ={1,−1, 1,−1, . . .}, {un} = {−1,−1,−1, . . .}, and {vn} = {1, 1, 1, . . .}. Then A({xn}) = AC({xn}) ={0}, A({un}) = {−1}, and A({vn}) = {1}. This means that {xn} ⇀ 0 but it does not have aΔ-limit.


Now, we extend Proposition 3.11 to the case of non-self-mappings. The proof closelyfollows the proof of Proposition 1 in [30]. For the convenience of the reader we include thedetails.

Proposition 3.14. Let C be a closed and convex subset of a complete CAT(0) spaceX and let T : C →X be an asymptotic pointwise nonexpansive mapping. Let {xn} be a bounded sequence in C such thatlimnd(xn, T(xn)) = 0 and {xn}⇀ w. Then T(w) = w.

Proof. As we have observed in the proof of Proposition 3.12, {xn} ⇀ w if and only ifAC({xn}) = {w}. By Lemma 3.3, we have A({xn}) = {w}. Since limnd(xn, T(xn)) = 0,then we have Φ(x) = lim supn→∞d(T

m(xn), x) for each x ∈ C and m ≥ 1. It follows thatΦ(Tm(x)) ≤ αm(x)Φ(x). In particular, we have Φ(Tm(w)) ≤ αm(w)Φ(w) for all m ≥ 1. Hence

lim supm→∞

Φ(Tm(w)) ≤ Φ(w). (3.14)

The (CN) inequality implies that

d

(xn,

w ⊕ Tm(w)2

)2

≤ 12d(xn,w)2 +

12d(xn, Tm(w))2 − 1

4d(w, Tm(w))2, (3.15)

for any n,m ≥ 1. By taking n → ∞, we get

Φ(w ⊕ Tm(w)

2

)2

≤ 12Φ(w)2 +

12Φ(Tm(w))2 − 1

4d(w, Tm(w))2, (3.16)

for any m ≥ 1. Since A({xn}) = {w}, we have

Φ(w)2 ≤ Φ(w ⊕ Tm(w)

2

)2

≤ 12Φ(w)2 +

12Φ(Tm(w))2 − 1

4d(w, Tm(w))2, (3.17)

for any m ≥ 1, which implies that

d(w, Tm(w))2 ≤ 2Φ(Tm(w))2 − 2Φ(w)2. (3.18)

By (3.14) and (3.18) we have limm→∞d(w, Tm(w)) = 0. Hence T(w) = w as desired.

As a consequence, we obtain the following corollary which is a generalization of [37,Proposition 3.7].

Corollary 3.15. Let C be a closed and convex subset of a complete CAT(0) space X and let T :C → X be an asymptotically nonexpansive mapping. Let {xn} be a bounded sequence in C suchthat limnd(xn, T(xn)) = 0 and Δ − limnxn = w. Then T(w) = w.


4. Hyperbolic Spaces

We begin this section by talking about hyperbolic spaces. This class contains the class ofCAT(0) spaces (see Lemma 4.4 below).

Definition 4.1 (see [41]). A hyperbolic space is a triple (X, d,W) where (X, d) is a metric spaceand W : X ×X × [0, 1] → X is such that

(W1) d(z,W(x, y, α)) ≤ (1 − α)d(z, x) + αd(z, y),(W2) d(W(x, y, α),W(x, y, β)) = |α − β|d(x, y),(W3) W(x, y, α) =W(y, x, 1 − α),(W4) d(W(x, z, α),W(y,w, α)) ≤ (1−α)d(x, y)+αd(z,w) for all x, y, z,w ∈ X, α, β ∈ [0, 1].

It follows from (W1) that, for each x, y ∈ X and α ∈ [0, 1],

d(x,W

(x, y, α

)) ≤ αd(x, y), d(y,W

(x, y, α

)) ≤ (1 − α)d(x, y). (4.1)

In fact, we have

d(x,W

(x, y, α

))= αd

(x, y), d

(y,W

(x, y, α

))= (1 − α)d(x, y), (4.2)

since if

d(x,W

(x, y, α

))< αd

(x, y)

or d(y,W

(x, y, α

))< (1 − α)d(x, y), (4.3)

then we get

d(x, y) ≤ d(x,W(x, y, α)) + d(W(x, y, α), y)< αd

(x, y)+ (1 − α)d(x, y)

= d(x, y),

(4.4)

which is a contradiction. By comparing between (2.2) and (4.2), we can also use the notation(1 − α)x ⊕ αy for W(x, y, α) in a hyperbolic space (X, d,W).

Definition 4.2 (see [41]). The hyperbolic space (X, d,W) is called uniformly convex if for anyr > 0 and ε ∈ (0, 2] there exists a δ ∈ (0, 1] such that, for all a, x, y ∈ X,

d(x, a) ≤ rd(y, a) ≤ r

d(x, y) ≥ εr

⎫⎪⎪⎬⎪⎪⎭

=⇒ d

(12x ⊕ 1

2y, a

)≤ (1 − δ)r. (4.5)

A mapping η : (0,∞) × (0, 2] → (0, 1] providing such a δ := η(r, ε) for given r > 0 andε ∈ (0, 2] is called a modulus of uniform convexity.


Lemma 4.3 (see [41, Lemma 7]). Let (X, d,W) be a uniformly convex hyperbolic space withmodulus of uniform convexity η. For any r > 0, ε ∈ (0, 2], λ ∈ [0, 1], and a, x, y ∈ X,

d(x, a) ≤ rd(y, a) ≤ r

d(x, y) ≥ εr

⎫⎪⎪⎬⎪⎪⎭

=⇒ d((1 − λ)x ⊕ λy, a) ≤ (1 − 2λ(1 − λ)η(r, ε))r. (4.6)

Lemma 4.4 (see [41, Proposition 8]). Assume that X is a CAT(0) space. Then X is uniformlyconvex, and

η(r, ε) =ε2

8(4.7)

is a modulus of uniform convexity.

The following result is a characterization of uniformly convex hyperbolic spaces whichis an analog of Schu [42, Lemma 1.3]. It can be applied to a CAT(0) space as well.

Lemma 4.5. Let (X, d,W) be a uniformly convex hyperbolic space with modulus of convexity η, andlet x ∈ X. Suppose that η increases with r (for a fixed ε), that {tn} is a sequence in [b, c] for someb, c ∈ (0, 1), and {xn}, {yn} are sequences inX such that lim supnd(xn, x) ≤ r, lim supnd(yn, x) ≤r, and limnd((1 − tn)xn ⊕ tnyn, x) = r for some r ≥ 0. Then

limn→∞

d(xn, yn

)= 0. (4.8)

Proof. The case r = 0 is trivial. Now suppose that r > 0. If it is not the case that d(xn, yn) → 0as n → ∞, then there are subsequences, denoted by {xn} and {yn}, such that

infnd(xn, yn

)> 0. (4.9)

Choose ε ∈ (0, 1] such that

d(xn, yn

) ≥ ε(r + 1) > 0 ∀n ∈ N. (4.10)

Since 0 < b(1 − c) ≤ 1/2 and 0 < η(r, ε) ≤ 1, 0 < 2b(1 − c)η(r, ε) ≤ 1. This implies that0 ≤ 1 − 2b(1 − c)η(r, ε) < 1. Choose R ∈ (r, r + 1) such that

(1 − 2b(1 − c)η(r, ε))R < r. (4.11)

Since

lim supn

d(xn, x) ≤ r, lim supn

d(yn, x

) ≤ r, r < R, (4.12)


there are further subsequences again denoted by {xn} and {yn} such that

d(xn, x) ≤ R, d(yn, x

) ≤ R, d(xn, yn

) ≥ εR ∀n ∈ N. (4.13)

Then by Lemma 4.3 and (4.11),

d((1 − tn)xn ⊕ tnyn, x

) ≤ (1 − 2tn(1 − tn)η(R, ε))R

≤ (1 − 2b(1 − c)η(r, ε))R < r,(4.14)

for all n ∈ N. Taking n → ∞, we obtain

limn→∞

d((1 − tn)xn ⊕ tnyn, x

)< r, (4.15)

which contradicts the hypothesis.

5. Δ-Convergence Theorem

We now give an application of Corollary 3.15. The following concept for Banach spaces isdue to Schu [42]. Let C be a nonempty closed convex subset of a CAT(0) space X and letT : C → C be an asymptotically nonexpansive mapping. The Krasnoselski-Mann iterationstarting from x1 ∈ C is defined by

xn+1 = αnTn(xn) ⊕ (1 − αn)xn, n ≥ 1, (5.1)

where {αn} is a sequence in [0, 1].Recall that a mapping T from a subset C of a CAT(0) space X into itself is called

uniformly Lipschitzian [43] if there exists L > 0 such that

d(Tn(x), Tn

(y)) ≤ Ld(x, y) ∀x, y ∈ C, n ∈ N. (5.2)

In this case, we call T a uniformly L-Lipschitzian mapping. We also note from (3.9) and (5.2)that every asymptotically nonexpansive mapping is uniformly L-Lipschitzian for some L > 0.


Lemma 5.1. LetC be a nonempty convex subset of a CAT(0) spaceX and let T : C → C be uniformlyL-Lipschitzian for some L > 0, {αn} ⊂ [0, 1], and x1 ∈ C. Suppose that {xn} is given by (5.1), andset cn = d(Tn(xn), xn) for all n ∈ N. Then

d(xn, T(xn)) ≤ cn + cn−1L(

1 + 3L + 2L2)∀n ∈ N. (5.3)



Lemma 5.2. Let {an} and {bn} be two sequences of nonnegative numbers such that

an+1 ≤ (1 + bn)an ∀n ≥ 1. (5.4)

If∑∞

n=1 bn converges, then limn→∞an exists. In particular, if there is a subsequence of {an} whichconverges to 0, then limn→∞an = 0.

The following lemmas are also needed.

Lemma 5.3. Let C be a nonempty bounded closed convex subset of a complete CAT(0) spaceX and letT : C → C be asymptotically nonexpansive with a sequence {kn} in [1,∞) for which

∑∞n=1(kn −1) <

∞. Suppose that x1 ∈ C, {αn} ⊂ [0, 1], and {xn} is given by (5.1). Then limn→∞d(xn, x) exists foreach x ∈ F(T).

Proof. Let x ∈ F(T), then we have

d(xn+1, x) ≤ d(αnTn(xn) ⊕ (1 − αn)xn, x)≤ αnd(Tn(xn), x) + (1 − αn)d(xn, x)≤ αnd(Tn(xn), Tn(x)) + (1 − αn)d(xn, x)≤ αnknd(xn, x) + (1 − αn)d(xn, x)≤ [αnkn + (1 − αn)]d(xn, x)≤ [1 + αn(kn − 1)]d(xn, x).

(5.5)

Hence

d(xn+1, x) ≤ [1 + αn(kn − 1)]d(xn, x). (5.6)

Since {d(xn, x)} is bounded and∑∞

n=1(kn−1) <∞, by Lemma 5.2, we get that limn→∞d(xn, x)exists. This completes the proof.

Lemma 5.4. Let C be a nonempty bounded closed and convex subset of a complete CAT(0) spaceX and let T : C → C be asymptotically nonexpansive with a sequence {kn} in [1,∞) for which∑∞

n=1(kn − 1) < ∞ and {αn} is a sequence in [a, b] for some a, b ∈ (0, 1). Suppose that x1 ∈ C andthat {xn} is given by (5.1). Then

limn→∞

d(xn, T(xn)) = 0. (5.7)

Proof. It follows from Theorem 3.8 that T has a fixed point x ∈ C. In view of Lemma 5.3 wecan let limnd(xn, x) = c for some c in R. Since

d(Tn(xn), x) = d(Tn(xn), Tn(x)) ≤ knd(xn, x), (5.8)


for all n ∈ N, then

lim supn→∞

d(Tn(xn), x) ≤ c. (5.9)

On the other hand, since

d(xn+1, x) ≤ αnd(Tn(xn), x) + (1 − αn)d(xn, x)≤ αnd(Tn(xn), Tn(x)) + (1 − αn)d(xn, x)≤ [αnkn + (1 − αn)]d(xn, x)≤ knd(xn, x),

(5.10)

then

d(xn+1, x) ≤ d(αnTn(xn) ⊕ (1 − αn)xn, x) ≤ knd(xn, x). (5.11)

Hence

limn→∞

(d(αnTn(xn) ⊕ (1 − αn)xn, x)) = c. (5.12)

By Lemma 4.5, we have limn→∞d(Tn(xn), xn) = 0. As we have observed, every asymptoticallynonexpansive mapping is also uniformly L-Lipschitzian for some L > 0; it follows fromLemma 5.1 that limn→∞d(T(xn), xn) = 0. This completes the proof.

Lemma 5.5 (see [26]). If {xn} is a bounded sequence in a complete CAT(0) space with A({xn}) ={x}, {un} is a subsequence of {xn} withA({un}) = {u}, and the sequence {d(xn, u)} converges, thenx = u.

Lemma 5.6. Let C be a closed convex subset of a complete CAT(0) space X and let T : C → Xbe an asymptotically nonexpansive mapping. Suppose that {xn} is a bounded sequence in C suchthat limnd(xn, T(xn)) = 0 and d(xn, v) converges for each v ∈ F(T), then ωw(xn) ⊂ F(T). Hereωw(xn) =

⋃A({un})where the union is taken over all subsequences {un} of {xn}. Moreover,ωw(xn)

consists of exactly one point.

Proof. Let u ∈ ωw(xn), then there exists a subsequence {un} of {xn} such that A({un}) = {u}.By Lemmas 3.5 and 3.3 there exists a subsequence {vn} of {un} such that Δ − limnvn = v ∈ C.By Corollary 3.15, we have v ∈ F(T). By Lemma 5.5, u = v. This shows that ωw(xn) ⊂ F(T).Next, we show that ωw(xn) consists of exactly one point. Let {un} be a subsequence of {xn}with A({un}) = {u} and let A({xn}) = {x}. Since u ∈ ωw(xn) ⊂ F(T), {d(xn, u)} converges.By Lemma 5.5, x = u. This completes the proof.

We are now ready to prove our main result.

Theorem 5.7. Let C be a bounded closed and convex subset of a complete CAT(0) space X and let T :C → C be asymptotically nonexpansive with a sequence {kn} ⊂ [1,∞) for which

∑∞n=1(kn − 1) <∞.


Suppose that x1 ∈ C and {αn} is a sequence in [a, b] for some a, b ∈ (0, 1). Then the sequence {xn}given by (5.1) Δ-converges to a fixed point of T.

Proof. It follows from Theorem 3.8 that F(T) is nonempty. By Lemma 5.3, {d(xn, v)} isconvergent for each v ∈ F(T). By Lemma 5.4, we have limn→∞d(xn, T(xn)) = 0. ByLemma 5.6, ωw(xn) consists of exactly one point and is contained in F(T). This shows that{xn} Δ-converges to an element of F(T).

Acknowledgments

The second author was supported by the Research Administration Center, Chiang MaiUniversity, Chiang Mai, Thailand. This work is dedicated to Professor W. A. Kirk.

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Research ArticleOn Some Properties of Hyperconvex Spaces

Marcin Borkowski,1 Dariusz Bugajewski,2 and Dev Phulara3

1 Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87,61-614 Poznan, Poland

2 Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore,MD 21251, USA

3 Department of Mathematics, Howard University, 2400 Sixth Street, NW, Washington, DC 20059, USA

Correspondence should be addressed to Marcin Borkowski, [email protected]

Received 13 September 2009; Accepted 13 January 2010


Copyright q 2010 Marcin Borkowski et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We are going to answer some open questions in the theory of hyperconvex metric spaces. Weprove that in complete R-trees hyperconvex hulls are uniquely determined. Next we show thathyperconvexity of subsets of normed spaces implies their convexity if and only if the spaceunder consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for R-trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse itsparticular cases to investigate hyperconvexity via measures of noncompactness.

1. Introduction

It is hard to believe that although hyperconvex metric spaces have been investigated formore that fifty years, some basic questions in their theory still remain open (let us recallthat hyperconvex metric spaces were introduced in [1] (see also [2]), but from formal pointof view it has to be emphasized that the notion of hyperconvexity was investigated earlierby Aronszajn in his Ph.D. thesis [3] which was never published). The main purpose of thispaper is to answer some of these questions.

Let us begin with the notion of hyperconvex hull which was introduced by Isbellin [4] (see Definition 2.7). This notion is more difficult to investigate than the classicalnotion of convex hull, since the former one is not uniquely determined (see Proposition 2.8).In Section 3 we are going to prove that in hyperconvex metric spaces with the uniquemetric segments property, hyperconvex hulls are uniquely determined. Let us recall thatsuch hyperconvex spaces were characterized by Kirk (see [5]) as complete R-trees (seeTheorem 2.15). This led to a surprising application of the theory of hyperconvex spaces tograph theory (see [6]).


Another interesting question is about the relation between the notion of convexity andhyperconvexity (cf. Remark 4.1). In particular, it is inspired by the following Sine’s remark[7, page 863], stated without a proof: “The term hyperconvex does have some unfortunateaspects. First, a hyperconvex subset of even R

2 (with the l∞ norm) need not be convex. Alsoconvex sets can fail to be hyperconvex (but for this one must go to at least R

3).” It turns outthat all hyperconvex subsets of a given normed space are convex if and only if the space inquestion is strictly convex; this fact is proved in Section 4.

In Section 5 we turn our attention to the classical Krein-Milman theorem (see [8]).We prove that a bounded complete R-tree is a convex hull of its extremal points (note that asimilar result, but with the assumption of compactness, is proved in [9]). Hence, in particular,such a property holds for bounded hyperconvex metric spaces with unique metric segments.

Let us denote by α and β the Kuratowski and Hausdorff measures of noncompactness,respectively, (see [10, 11] for the definition and basic properties). It was noticed by Espınola(see [12]) that if a metric space is hyperconvex, then α(A) = 2β(A) for all its bounded subsetsA. The question is about the inverse implication. More precisely, assume that α(A) = 2β(A)for every bounded subset of a given metric space X. Does this equality imply that X ishyperconvex? (Obviously, we mean nontrivial cases, i.e., we exclude spaces in which everybounded set is relatively compact.) In Sections 6 and 7 we introduce a few metric spaceswhich are not hyperconvex, but α(A) = 2β(A) for all their bounded subsets. Hence the answerto the above question is negative. Let us emphasize that the metrics considered in Sections 6and 7 are extensions and generalizations of commonly known radial metric and river metric,which were proved in [13] to be hyperconvex.

Let us notice that in general it is not easy to provide explicit formulae which wouldallow to evaluate the measures of noncompactness in particular spaces. We are going to statesuch formulae for the metric spaces considered in Sections 6 and 7.

Let us emphasize that another motivation to consider those metrics comes from thereal world. Let us consider an example of the transmission of phone signals, when one person(say, v1) calls another (say, v2), assuming there are two base transceiver stations (say, A andB). We may have two cases. If v1 and v2 are in the range of one of the BTS’s, say A, then thesignal is first transmitted from v1 to A and then from A to v2—even if v1 and v2 are “close”to each other. If v1 and v2 are located in the ranges of A and B, respectively, then the signalis transmitted from v1 to A, then from A to B and finally from B to v2. Hence we have themetric considered in Definition 7.4.

In Section 8 we provide a general scheme to construct metrics similar to these ofSections 6 and 7. This scheme is a generalization of a construction from [14].

For completeness, in Section 2 we collect some basic definitions and facts used in thesequel.

2. Preliminaries

In what follows we will denote the Euclidean metric on Rn by ρ and a “maximum” norm on

any suitable space by ‖ · ‖∞.Let us begin with some classical definitions and facts.

Definition 2.1. Let (X, d) be a metric space. We call a set S ⊂ X a metric segment (joining thepoints p, q ∈ X) if there exists an isometric embedding i : [0, d(p, q)] → X such that i(0) = pand i(d(p, q)) = q.


Definition 2.2 (see [1, page 410, Definition 1]). We call a metric space (X, d) hyperconvex, if anyfamily of closed balls {B(xi, ri)}i∈I with centers at xi’s and radii of ri’s, respectively, such thatd(xi, xj) ≤ ri + rj for any i, j ∈ I has a nonempty intersection.

Hyperconvex spaces possess—among others—the following properties.

Proposition 2.3 (see [1, page 417, Theorem 1’]). A hyperconvex space is complete.

Proposition 2.4 (see [1, page 423, Theorem 9]). A nonexpansive retract (i.e., a retract by anonexpansive retraction) of a hyperconvex space is hyperconvex.

Proposition 2.5 (see [1, page 422, Corollary 4]). Each hyperconvex metric space is an absolutenonexpansive retract, that is, it is a nonexpansive retract of any metric space it is isometricallyembedded in. In particular, hyperconvex spaces are absolute retracts.

The following theorem gives a characterization of hyperconvex real Banach spaces.

Theorem 2.6 (Nachbin-Kelley, see [15, 16]). A real Banach space is hyperconvex if and only ifit is isometrically isomorphic to some space CR(K) of all real continuous functions on a Hausdorff,compact and extremally disconnected topological spaceK with the “sup′′ norm.

Now let us state the definition of a hyperconvex hull. We will not need the generalversion of this notion, investigated by Isbell in [4]; instead, the notion of a hyperconvex hullof a subset of a hyperconvex space will suffice for our considerations.

Definition 2.7 (see, e.g., [17, page 408]). Let A ⊂ H be a nonempty subset of a hyperconvexspace H. We call B ⊂ H a hyperconvex hull of A (inH) if A ⊂ B, the set B is hyperconvex (as ametric subspace) and there exists no hyperconvex B′ ⊂ H such that A ⊂ B′ � B.

A hyperconvex hull always exists, but needs not to be unique. It is, however, uniqueup to an isometry. To be more precise, the following holds.

Proposition 2.8 (cf. [17, page 408, Proposition 5.6]). Each nonempty subset of a hyperconvexmetric space possesses a hyperconvex hull. If (X, dX) and (Y, dY ) are hyperconvex spaces, AX ⊂ X,AY ⊂ Y are isometric and i : AX → AY is an isometry, then for any hyperconvex hulls HX ⊂ X,HY ⊂ Y of AX and AY , respectively, the isometry i extends to an isometry ı : HX → HY .

In what follows, we will also need the definitions of total and strict convexity.

Definition 2.9 (see, e.g. [1, page 407] and [18, page 6, Definition 2.1]). A metric space (X, d)is called totally convex if for any two points p, q ∈ X and for all α, β ∈ [0, 1] such that α + β = 1there exists a point r ∈ X satisfying the equalities d(p, r) = αd(p, q) and d(r, q) = βd(p, q).If this point is unique for all possible combinations of p, q, α, β, we call the space X strictlyconvex and denote this point by αp + βq.

Remark 2.10 (see [1, page 410]). A hyperconvex space is totally convex.

Remark 2.11 (see, e.g., [18, page 7]). For normed spaces, the above definition of strictconvexity (Definition 2.9) coincides with the usual one.

Proposition 2.12 (see, e.g., [18, page 7]). In a strictly convex metric space, intersection of anyfamily of totally convex subsets is itself totally convex.


The above proposition lets us define the notion of a convex hull in any strictly convexmetric space in a natural way.

Definition 2.13. Let A be a nonempty subset of a strictly convex metric space X. The convexhull of A (in X) is the set

convXA :=⋂{

C ⊂ X | A ⊂ C and the subspace C is totally convex}. (2.1)

When the underlying space X is obvious from the context, we will usually write convAinstead of convXA.

Now, let us recall the definition of an R-tree.

Definition 2.14 (see, e.g., [5, page 68, Definition 1.2]). A metric space (T, d) is called an R-tree,if the following conditions are satisfied:

(1) any two points p, q ∈ T are joined by a unique metric segment (denoted by [p, q]d);

(2) if p, q, r ∈ T and [p, q]d ∩ [q, r]d = {q}, then [p, q]d ∪ [q, r]d = [p, r]d;

(3) for any p, q, r ∈ T there exists s ∈ T such that [p, q]d ∩ [p, r]d = [p, s]d.

(Let us note that (3) follows from (1); it is, however, useful to have it among the basicproperties of R-trees.) We will also use the notation (p, q)d := [p, q]d \ {p, q} for an open metricsegment joining p and q and (p, q]d := [p, q]d \ {p} for a left-open one.

Theorem 2.15 (see [5, Theorem 3.2]). For a metric spaceX the following conditions are equivalent:

(1) X is a complete R-tree;

(2) X is hyperconvex and any two points in X are joined by a unique metric segment.

In what follows, we will also use the classical notions of Chebyshev subset of a metricspace, a metric projection onto such a set C (which we will denote by PC), Kuratowski andHausdorff measures of noncompactness (which we will denote by α and β, resp.), and theradial and river metrics (which we will denote by dr and dri, resp.). The reader may find therelevant definitions, for instance, in the papers [11, 19, 20].

3. R-Trees

Let us begin this section with the following three simple propositions, which will enable usto characterize R-trees as exactly these hyperconvex spaces in which hyperconvex hulls areunique.

Proposition 3.1. A hyperconvex hull of a two-point subset {p, q} of a hyperconvex metric space is ametric segment joining p and q.

Proof. It is enough to consider {p, q} as a subset of R and apply the uniqueness (up toisometry) of hyperconvex hulls (Proposition 2.8).

Proposition 3.2. R-trees are strictly convex.


Proof. Let (T, d) be an R-tree. Assume that x, y ∈ T , α, β ≥ 0, α + β = 1, z1, z2 ∈ T , z1 /= z2 andd(x, zi) = αd(x, y), d(zi, y) = βd(x, y) for i ∈ {1, 2}. Then [x, z1]d /= [x, z2]d, [z1, y]d /= [z2, y]d.But we have [x, zi]d ∩ [zi, y]d = {zi} for i ∈ {1, 2} and therefore [x, y]d = [x, z1]d ∪[z1, y]d /= [x, z2]d ∪ [z2, y]d = [x, y]d, which is a contradiction.

Proposition 3.3. For a subset A of an R-tree, the following conditions are equivalent:

(1) A is hyperconvex;

(2) A is closed and totally convex.

Proof. For (1)⇒(2), it is enough to use Proposition 2.3 and Remark 2.10. On the other hand, ifa subset C of an R-tree T is closed and totally convex, it is a complete sub-R-tree of T . Indeed,it is enough to show that for each p, q ∈ C, the metric segment [p, q]d ⊂ C. But in view of thestrict convexity of T , we have [p, q]d = {αp + βq | α, β ≥ 0, α + β = 1} ⊂ C. Now, in view ofTheorem 2.15, C is hyperconvex.

A natural question to ask is: in which hyperconvex metric spaces the hyperconvexhulls are unique? The following theorem answers this question.

Theorem 3.4. Let (H,d) be a hyperconvex metric space. The following conditions are equivalent:

(1) for each A ⊂ H there exists exactly one hyperconvex hull of A inH;

(2) H is an R-tree.

Proof. Necessity follows easily from Proposition 3.1 and Theorem 2.15. Sufficiency. Let A bea subset of an R-tree H. Notice that Σ := {B ⊂ H | A ⊂ B, B hyperconvex} = {B ⊂ H |A ⊂ B, B closed and totally convex}. Using Propositions 3.2, 2.12 and 3.3, we arrive at theconclusion that

⋂Σ is the hyperconvex hull of A in H.

4. Normed Spaces

In the first part of this section we will give an answer to the following question: In whichspaces closed and convex subsets are hyperconvex?

Remark 4.1. Note that the question whether all closed and convex subsets of some normedspace are hyperconvex makes sense only in spaces which are themselves hyperconvex, so wewill now restrict our attention to such spaces.

Theorem 4.2 (see [21, page 474, Theorem 1]). If E is a two-dimensional real normed space, theneach nonempty, closed, and convex subset of E is a nonexpansive retract of E.

Corollary 4.3. Each nonempty, closed and convex subset of R2 endowed with any hyperconvex norm

is hyperconvex.

Remark 4.4. Notice that “any hyperconvex norm on R2” means essentially (i.e., up to an

isometric isomorphism) the maximum norm; this follows from Theorem 2.6 and can also beproved using a geometric argument (see [19, Theorem 4.1]).

Theorem 4.5. Let E be a hyperconvex normed space. If E is not isometrically isomorphic to R1 or

(R2, ‖ · ‖∞), then there exists a two-dimensional linear subspace of E which is not hyperconvex.


Proof. Since E is not isometrically isomorphic to R1, its dimension must be at least 2. Further,

since the only (up to an isometric isomorphism) two-dimensional hyperconvex space is(R2, ‖ · ‖∞), we may assume dimE ≥ 3. By Theorem 2.6 we may assume that E is the spaceCR(K) for some Hausdorff, compact and extremally disconnected topological space K. SincedimE ≥ 3, the space K has at least three points, so CR(K) includes a copy of (R3, ‖ · ‖∞). Thismeans that it is enough to prove the theorem in case of E = R

3 with the “maximum” norm.For simplicity, we will construct an affine non-hyperconvex subspace of E; by an

appropriate translation one can obtain a linear one. Let V := {(x1, x2, x3) ∈ E | x1 +x2 + x3 = 1}. Consider the following three balls in V : BV ((−1, 1, 1), 1), BV ((1,−1, 1), 1),BV ((1, 1,−1), 1). Since the corresponding balls in E intersect only at (0, 0, 0)/∈V, the spaceV is not hyperconvex.

Corollary 4.3 and Theorem 4.5 yield the following characterization.

Corollary 4.6. Let E be a real normed space. The following conditions are equivalent:

(1) each nonempty, closed, and convex subset of E is hyperconvex;

(2) E is isometrically isomorphic to R1 or (R2, ‖ · ‖∞).

We will now turn our attention to the problem of describing the spaces in whichhyperconvexity implies convexity. We will start with an observation suggested to us byGrzybowski [22].

Proposition 4.7. If a real normed space E is strictly convex, then all its hyperconvex subsets areone-dimensional.

Proof. Let A ⊂ E be at least two-dimensional. Therefore there exist three noncollinear pointsa, b, c ∈ A. Put p := (1/2)(‖a − b‖ + ‖b − c‖ + ‖a − c‖) and let ra := p − ‖b − c‖, rb := p − ‖a − c‖,rc := p − ‖a − b‖. It is clear that ‖a − b‖ = ra + rb and similarly for other distances. But Eis strictly convex, so we have BE(a, ra) ∩ BE(b, rb) = {(rb/(ra + rb))a + (ra/(ra + rb))b} andBE(a, ra)∩BE(c, rc) = {(rc/(ra + rc))a+ (ra/(ra + rc))c}, so BE(a, ra)∩BE(b, rb)∩BE(c, rc) = ∅.It must be therefore BA(a, ra) ∩ BA(b, rb) ∩ BA(c, rc) = ∅, which finishes the proof.

Corollary 4.8. If a real normed space E is strictly convex, then all its hyperconvex subsets are convex.

Proof. From Proposition 4.7 we know that hyperconvex subsets of E are one dimensional;but from Proposition 2.5 we infer that hyperconvex sets are connected, which for one-dimensional sets is equivalent to their convexity.

To prove the inverse implication, we will need a simple lemma.

Lemma 4.9 (see [23, page 44, Lemma 15.1]). Let X be a metric space and a, b, c ∈ X be such thatd(a, c) + d(c, b) = d(a, b). If there exist metric segments: Sac, joining the points a and c and Scb,joining the points c and b, then Sac ∪ Scb is a metric segment joining the points aand b.

Now we are ready to prove the following theorem.

Theorem 4.10. If all hyperconvex subsets of a real normed space E are convex, then E is strictlyconvex.


Proof. Assume that E is not strictly convex; we will construct a nonconvex, hyperconvexsubset of E. There exist points a, b, c1, c2 ∈ E and positive numbers α, β such that c1 /= c2,α + β = 1 and the equalities d(a, c1) = d(a, c2) = αd(a, b) and d(c1, b) = d(c2, b) = βd(a, b)hold. From Lemma 4.9, both sets [a, c1] ∪ [c1, b] and [a, c2] ∪ [c2, b], where [x, y] meansan affine segment with endpoints x, y, are metric segments joining a and b (and hencehyperconvex sets). They cannot be, however, both convex, so at least one of them is thedesired counterexample.

Again, combining Corollary 4.8 and Theorem 4.10, we obtain the following character-ization of strictly convex normed spaces.

Theorem 4.11. A normed space is strictly convex if and only if each its hyperconvex subset is convex.

5. Krein-Milman Type Theorem

In this short section, we will show that a Krein-Milman type theorem holds for R-trees. Itturns out that instead of compactness we only need a weaker boundedness condition.

For completeness, let us state the definition of an extremal point in the setting of R-trees.

Definition 5.1. Let X be a subset of an R-tree T . We call a point x ∈ X an extremal point of X ifno open metric segment included in X contains x.

Theorem 5.2. A complete and bounded R-tree is a convex hull of the set of its extremal points.

Proof. It is enough to show that each point of X lies on a metric segment joining some twoextremal points of X. Let x ∈ X. We may assume that x is not extremal; let x ∈ (a, b)d. Thefamily of all metric segments having x as one of its endpoints satisfies the assumptions ofthe Kuratowski-Zorn lemma. Let [x, c]d ⊃ [x, a]d and [x, d]d ⊃ [x, b]d be maximal metricsegments containing the respective given metric segments. We will first show that c and d areextremal points.

If, say, c were not extremal, we would have c ∈ (e, f)d for some e, f ∈ X, e /= f . Let[c, x]d ∩ [c, e]d = [c, e′]d and [c, x]d ∩ [c, f]d = [c, f ′]d. If e′ /= c /= f ′, we would have (c, e′]d ⊂(c, x]d and (c, f ′]d ⊂ (c, x]d, so c /∈ [e′, f ′]d; but [c, e′]d ∩ [c, f ′]d ⊂ [c, e]d ∩ [c, f]d = {c}, so[c, e′]d ∪ [c, f ′]d = [e′, f ′]d—contradiction. This means that c = e′ or c = f ′; assume c = e′.Now [c, x]d ∩ [c, e]d = {c}, so [c, x]d ∪ [c, e]d = [x, e]d, which contradicts the maximality of[x, c]x.

Now let us show that x ∈ [c, d]d. We will prove that [x, c]d ∩ [x, d]d = {x}. Assume[x, c]d ∩ [x, d]d = [x, y]d and x /=y. Let ε = min{d(x, a), d(x, b)}. Choose w ∈ (x, y]dsuch that d(x,w) < ε. We have [x,w]d ⊂ [x, c]d and hence [x,w]d ⊂ [x, a]d; analogously,[x,w]d ⊂ [x, b]d. This means that w ∈ [x, a]d ∩ [x, b]d and w/=x; but [x, a]d ∩ [x, b]d = {x}—contradiction.

Since closed and convex subsets of an R-tree are hyperconvex (Proposition 3.3),Corollary 4.6 might give the impression that R-trees are somehow similar to 1- or 2-dimensional vector spaces and that completeness and boundedness of an R-tree imply itscompactness. As the following example shows, this analogy is misleading.

Example 5.3. Let T be R2 with the radial metric. It is easy to see that X is an R-tree and so is

BX((0, 0), 1), which is both complete and bounded, but not compact.


6. Hyperconvexity and Measures of Noncompactness

Let us begin this section with the following definition.

Definition 6.1. Let A ∈ R2 be some point in the Euclidean plane. Let us define a function

dAr : R2 × R

2 → [0,+∞) as follows:

dr(v1, v2) =

⎧⎨⎩ρ(v1, A) + ρ(v2, A) if v1 /=v2,

0 if v1 = v2,(6.1)

for all v1, v2 ∈ R2. If A = (0, 0), we will write dr instead of dAr .

It is easy to prove the following lemma.

Lemma 6.2. (R2, dr) is a complete metric space.

We will call the function dr (resp., dAr ) introduced in Definition 6.1, the modified radialmetric (resp., centered at A).

Remark 6.3. The topology of R2 with the metric dr is strictly stronger than the topology of the

same space induced by the radial metric.

Lemma 6.4. The space R2 with the metric dr is not hyperconvex.

Proof. Let us consider two closed balls B((0, 0), 1) and B((√

2,√

2), 1). Then

dr((0, 0),

(√2,√

2))

= 2, (6.2)

but

B((0, 0), 1) ∩ B((√

2,√

2), 1)= ∅. (6.3)

This shows that the metric dr fails to be hyperconvex.

Now we are going to examine the measures of noncompactness in the space (R2, dr).For this purpose we are going to use a similar approach as in the case of the measures ofnoncompactness in R

2 with the radial metric (cf. [20, Theorem 4]). First let us introduce thefollowing definition.

Definition 6.5. Let D be a bounded subset of (R2, dr). We say that w′ ∈ R+ satisfies

(1) V ∗(D) condition, if for every w < w′, there exist infinitely many pairwise distinctpoints v ∈ D such that w < ρ(v, (0, 0)) ≤ w′;

(2) V∗(D) condition, if for every w > w′, there exist infinitely many pairwise distinctpoints v ∈ D such that w > ρ(v, (0, 0)) ≥ w′.

Let us put v∗(D) = sup{0} ∪ {w′ : w′ satisfies V ∗(D) or V∗(D)}.


Using above conditions we can prove the following theorem.

Theorem 6.6. For any bounded subset D of R2 with the metric dr we have α(D) = 2v∗(D) and

β(D) = v∗(D).

Proof. If there exists no nonnegative number w′ satisfying either V ∗(D) or V∗(D), then clearlyD consists of a finite number of points. Hence α(D) = β(D) = 0 in this case.

Now consider a bounded set D such that there exists a w′ satisfying V ∗(D) or V∗(D)condition. To prove that α(D) = 2v∗(D), let us first show that α(D) ≥ 2v∗(D). For this,consider a covering (Dj)j=1,2,...,m of D such that

maxj=1,...,m

δ(Dj

) ≤ ε (6.4)

for some ε > 0. Consider the setsAn = {v = (x, y) ∈ D :√x2 + y2 ≥ v∗(D)−1/n}, where n ∈ N.

Then for every n ∈ N there exists a jn ∈ {1, 2, . . . , m} and vn1 , vn2 ∈ D such that vn1 /=v

n2 , v

n1 , v

n2 ∈

Djn ∩ An. Since dr(vn1 , vn2 ) = ρ(vn1 , 0) + ρ(v

n2 , 0) ≥ 2v∗(D) − 2/n for every n ∈ N, ε ≥ 2v∗(D).

Hence α(D) ≥ 2v∗(D).Next we prove that β(D) ≤ v∗(D). Obviously, if

v∗(D) = sup(x,y)∈D

ρ((x, y), (0, 0)

), (6.5)

then D is contained in the closed ball of center (0, 0) and radius v∗(D). So in this case β(D) ≤v∗(D).

Let

v∗(D) < sup(x,y)∈D

ρ((x, y), (0, 0)

), (6.6)

then according to Definition 6.5, for every ε > 0, there exist at most finitely many points(x, y) ∈ D with the property ρ((x, y), (0, 0)) > v∗(D) + ε. Hence β({(x, y) ∈ D :ρ((x, y), (0, 0)) > v∗(D) + ε}) = 0. Moreover,

β({(

x, y) ∈ D : ρ

((x, y), (0, 0)

) ≤ v∗(D) + ε}) ≤ v∗(D) + ε. (6.7)

Since ε > 0 is arbitrary, we get β(D) ≤ v∗(D) in this case. Finally, we get v∗(D) ≤ (1/2)α(D) ≤β(D) ≤ v∗(D). This implies α(D) = 2v∗(D) and β(D) = v∗(D).

Example 6.7. Using the previous formulae, we can calculate that in (R2, dr) we haveα(B((0, 0), 1)) = 2β(B((0, 0), 1)) = 2; in particular, the closed unit ball is noncompact.

Remark 6.8. It is known (see [12, page 135] for the details) that if a space is hyperconvex, thenfor any of its bounded subset D, the following equality holds

α(D) = 2β(D). (6.8)


The above theorem shows that even in the nontrivial cases (i.e., in cases, when bounded setsare not necessarily relatively compact), the above equality does not have to imply that thespace in question is hyperconvex.

Definition 6.1 can be slightly modified. Namely, let us introduce the followingdefinition.

Definition 6.9. Let us define a function dr : R2 × R


dr(v1, v2) =

⎧⎨⎩|x1| +

∣∣y1∣∣ + |x2| +

∣∣y2∣∣ if v1 /=v2,

0 if v1 = v2,(6.9)

for all v1 = (x1, y1), v2 = (x2, y2) ∈ R2.

Remark 6.10. It can be easily checked that (R2, dr) is a complete metric space. Its topology isalso stronger than the topology of R

2 with the radial metric. On the other hand this topologyis obviously equivalent to the topology induced by the metric dr .

Lemma 6.11. The space R2 with the metric dr is not hyperconvex.

Proof. Let us consider two closed balls B((0, 0), 1) and B((2, 0), 1). Then

dr((0, 0), (2, 0)) = 2 but B((0, 0), 1) ∩ B((2, 0), 1) = ∅. (6.10)

It shows that the metric dr fails to be hyperconvex.

For the measures of noncompactness in the space of bounded subsets in the space(R2, dr) we have similar formulas to those given in Theorem 6.6.

Definition 6.12. Let D be a bounded subset of R2 with the metric dr . We say that w′ ∈ R+

satisfies

(1) U∗(D) condition, if for every w < w′, there exist infinitely many pairwise distinctpoints u = (ux, uy) ∈ D such that w < |ux| + |uy| ≤ w′;

(2) U∗(D) condition, if for every w > w′, there exist infinitely many pairwise distinctpoints u = (ux, uy) ∈ D such that w > |ux| + |uy| ≥ w′.

Let us put u∗(D) = sup{0} ∪ {w′ : w′ satisfies U∗(D) or U∗(D)}.

Theorem 6.13. For any bounded subset D of R2 with the metric dr one has α(D) = 2u∗(D) and

β(D) = u∗(D).

The proof of Theorem 6.13 is similar to the proof of Theorem 6.6 and therefore we omitit.

The metric we are going to consider to the end of this section is, roughly speaking, likebetween the radial metric and the river metric. We will call it a modified river metric.


Definition 6.14. Let A = (ax, ay) ∈ R2. Define a function dAri : R

2 × R2 → [0,+∞) as follows:

dAri (v1, v2) =

⎧⎨⎩∣∣y1 − y2

∣∣, if x1 = x2,

|x1 − ax| +∣∣y1 − ay

∣∣ + |x2 − ax| +∣∣y2 − ay

∣∣, otherwise,(6.11)

for all v1 = (x1, y1), v2 = (x2, y2) ∈ R2. If A = (0, 0), we will write dmri instead of dAri .

The following fact can be easily checked.

Lemma 6.15. (R2, dmri ) is a complete metric space.

Remark 6.16. The topology of (R2, dmri ) is strictly stronger than the topology of R2 induced by

the river metric.

It is interesting to consider a closed ball B((a, b); r) ⊂ (R2, dmri ), where a ∈ R \ {0} and|a|+ |b| < r < 2|a|+ |b|. Such a ball consists of two disjoint closed sets (a square and a segment)which, in particular, means that it is not connected.

Lemma 6.17. The space R2 with the metric dmri is not hyperconvex.

Proof. Let us consider two closed balls B1((1, 1), 3/2) and B2((0, 0), 1/2). Thendmri ((0, 0), (1, 1)) = 2 but B2((0, 0), 1/2) ∩ B1((1, 1), 3/2) = ∅. This shows that (R2, dmri ) isnot hyperconvex.

To evaluate the measures of noncompactness of any bounded subset of (R2, dmri ) onecan use a similar approach as in the case of (R2, dr) (cf. Definition 6.12 and Theorem 6.13).

In connection with Remark 6.8 let us notice that (R2, dr) as well as (R2, dmri ) are alsoexamples of metric spaces such that α(D) = 2β(D) for any bounded subset D ⊂ (R2, dr) orD ⊂ (R2, dmri ), but those spaces are not hyperconvex.

7. Generalized Modified Radial and River Metrics

The metric spaces (R2, dr) as well as (R2, dri) are special cases of a general constructionprovided in [19]. More precisely, let E be a normed space and C ⊂ E its Chebyshev subset.

Definition 7.1. Let C ⊂ E be a Chebyshev set in a normed space E and let dC be any metricdefined on C. Let us define d : E × E → [0,+∞) by the formula

d(x, y)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

∥∥x − y∥∥, if PC(x) = PC(y), and x, PC(x), y are collinear,

‖x−PC(x)‖+dC(PC(x), PC

(y))

+∥∥PC(y)−y∥∥, otherwise.

(7.1)


The above defined function d is a metric (see [19, Lemma 3.1]). Now, the followingquestion can be risen. Is it possible to consider two disjoint Chebyshev sets, instead of oneChebyshev set C, in such a way to get a variant of the metric defined above? The followingtwo examples show that in the case of classical hyperconvex metrics: the radial metric as wellas the river metric, this problem seems not to be easy.

Example 7.2. LetAB be a fixed segment in R2 and L the perpendicular bisector ofAB dividing

the whole plane R2 into two open half-planes II � A and I � B. Let us define a function

d : R2 × R


d(v1, v2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

dAr (v1, v2) if v1, v2 ∈ II ∪ L,

dBr (v1, v2) if v1, v2 ∈ I ∪ L,

ρ(v1, A) + ρ(A,B) + ρ(B, v2) if v1 ∈ II, v2 ∈ I,

ρ(v2, A) + ρ(A,B) + ρ(B, v1) if v1 ∈ I, v2 ∈ II,

(7.2)

for all v1, v2 ∈ R2, where dAr , dBr are the radial metrics on the plane centered at A and B,

respectively. Then this d is not a metric. Indeed it does not satisfy the triangle inequality inthe following case.

Let us consider three points v1, v2, v3 ∈ R2 such that v1 ∈ II, v2 ∈ L, v3 ∈ I; v2, v1,

and A are collinear; v2, v3, and B are collinear; ρ(v2, v1) < ρ(v1, A) and ρ(v2, v3) < ρ(v3, B).Then d(v1, v2) + d(v2, v3) < d(v1, v3).

Example 7.3. Let A := (−a, 0) and B := (a, 0), where a > 0, be two points in R2. Let L be the

perpendicular bisector of AB; it divides the whole plane R2 into two open half-planes II � A

and I � B. Let us define a function d : R2 × R


d(v1, v2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

dri(v1, v2) if v1, v2 ∈ I ∪ Lor v1, v2 ∈ II ∪ L,

dri(v1, A) + ρ(A,B) + dri(B, v2) if v1 ∈ II, v2 ∈ I,dri(v2, A) + ρ(A,B) + dri(B, v1) if v1 ∈ I, v2 ∈ II,

(7.3)

for all v1, v2 ∈ R2, where dri denotes the river metric. Then this d is not a metric. Indeed, it

does not satisfy the triangle inequality in the following case. LetA = (−2, 0), B = (2, 0), and letus take three points v1 = (−1, 1), v2 = (0, 0), v3 = (1, 1). Then, by the definition d(v1, v2) = 2and d(v2, v3) = 2 but d(v1, v3) = 8, which shows d(v1, v2) + d(v2, v3) < d(v1, v3).

However, it appears that all the metrics introduced in Section 6 (Definitions 6.1, 6.9and 6.14) are appropriate to define new metrics using the idea described at the beginning ofthis section.

Let us begin with the following definition.


Definition 7.4. Let AB be a segment in R2 and L be the perpendicular bisector of AB dividing

the whole plane R2 into two open half-planes II � A and I � B. Let us define a function

d1 : R2 × R


d1(v1, v2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dAr (v1, v2), if v1, v2 ∈ II ∪ L,

dBr (v1, v2), if v1, v2 ∈ I ∪ L,

ρ(v1, A) + ρ(A,B) + ρ(B, v2), if v1 ∈ II, v2 ∈ I,

ρ(v2, A) + ρ(A,B) + ρ(B, v1), if v1 ∈ I, v2 ∈ II,

(7.4)

for all v1, v2 ∈ R2, where dAr and dBr denote modified radial metrics centered at A and B,

respectively.

Let us note that if v1 and v2 both are in L, then d1(v1, v2) = dAr (v1, v2) = dBr (v1, v2), sod1 is well-defined.

Lemma 7.5. (R2, d1) is a complete metric space.

Proof. It is easy to check that d1 is a metric. Now to verify that it is complete, let us considera Cauchy sequence vn in the space (R2, d1). Then there exists N ∈ N such that for all n ≥ N,the points vn belong to the same closed half-plane I ∪ L or II ∪ L. Hence, by Lemma 6.2, (vn)is convergent, which completes the proof.

Remark 7.6. It is clear that the topologies of R2 induced by the metric d1 and the modified

radial metric are not comparable.

Lemma 7.7. The space R2 with the metric d1 is not hyperconvex.

Proof. For convenience consider A = (1, 0), B = (−1, 0) and consider two closed ballsB((1, 0), 1/2) and B((1, 1), 1/2). Then d1((1, 0), (1, 1)) = 1 but B((1, 0), 1/2) ∩ B((1, 1), 1/2) =∅.

Remark 7.8. It is easy to evaluate the Kuratowski and Hausdorff measures of noncompactnessof bounded sets in R

2 with the metric d1.Indeed, let us consider a bounded set D in R

2 with this metric. Then we can write D asthe union of two sets U and V , where

U = D ∩ (I ∪ L), V = D ∩ (II). (7.5)

Then, by the maximum property of the measures of noncompactness, we get

α(D) = α(U ∪ V ) = max(α(U), α(V )). (7.6)


To evaluate α(U) and α(V ) it is enough to apply formulas similar to the one given inTheorem 6.6.

Remark 7.9. It is clear that in Definition 7.4 one can replace dAr , dBr by d

A

r , dB

r , respectively,(cf. Definition 6.9) getting again a complete metric space which is not hyperconvex.

Now, using the metric from Definition 6.14, let us introduce the following metric.

Definition 7.10. Let AB be a fixed segment in R2 parallel to the x-axis and L perpendicular

bisector of AB dividing the whole plane into two open half-planes I and II. Let us define afunction d2 : R

2 × R2 → [0,+∞) as follows:

d2(v1, v2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dAri (v1, v2) if v1, v2 ∈ II ∪ L,

dBri(v1, v2) if v1, v2 ∈ I ∪ L,

dAri (v1, A) + ρ(A,B) + dBri(B, v2) if v1 ∈ II, v2 ∈ I,

dBri(v1, B) + ρ(A,B) + dAri (A, v2) if v1 ∈ I, v2 ∈ II,

(7.7)

for all v1, v2 ∈ R2, where dAri and dBri denote the metrics from Definition 6.14.

One can prove the following lemma.


The proof of this Lemma is similar to the proof of Lemma 7.5 and therefore we omit it.

Remark 7.12. The metric d2 is a variant of the metric dmri defined in Definition 6.14.The topologies induced by these metrics are not comparable. The space (R2, d2) is nothyperconvex, either. Finally, to find the Kuratowski and the Hausdorff measures ofnoncompactness of bounded sets in R

2 with the metric d2, it is enough to use the sameapproach as in Remark 7.8.

In Definitions 7.4 and 7.10, we considered two Chebyshev sets. Now one can think ofthe following question. Is it possible to increase the number of suitably chosen Chebyshevsets? The answer is “yes.” Let us introduce the following definition.

Definition 7.13. Let us consider the square ABCD in R2 with vertices: A := (a, a), B := (−a, a),

C := (−a,−a), D := (a,−a), where a > 0. Denote L1 := {(x, y) ∈ R2 | y = 0}, L2 := {(x, y) ∈ R

2 |x = 0}, L+

1 := {(x, y) ∈ L1 | x ≥ 0}, L−1 := {(x, y) ∈ L1 | x ≤ 0}, L+2 := {(x, y) ∈ L2 | y ≥ 0},

L−2 := {(x, y) ∈ L2 | y ≤ 0}. Let dm be the “maximum” metric on R2. By dAB1 , dAC1 , and so

forth, we will mean a metric defined as in Definition 7.4, but using dm(A,B), dm(A,C), andso forth, instead of ρ(A,B), ρ(A,C), and so forth. Denote the four open quadrants by I � A,II � B, III � C and IV � D. Let us define a function d4 : R

2 × R2 → [0,+∞) as follows:


d3(v1, v2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dAr (v1, v2) if v1, v2 ∈ Ic,

dBr (v1, v2) if v1, v2 ∈ IIc,

dCr (v1, v2) if v1, v2 ∈ IIIc,

dD1 (v1, v2) if v1, v2 ∈ IV c,

dAB1 (v1, v2) if v1 ∈ I, v2 ∈ II or vice versa,

dBC1 (v1, v2) if v1 ∈ II, v2 ∈ III or vice versa,

dCD1 (v1, v2) if v1 ∈ III, v2 ∈ IV or vice versa,

dDA1 (v1, v2) if v1 ∈ IV, v2 ∈ I or vice versa,

dAC1 (v1, v2) if v1 ∈ I, v2 ∈ III or vice versa,

dBD1 (v1, v2) if v1 ∈ II, v2 ∈ IV or vice versa,

dAr (v1, A) + dm(A,B) + dBr (B, v2),

or dAr (v1, A) + dm(A,C) + dCr (C, v2),

if v1 ∈ I, v2 ∈ L1−,

dDr (v1, D) + dm(D,B) + dBr (B, v2),

or dDr (v1, D) + dm(D,C) + dCr (C, v2),

if v1 ∈ IV, v2 ∈ L1−,

(7.8)

and eight more similar expressions involving L+1 , L+

2 , and L−2 for all v1, v2 ∈ R2, where

Ic, IIc, IIIc and IV c denote the closed quadrants and dAr , dBr , d

cr , and dDr denote the

modified radial metrics defined in Definition 6.1.


Proof. To prove that d3 is a metric on R2 is straightforward, although quite long, so we omit

this proof. To prove that (R2, d3) is complete, let us consider a Cauchy sequence (vn) in thespace (R2, d3). Then for every ε > 0, there exists N ∈ N such that d3(vm, vn) < ε for everym,n ≥ N. It means there exists N ∈ N such that for every n ≥ N,vn belongs to the sameclosed quadrant, because if vn and vm were in different quadrants (without loss of generalitysuppose vm ∈ I and vn ∈ II), then

d3(vm, vn) = dAr (vm,A) + dm(A,B) + dBr (B, vn). (7.9)

So, if we choose ε < dm(A,B), then d3(vn, vm) > ε which contradicts that (vn) is a Cauchysequence. Hence almost all the terms of any Cauchy sequence must be in the same closedquadrant. Thus by Lemma 6.2, (vn) is convergent, which shows that the space (R2, d3) iscomplete.


For the convenience of the reader, let us present a figure of a closed ball BB(P, r) in(R2, d3), where A = (1, 1), B = (−1, 1), C = (−1,−1), D = (1,−1), P = (a, b), a > 1, b > 1,and ρ(A,P) + dm(A,B) < r < ρ(A,P) + (3/2)dm(A,B).

Obviously, the following lemma holds.

Lemma 7.15. The space R2 with the metric d3 is not hyperconvex.

Proof. For convenience let us consider A = (1, 1), B = (−1, 1), C = (−1,−1), and D =(1,−1) and two closed balls B((1, 1), 1/2) and B((2, 1), 1/2). Then d3((1, 1), (2, 1)) = 1 butB((1, 1), 1/2) ∩ B((2, 1), 1/2) = ∅.

Remark 7.16. It is easy to evaluate the Kuratowski and Hausdorff measures of noncompact-ness of bounded sets in (R2, d3). Indeed, one can use a similar approach as in Remark 7.8.

8. Linking Construction

In this section we will give a slight generalization of the so-called linking constructiondescribed by Aksoy and Maurizi in [14] and show how this generalization includes themetrics of Section 7. Notice that a similar concept appears in [24], where it is used to studyexistence of certain mappings between Banach spaces.

Definition 8.1 (cf. [14, page 221, Theorem 2.1]). Let (X, d) be a metric space and {Wλ, dλ}λ∈Λa collection of pairwise disjoint metric spaces, each disjoint with X. Let f : Λ → X be anarbitrary function and let g : Λ → ⋃

λ∈ΛWλ be a function satisfying g(λ) ∈ Wλ for eachλ ∈ Λ. Define Wλ := Wλ \ {g(λ)} for λ ∈ Λ. Let Z := X ∪ ⋃λ∈Λ Wλ. Define the functiondZ : Z × Z → [0,+∞) by the formula

dZ(x, y)

:=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

d(x, y)

if x, y ∈ X,dλ(x, y)

if x, y ∈Wλ for some λ ∈ Λ,

dλ1

(x, g(λ1)

)+ d(f(λ1), f(λ2)

)+ dλ2

(g(λ2), y

)if x ∈ Wλ1 , y ∈ Wλ2 , λ1 /=λ2

d(x, f(λ)

)+ dλ

(g(λ), y

)if x ∈ X, y ∈ Wλ for some λ ∈ Λ,

d(y, f(λ)

)+ dλ

(g(λ), x

)if y ∈ X and x ∈ Wλ for some λ ∈ Λ.

(8.1)

Theorem 8.2 (cf. [14, page 221, Theorem 2.1]). The function dZ defined above is a metric on Z. Ifall the metric spaces X,Wλ for λ ∈ Λ are hyperconvex, then so is (Z, dZ).

Remark 8.3. The paper [14] contains the above theorem only for hyperconvex spaces. It isobvious that dZ is a metric also in the general case.


C

0

D

B A

P(a, b)

Figure 1: An example of a ball in the metric d3.

Remark 8.4. The authors of the paper [14] applied their version of Theorem 8.2 to obtain thehyperconvexity of the metric of Definition 7.1 (see [14, Theorem 2.2]). Let us notice that anidentical result was given in an earlier work [19].

Proposition 8.5. The metric dZ from Definition 8.1 is complete if all the spaces Wλ and X arecomplete.

Proof. Let (xn) be a Cauchy sequence in (Z, dZ). We will show that (xn) has a convergentsubsequence. If (xn) has infinitely many terms in Z, we are done. If (xn) has infinitely manyterms in some Wλ, it must be convergent in Wλ to some x ∈ Wλ; if x /= g(λ), the proof iscomplete, and if x = g(λ), it is easily seen that xn → f(λ) in Z as n → ∞. Therefore we mayassume that (xn) includes only a finite number (possibly zero) of points from Z and each Wλ.Define PX : Z → X by

PX(x) :=

⎧⎨⎩x if x ∈ X;

f(λ) if x ∈ Wλ for some λ ∈ Λ.(8.2)

Observe that limn→∞dZ(xn, PX(xn)) = 0; for if that were not the case, there would exista subsequence (xnk) and an ε > 0 such that each xnk would lie in different Wλ anddZ(xnk , PX(xnk)) > ε; this would mean that dZ(xnk , xnl)) > 2ε for all k, l ∈ N—contradictionwith (xn) being Cauchy.

Now notice that dZ(PX(xm), PX(xn)) ≤ dZ(xm, xn) for m,n ∈ N, so the sequence(PX(xn)) is also Cauchy and hence convergent to some x ∈ X. We have dZ(x, xn) ≤dZ(x, PX(xn)) + dZ(PX(xn), xn) → 0 as n → ∞ and the proof is complete.

Remark 8.6. To evaluate the Kuratowski and Hausdorff measures of noncompactness ofbounded sets in Z with the metric dZ, when the set Λ is finite, we use following procedure.

Let us consider a bounded set D in Z with the metric dZ. Then we can write D as thefollowing union:

D = (X ∩D) ∪(⋃

λ∈Λ

(Wλ ∩D

)). (8.3)


Then, by the maximum property of the measures of noncompactness, we get

α(D) = α

((X ∩D) ∪

(⋃λ∈Λ

(Wλ ∩D

)))

= maxλ∈Λ

{α(X ∩D), α

(Wλ ∩D

)}.

(8.4)

Example 8.7. Notice that the metric from Definition 7.4 can be obtained as a special case ofDefinition 8.1. Indeed, put X := {A,B} and Λ := R

2 \X. For each λ ∈ Λ, define

f(λ) :=

⎧⎨⎩A if λ ∈ II,B if λ ∈ I ∪ L,

(8.5)

Wλ := {f(λ), λ} × {0} for λ ∈ Λ and g(λ) := (f(λ), 0) for λ ∈ Λ.

In a similar way, other metrics from Sections 6 and 7 are special cases of Definition 8.1.As an example, let us provide a way to construct the metric dmri from Definition 6.14.

Example 8.8. Let X := R and Wλ := {λ} × R for λ ∈ Λ := R. Define the metric d : X × X →[0,+∞) by the formula

d(x, y)

:=

⎧⎨⎩|x| + ∣∣y∣∣ if x /=y;

0 if x = y.(8.6)

For each λ ∈ Λ, let dλ : Wλ×Wλ → [0,+∞) be the metric defined by dλ((λ, x), (λ, y)) := |x−y|.Further, let f : Λ → X be an identity mapping and g : Λ → ⋃

λ∈ΛWλ : λ �→ (λ, 0). It is easilyseen that applying Definition 8.1 we obtain the metric space dmri .

At the beginning of Section 7 we posed a question whether it is possible to constructa metric analogous to that from Definition 7.1, but with more than one Chebyshev subset.In all our examples, however, these subsets were singletons. Let us now show an exampleof two similar metrics constructed using two disjoint Chebyshev subsets consisting of morethan one point.

Example 8.9. Define the following two Chebyshev subsets of the Euclidean plane: C− :=conv{(−1,−1), (−1, 1)} and C+ := conv{(1,−1), (1, 1)}. Put Λ := X := C− ∪ C+. Let H− :={(x, y) ∈ R

2 | x < 0} and H+ := {(x, y) ∈ R2 | x ≥ 0}. Let P− : H− → C− and P+ : H+ → C+ be

metric projections and define P : R2 → X by the formula

P(x) :=

⎧⎨⎩P−(x) if x ∈ H−,P+(x) if x ∈ H+.

(8.7)

For each λ ∈ Λ, let Wλ := {x ∈ R2 | P(x) = λ} × {0}. Let f : Λ → X the identity map and

g : Λ → R2 × {0} be defined by g(λ) := (λ, 0) for λ ∈ Λ. The metrics on X and Wλ’s are


inherited from R2. Applying Definition 8.1 we obtain a certain metric on R

2. Let us noticethat it is not complete; taking Λ := R

2 and Wλ := {λ, P(λ)} × {0} for λ ∈ Λ, f := P and g asbefore we obtain another metric, this time complete. Let us finish by observing that since X,and hence Z, is disconnected, in both cases Z cannot be hyperconvex.

References

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[2] N. Aronszajn and P. Panitchpakdi, “Correction to: “Extension of uniformly continuous transforma-tions in hyperconvex metric spaces”,” Pacific Journal of Mathematics, vol. 7, p. 1729, 1957.

[3] N. Aronszajn, On metric and metrization, Ph.D. thesis, Warsaw University, Warsaw, Poland, 1930.[4] J. R. Isbell, “Six theorems about injective metric spaces,” Commentarii Mathematici Helvetici, vol. 39,

pp. 65–76, 1964.[5] W. A. Kirk, “Hyperconvexity of R-trees,” Fundamenta Mathematicae, vol. 156, no. 1, pp. 67–72, 1998.[6] R. Espınola and W. A. Kirk, “Fixed point theorems in R-trees with applications to graph theory,”

Topology and Its Applications, vol. 153, no. 7, pp. 1046–1055, 2006.[7] R. Sine, “Hyperconvexity and approximate fixed points,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 13, no. 7, pp. 863–869, 1989.[8] M. Krein and D. Milman, “On extreme points of regular convex sets,” Studia Mathematica, vol. 9, pp.

133–138, 1940.[9] A. G. Aksoy, M. S. Borman, and A. L. Westfahl, “Compactness and measures of noncompactness in

metric trees,” in Banach and Function Spaces II, pp. 277–292, Yokohama, Japan, 2008.[10] K. Kuratowski, “Sur les espaces complets,” Fundamenta Mathematicae, vol. 15, pp. 301–309, 1930.[11] D. Bugajewski, “Some remarks on Kuratowski’s measure of noncompactness in vector spaces with a

metric,” Commentationes Mathematicae. Prace Matematyczne, vol. 32, pp. 5–9, 1992.[12] R. Espınola, “Darbo-Sadovski’s theorem in hyperconvex metric spaces,” Rendiconti del Circolo

Matematico di Palermo. Serie II. Supplemento, no. 40, pp. 129–137, 1996.[13] D. Bugajewski and E. Grzelaczyk, “A fixed point theorem in hyperconvex spaces,” Archiv der

Mathematik, vol. 75, no. 5, pp. 395–400, 2000.[14] A. G. Aksoy and B. Maurizi, “Metric trees, hyperconvex hulls and extensions,” Turkish Journal of

Mathematics, vol. 32, no. 2, pp. 219–234, 2008.[15] J. L. Kelley, “Banach spaces with the extension property,” Transactions of the American Mathematical

Society, vol. 72, pp. 323–326, 1952.[16] L. Nachbin, “A theorem of the Hahn-Banach type for linear transformations,” Transactions of the

American Mathematical Society, vol. 68, pp. 28–46, 1950.[17] R. Espınola and M. A. Khamsi, “Introduction to hyperconvex spaces,” in Handbook of Metric Fixed

Point Theory, W. A. Kirk and B. Sims, Eds., pp. 391–435, Kluwer Academic Publishers, Dordrecht, TheNetherlands, 2001.

[18] I. Bula, “Strictly convex metric spaces and fixed points,” Mathematica Moravica, vol. 3, pp. 5–16, 1999.[19] M. Borkowski, D. Bugajewski, and H. Przybycien, “Hyperconvex spaces revisited,” Bulletin of the

Australian Mathematical Society, vol. 68, no. 2, pp. 191–203, 2003.[20] D. Bugajewski and E. Grzelaczyk, “On the measures of noncompactness in some metric spaces,” New

Zealand Journal of Mathematics, vol. 27, no. 2, pp. 177–182, 1998.[21] L. A. Karlovitz, “The construction and application of contractive retractions in 2-dimensional normed

linear spaces,” Indiana University Mathematics Journal, vol. 22, no. 5, pp. 473–481, 1972.[22] J. Grzybowski, personal communication.[23] L. M. Blumenthal, Theory and Applications of Distance Geometry, Clarendon Press, Oxford, UK, 1953.[24] W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, “Lipschitz quotients from metric trees

and from Banach spaces containing l1,” Journal of Functional Analysis, vol. 194, no. 2, pp. 332–346, 2002.


Research ArticleFixed Point Theorems for Set-Valued ContractionType Maps in Metric Spaces

A. Amini-Harandi1 and D. O’Regan2

1 Department of Mathematics, University of Shahrekord, Shahrekord, 88186-34141, Iran2 Department of Mathematics, National University of Ireland, Galway, Ireland

Correspondence should be addressed to A. Amini-Harandi, aminih [email protected]

Received 13 August 2009; Revised 15 October 2009; Accepted 13 January 2010


Copyright q 2010 A. Amini-Harandi and D. O’Regan. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We first give some fixed point results for set-valued self-map contractions in complete metricspaces. Then we derive a fixed point theorem for nonself set-valued contractions which aremetrically inward. Our results generalize many well-known results in the literature.


Let (X, d) be a metric space and let CB(X) denote the class of all nonempty bounded closedsubsets of X. Let H be the Hausdorff metric with respect to d, that is,

H(A,B) = max

{supu∈A

d(u, B), supv∈B

d(v,A)

}(1.1)

for every A,B ∈ CB(X), where d(u, B) = inf{d(u, y) : y ∈ B}. In 1969, Nadler [1] extendedthe Banach contraction principle [2] to set-valued mappings.

Theorem 1.1 (Nadler [1]). Let (X, d) be a complete metric space and let T : X →CB(X) be aset-valued map. Assume that there exists r ∈ [0, 1) such that

H(Tx, Ty

) ≤ rd(x, y) (1.2)

for all x, y ∈ X. Then T has a fixed point.

Mizoguchi and Takahashi [3] proved the following generalization of Theorem 1.1.


Corollary 1.2 (Mizoguchi and Takahashi [3]). Let (X, d) be a complete metric space and let T :X →CB(X) be a set-valued map satisfying

H(Tx, Ty

) ≤ α(d(x, y))d(x, y), for each x, y ∈ X, (1.3)

where α : [0,∞) → [0, 1) satisfies lim sups→ t+α(s) < 1 for each t ∈ [0,∞). Then T has a fixedpoint.

Also, Reich [4] has proved that if for each x ∈ X, Tx is nonempty and compact, thenthe above result holds under the weaker condition lim sups→ t+α(s) < 1 for each t > 0. To setup our results in the next section, we introduce some definitions and facts.

Definition 1.3. Throughout the paper, let Ψ be the family of all functions ψ : [0,∞) → [0,∞)satisfying the following conditions:

(a) ψ(s) = 0⇔ s = 0;

(b) ψ is lower semicontinuous and nondecreasing;

(c) lim sups→ 0+(s/ψ(s)) <∞.

Theorem 1.4 (Bae [5]). Let (M,ρ) be a complete metric space, φ : M → [0,∞) a lowersemicontinuous function, and ϕ : [0,∞) → [0,∞) a lower semicontinuous function such thatϕ(t) > 0 for t > 0 and

lim sups→ 0+

s

ϕ(s)<∞. (1.4)

Let g : M → M be a map such that for any x ∈M, ρ(x, gx) ≤ φ(x) and

ϕ(ρ(x, gx

)) ≤ φ(x) − φ(g(x)) (1.5)

hold. Then g has a fixed point inM.

Definition 1.5. Let (X, d) be a complete metric space and D be a nonempty closed subset ofX.

(i) Set

MID(x) ={z ∈ X : z = x or there exits y ∈ D satisfyingy /=x,

d(x, z) = d(x, y

)+ d

(y, z

)}.

(1.6)

Then MID(x) is called the metrically inward set of D at x (see [5]);

(ii) Let T : D → CB(X) be a set-valued map. T is said to be metricaly inward, if for eachx ∈ D,

Tx ⊆MID(x). (1.7)

In Section 2 we generalize Corollary 1.2 and Theorem 1.4.


2. Extension of Mizoguchi-Takahashi’s Theorem

In the first result of this section, we use the technique in [6] to extend Corollary 1.2.

Theorem 2.1. Let (X, d) be a complete metric space and let T : X →CB(X) be a set-valued mapsatisfying

ψ(H(Tx, Ty

)) ≤ α(ψ(d(x, y)))ψ(d(x, y)), for each x, y ∈ X, (2.1)

where α : [0,∞) → [0, 1) satisfies lim sups→ t+α(s) < 1 for each t ∈ [0,∞) and ψ ∈ Ψ. Then T hasa fixed point.

Proof . Define a function β : [0,∞) → [0, 1) by β(t) = (α(t) + 1)/2. Then α(t) < β(t) andlim sups→ t+β(s) < 1 for all t ∈ [0,∞). Since ψ is nondecreasing, then from (1.3), for each x /=y,we have

max

{supu∈Tx

ψ(d(u, Ty

)), supv∈Ty

ψ(d(v, Tx))

}

= max

{ψ

(supu∈Tx

d(u, Ty

)), ψ

(supv∈Ty

d(v, Tx)

)}

= ψ(H(Tx, Ty

))< β

(ψ(d(x, y

)))ψ(d(x, y

)).

(2.2)

Hence for each x ∈ X and y ∈ Tx, there exists an element z ∈ Ty such that ψ(d(y, z)) ≤β(ψ(d(x, y)))ψ(d(x, y)). Thus we can define a sequence {xn} in X satisfying

xn+1 ∈ Txn, ψ(d(xn+1, xn+2)) ≤ β(ψ(d(xn, xn+1))

)ψ(d(xn, xn+1)), (2.3)

for each n ∈ N. Let us show that {xn} is convergent. Since β(t) < 1 for each t ∈ [0,∞), then{ψ(d(xn, xn+1))} is a nonincreasing sequence of non-negative numbers and so is convergent toa real number, say r0. Since lim sups→ r0

+β(s) < 1 and β(r0) < 1, there exist r ∈ [0, 1) and ε > 0such that β(s) ≤ r for all s ∈ [r0, r0+ε]. We can take n0 ∈ N such that r0 ≤ ψ(d(xn, xn+1)) ≤ r0+εfor all n ∈ N with n ≥ n0. Since

ψ(d(xn+1, xn+2)) ≤ β(ψ(d(xn, xn+1))

)ψ(d(xn, xn+1)) ≤ rψ(d(xn, xn+1)) (2.4)

for all n ≥ n0, then we have r0 ≤ rr0 and so r0 = 0 (note that r < 1). If d(xm, xm+1) = 0for some m ∈ N, then d(xn, xn+1) = 0 for each n ≥ m (note that {ψ(d(xn, xn+1))} isnonincreasing). Thus {xn} is eventually constant, so we have a fixed point of T (note thatxn+1 ∈ Txn). Now, we assume that d(xn, xn+1)/= 0 for each n ∈ N. Since {ψ(d(xn, xn+1))} isdecreasing and ψ is nondecreasing, then the nonnegative sequence d(xn, xn+1) converges tosome nonnegative real number τ . Since ψ is nondecreasing and d(xn, xn+1) is nonincreasing,then ψ(τ) ≤ ψ(d(xn, xn+1)) for each n ∈ N. Thus

ψ(τ) ≤ limn→∞

ψ(d(xn, xn+1)) = r0 = 0. (2.5)


Thus τ = 0 (note that ψ(τ) = 0 implies τ = 0). Also we have (note ψ(d(xn+1, xn+2)) ≤rψ(d(xn, xn+1)) for n ≥ n0)

∞∑1

ψ(d(xn, xn+1)) ≤n0∑1

ψ(d(xn, xn+1)) +∞∑1

rnψ(d(xn0 , xn0+1)) <∞. (2.6)

Since

lim supn→∞

d(xn, xn+1)ψ(d(xn, xn+1))

≤ lim sups→ 0+

s

ψ(s)<∞, (2.7)

then∑∞

1 d(xn, xn+1) < ∞. Hence {xn} is a Cauchy sequence. Since X is complete, {xn}converges to some point x0 ∈ X. Since ψ is lower semicontinuous and nondecreasing (recallalso from above that limn→∞ψ(d(xn, xn+1)) = 0), then

ψ(d(x0, Tx0)) ≤ lim infn→∞

ψ(d(xn+1, Tx0)) ≤ lim infn→∞

ψ(H(Txn, Tx0))

≤ lim infn→∞

β(ψ(d(xn, x0))

)ψ(d(xn, x0)) ≤ lim inf

n→∞ψ(d(xn, x0))

= lims→ 0+

ψ(s) = limn→∞

ψ(d(xn, xn+1)) = 0,

(2.8)

and this with Tx0 closed and (a) of Definition 1.3 implies x0 ∈ Tx0.

Corollary 2.2. Let (X, d) be a complete metric space and let T : X →CB(X) be a set-valued mapsatisfying

ψ(H(Tx, Ty

)) ≤ ψ(d(x, y)) − ϕ(ψ(d(x, y))), for each x, y ∈ X, (2.9)

where ψ ∈ Ψ and ϕ : [0,∞) → [0,∞) satisfying lim infs→ t+(ϕ(s)/ψ(s)) > 0 for each t ∈ [0,∞).Then T has a fixed point.

Proof. Let α(s) = 1 − ϕ(s)/ψ(s) and apply Theorem 2.1.

In the following, we present a fixed point theorem for nonself set-valued contractiontype maps which are metrically inward.

Theorem 2.3. Let D be a nonempty closed subset of a complete metric space (X, d) and T : D →CB(X) be a set-valued map satisfying

ψ(H(Tx, Ty

)) ≤ ψ(d(x, y)) − ϕ(ψ(d(x, y))), for each x, y ∈ X, (2.10)

for which ψ ∈ Ψ is continuous and

ψ(r − s) + ψ(s + t) ≤ ψ(r) + ψ(t), for each 0 ≤ s ≤ r ≤ s + t. (2.11)


Assume that ϕ : [0,∞) → [0,∞) is a lower semicontinuous function satisfying lim infs→ 0+(ϕ(s)/ψ(s)) > 0 and ϕ(s) > 0 for s > 0. Suppose that T is metrically inward onD. Then T has a fixed pointin D.

Proof . We first show that lim sups→ 0+(s/ϕ(s)) < ∞. On the contrary, we assume that thereexists a sequence sn → 0+ for which

lim supn→∞

snϕ(sn)

= lim supn→∞

sn/ψ(sn)ϕ(sn)/ψ(sn)

=∞. (2.12)

Since lim infn→∞(ϕ(sn)/ψ(sn)) > 0, then we get lim supn→∞(sn/ψ(sn)) = ∞, which con-tradicts our assumption on ψ. Let M = {(x, y) : x ∈ X, y ∈ Tx} be the graph of T . Letρ : M ×M → [0,∞) be given by

ρ((x, z), (u, v)) = max{ψ(d(x, u)), ψ(d(z, v))

}. (2.13)

We show that (M,ρ) is a complete metric space. First note that since ψ(s) = 0 ⇔ s = 0 thenρ((x, z), (u, v)) = 0⇔ (x, z) = (u, v). Clearly, ρ((x, z), (u, v)) = ρ((u, v), (x, z)). Now we showthe triangle inequality. From (2.11), we have ψ(r + t) ≤ ψ(r) + ψ(t), ∀r, t ≥ 0. Hence,

ρ((x, z), (r, s)) + ρ((r, s), (u, v))

= max{ψ(d(x, r)), ψ(d(z, s))

}+ max

{ψ(d(r, u)), ψ(d(s, v))

}≥ max

{ψ(d(x, r)) + ψ(d(r, u)), ψ(d(z, s)) + ψ(d(s, v))

}≥ max

{ψ(d(x, r) + d(r, u)), ψ(d(z, s) + d(s, v))

}≥ max

{ψ(d(x, u)), ψ(d(z, v))

}= ρ((x, z), (u, v)).

(2.14)

To prove the completeness of ρ, we first need to show that T is Hausdorff continuous. Toprove this, let (xn) be a sequence in D such that xn → x ∈ D. Since ψ is continuous at 0,then limn→∞ψ(d(xn, x)) = ψ(0) = 0. Hence from (2.10), we get limn→∞ψ(H(Txn, Tx)) = 0.We claim that limn→∞H(Txn, Tx) = 0 (and then we are finished). On the contrary, assumethat there exist ε > 0 and a subsequence xnk such that H(Txnk , Tx) ≥ ε, k=1,2,3,. . . . Sinceψ is nondecreasing, then ψ(H(Txnk , Tx)) ≥ ψ(ε) > 0, a contradiction. Now, let (xn, zn) bea Cauchy sequence in M with respect to ρ. Then {xn} and {zn} are Cauchy sequences inthe complete metric space (X, d). Then there exist x, z ∈ X such that d(xn, x) → 0 andd(zn, z) → 0. Since zn ∈ Txn and T is Hausdorff continuous, then z ∈ Tx. Thus (x, z) ∈ Mand ρ((xn, zn), (x, z)) → 0. Therefore, (M,ρ) is a complete metric space. Suppose that T hasno fixed point. Then for each (x, z) ∈ M, we have x /= z. Since z ∈ Tx ⊆ MID(x), we canchoose u ∈ D such that u/=x and

d(x, z) = d(x, u) + d(u, z). (2.15)


Since T satisfies (2.10) and ψ is continuous, then we can choose v ∈ Tu such that

ψ(d(z, v)) ≤ ψ(d(x, u)) − 12ϕ(ψ(d(x, u))

). (2.16)

Let ϕ(t) = ϕ(t)/2. Then by combining (2.15) and (2.16), we get

ϕ(ψ(d(x, u))

) ≤ ψ(d(x, u)) − ψ(d(z, v))= ψ(d(x, z) − d(u, z)) − ψ(d(z, v)).

(2.17)

From (2.11), we have (note that ψ is nondecreasing)

ψ(d(x, z) − d(u, z)) − ψ(d(z, v)) ≤ ψ(d(x, z)) − ψ(d(z, v) + d(u, z))≤ ψ(d(x, z)) − ψ(d(u, v)).

(2.18)

Thus (2.17) and (2.18) yield

ϕ(ψ(d(x, u))

) ≤ ψ(d(x, z)) − ψ(d(u, v)). (2.19)

Since ρ((x, z), (u, v)) = max{ψ(d(x, u)), ψ(d(z, v))} = ψ(d(x, u)) ≤ ψ(d(x, z)) ≡ φ(x, z), bydefining g : M → M by g(x, z) = (u, v), from Theorem 1.4, g must have a fixed point, say(x0, z0). Then (x0, z0) = g(x0, z0) = (u0, v0). Hence x0 = u0. This is a contradiction. Therefore,T has a fixed point.

Remark 2.4. Note that Theorem 2.3 does not follow from Theorem 3.3 of Bae [5] by replacingthe metric d by ψ(d). In Theorem 2.3, we assume T is metrically inward with respect to d butto apply Theorem 3.3 of [5] with ψ(d) rather than d, we need T to be metrically inward with

respect to ψ(d).

Letting ψ(s) = s for each s ∈ [0,∞), we get the following corollary due to Bae [5].

Corollary 2.5. Let D be a nonempty closed subset of a complete metric space (X, d) and T : D →CB(X) be a set-valued map satisfying

H(Tx, Ty

) ≤ d(x, y) − ϕ(d(x, y)), for each x, y ∈ X, (2.20)

for which ϕ : [0,∞) → [0,∞) is a lower semicontinuous function satisfying lim infs→ 0+(ϕ(s)/s) >0. Suppose that T is metrically inward on D. Then T has a fixed point in D.

Examples 2.6. Let ψ : [0,∞) → [0,∞) be a differentiable function with ψ(0) =0 such that ψ ′ is positive and decreasing in (0,∞) and lims→ 0+ψ

′(s) = ∞. Now we showthat (ψ) satisfies all the conditions of Theorem 2.3. Obviously, ψ is continuous andincreasing. Since lims→ 0+(1/ψ ′(s)) = 0, then by L’Hopital’s rule lims→ 0+(s/ψ(s)) = 0.Thus lim sups→ 0+(s/ψ(s)) <∞. Now we prove that for each 0 ≤ t ≤ r, ψ(r + t) ≤ ψ(r) +ψ(t).To show this let h(t) = ψ(r) + ψ(t) − ψ(r + t) for 0 ≤ t ≤ r. Then h′(t) = ψ ′(t) − ψ ′(r + t) > 0.


Since h(0) = 0 and h is increasing, we get h(t) ≥ 0 for each 0 ≤ t ≤ r and we are done. Finally,we show that for each 0 ≤ s ≤ r ≤ s + t, we have ψ(r − s) + ψ(s + t) ≤ ψ(r) + ψ(t). Let k(s) =ψ(r)+ψ(t)−ψ(r−s)+ψ(s+t) for 0 ≤ s ≤ r. Then k′(s) = ψ ′(r−s)−ψ ′(s+t). If r ≤ t, then k′(s) >0. Since k(0) = 0, we obtain k(s) ≥ 0 for each 0 ≤ s ≤ r and we are finished. In thecase, r > t, k′(s) = 0 if and only if s = (r−t)/2. Since k′(s) > 0 for 0 < s < (r−t)/2 and k′(s) <0 for (r − t)/2 < s ≤ t, then inf0≤s≤rk(s) = min(k(0), k(r)) = min(0, ψ(r) + ψ(t) − ψ(r + t)) =0, and we are finished (note that we proved above that ψ(r) + ψ(t) − ψ(r + t) ≥ 0).

Acknowledgments

The authors would like to thank the referees for careful reading and giving valuablecomments. This work was supported in part by the Shahrekord University. The first authorwould like to thank this support.

References

[1] S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.

[2] S. Banach, “Sur les operation dans les ensembles et leur application aux equations integrales,”Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.

[3] N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metricspaces,” Journal of Mathematical Analysis and Applications, vol. 141, no. 1, pp. 177–188, 1989.

[4] S. Reich, “Fixed points of contractive functions,” Bollettino dell’Unione Matematica Italiana, vol. 5, no. 4,pp. 26–42, 1972.

[5] J. S. Bae, “Fixed point theorems for weakly contractive multivalued maps,” Journal of MathematicalAnalysis and Applications, vol. 284, no. 2, pp. 690–697, 2003.

[6] T. Suzuki, “Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s,” Journal ofMathematical Analysis and Applications, vol. 340, no. 1, pp. 752–755, 2008.


Research ArticleGeneralized IFSs on Noncompact Spaces

Alexandru Mihail and Radu Miculescu

Faculty of Mathematics and Computer Science, University of Bucharest, Academiei Street 14,010014 Bucharest, Romania

Correspondence should be addressed to Radu Miculescu, [email protected]



Copyright q 2010 A. Mihail and R. Miculescu. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The aim of this paper is to continue the research work that we have done in a previous paperpublished in this journal (see Mihail and Miculescu, 2008). We introduce the notion of GIFS, whichis a family of functions f1, . . . , fn : Xm → X, where (X, d) is a complete metric space (in the abovementioned paper the case when (X, d) is a compact metric space was studied) andm,n ∈ N. In casethat the functions fk are Lipschitz contractions, we prove the existence of the attractor of such aGIFS and explore its properties (among them we give an upper bound for the Hausdorff-Pompeiudistance between the attractors of two such GIFSs, an upper bound for the Hausdorff-Pompeiudistance between the attractor of such a GIFS, and an arbitrary compact set of X and we prove itscontinuous dependence in the fk’s). Finally we present some examples of attractors of GIFSs. Thelast example shows that the notion of GIFS is a natural generalization of the notion of IFS.

1. Introduction

1.1. The Organization of the Paper

The paper is organized as follows. Section 2 contains a short presentation of the notion of aniterated function system (IFS), one of the most common and most general ways to generatefractals. This will serve as a framework for our generalization of an iterated function system.

Then, we introduce the notion of a GIFS, which is a finite family of Lipschitzcontractions fk : Xm → X, where (X, d) is a complete metric space and m ∈ N.

In Section 3 we prove the existence of the attractor of such a GIFS and explore itsproperties (among them we give an upper bound for the Hausdorff-Pompeiu distancebetween the attractors of two such GIFSs, an upper bound for the Hausdorff-Pompeiudistance between the attractor of such a GIFS, and an arbitrary compact set of X and weprove its continuous dependence in the fk’s).

Section 4, the last one, contains some examples and remarks. The last example showsthat the notion of GIFS is a natural generalization of the notion of IFS.


1.2. Some Generalizations of the Notion of IFS

IFSs were introduced in their present form by John Hutchinson and popularized by Barnsley(see [1]). There is a current effort to extend Hutchinson’s classical framework for fractals tomore general spaces and infinite IFSs. Some papers containing results on this direction are[2–7].

1.3. Some Physical Applications of IFSs

In the last period IFSs have attracted much attention being used by researchers who work onautoregressive time series, engineer sciences, physics, and so forth. For applications of IFSsin image processing theory, in the theory of stochastic growth models, and in the theory ofrandom dynamical systems one can consult [8–10]. Concerning the physical applications ofiterated function systems we should mention the seminal paper [11] of El Naschie whichdraws attention to an informal but instructive analogy between iterated function systemsand the two-slit experiment which is quite valuable in illuminating the role played bythe possibly DNA-like Cantorian nature of microspacetime and clarifies the way in whichprobability enters into this subject. We also mention the paper [12] of Słomczynski wherea new definition of quantum entropy is introduced and one method (using the theory ofiterated function systems) of calculating coherent states entropy is presented. The coherentstates entropy is computed as the integral of the Boltzmann-Shannon entropy over a fractalset.

In [13], Bahar described bifurcation from a fixed-point generated by an iteratedfunction system (IFS) as well as the generation of “chaotic” orbits by an IFS, and in [14]unusual and quite interesting patterns of bifurcation from a fixed-point in an IFS system, aswell as the routes to chaos taken by IFS-generated orbits, are discussed. Moreover, in [15]it is shown that random selection of transformation in the IFS is essential for the generationof a chaotic attractor. In [16, Section 6.4], one can find a lengthy but elementary explanationwhich features of randomness play the main role.

2. Preliminaries

Notations. Let (X, dX) and (Y, dY ) be two metric spaces.As usual, C(X,Y ) denotes the set of continuous functions from X to Y , and d :

C(X,Y ) × C(X,Y ) → R+ = R+ ∪ {∞} defined by

d(f, g

)= sup

x∈XdY

(f(x), g(x)

)(2.1)

is the generalized metric on C(X,Y ).For a sequence (fn)n of elements of C(X,Y ) and f ∈ C(X,Y ), fn

s→ f denotes thepunctual convergence, fn

u.c→ f denotes the uniform convergence on compact sets, and fnu→

f denotes the uniform convergence, that is, the convergence in the generalized metric d.


Definition 2.1. Let (X, d) be a complete metric space and m ∈ N.

For a function f : Xm = ×mk=1X → X, the number

inf{c : d

(f(x1, . . . , xm), f

(y1, . . . , ym

)) ≤ cmax{d(x1, y1

), . . . , d

(xm, ym

)},

∀x1, . . . , xm, y1, . . . , ym ∈ X},

(2.2)

which is the same with

supx1,...,xm,y1,...,ym∈X;max{d(x1,y1),...,d(xm,ym)}>0

d(f(x1, . . . , xm), f

(y1, . . . , ym

))max

{d(x1, y1

), . . . , d

(xm, ym

)} , (2.3)

is denoted by Lip(f) and it is called the Lipschitz constant of f .A function f : Xm → X is called a Lipschitz function if Lip(f) < ∞ and a Lipschitz

contraction if Lip(f) < 1.We will use the notation LConm(X) for the set {f : Xm → X : Lip(f) < 1}.

Notations. P(X) denotes the subsets of a given set X and P ∗(X) denotes the set P(X) − {∅}.For a subset A of P(X), by A∗ we mean A − {∅}.Given a metric space (X, d), K(X) denotes the set of compact subsets of X and B(X)

denotes the set of closed bounded subsets of X.

Remark 2.2. It is obvious that K(X) ⊆ B(X) ⊆ P(X).

Definition 2.3. For a metric space (X, d), we consider on P ∗(X) the generalized Hausdorff-Pompeiu pseudometric h : P ∗(X) × P ∗(X) → [0,+∞] defined by h(A,B) = max(d(A,B),d(B,A)) = inf{r ∈ [0,∞] : A ⊆ B(B, r) and B ⊆ B(A, r)}, where B(A, r) = {x ∈ X : d(x,A) <r} and d(A,B) = supx∈Ad(x, B) = supx∈A(infy∈Bd(x, y)).

Remark 2.4. The Hausdorff-Pompeiu pseudometric is a metric on B∗(X) and, in particular, onK∗(X).

Remark 2.5. The metric spaces (B∗(X), h) and (K∗(X), h) are complete, provided that (X, d) isa complete metric space (see [1, 7, 17]).

The following proposition contains the important properties of the Hausdorff-Pompeiu semimetric (see [1, 17] or [18]).

Proposition 2.6. Let (X, dX) and (Y, dY ) be two metric spaces. Then one has the following:

(i) ifH and K are two nonempty subsets of X, then

h(H,K) = h(H,K

); (2.4)


(ii) if (Hi)i∈I and (Ki)i∈I are two families of nonempty subsets of X, then

h

(∪i∈IHi, ∪

i∈IKi

)= h

(∪i∈IHi, ∪

i∈IKi

)≤ sup

i∈Ih(Hi,Ki); (2.5)

(iii) ifH and K are two nonempty subsets of X and f : X → X is a Lipschitz function, then

h(f(K), f(H)

) ≤ Lip(f)h(K,H). (2.6)

Definition 2.7. An iterated function system on X consists of a finite family of Lipschitzcontractions (fk)k=1,n on X and is denoted S = (X, (fk)k=1,n).

Theorem 2.8. Let (X, d) be a complete metric space, let S = (X, (fk)k=1,n) be an IFS. Then thereexists a unique A(S) ∈ K∗(X) such that

FS(A(S)) def= f1(A(S))⋃. . .

⋃fn(A(S)) = A(S). (2.7)

The set A(S) is called the attractor of the IFS S = (X, (fk)k=1,n).

Given a metric space (X, d), the idea of our generalization of the notion of an IFS is toconsider contractions from Xm = ×mk=1X to X, rather than contractions from X to itself.

Definition 2.9. Let (X, d) be a complete metric space and m ∈ N. A generalized iteratedfunction system on X of order m (for short a GIFS or a GmIFS), denoted S = (X, (fk)k=1,n),consists of a finite family of functions (fk)k=1,n, fk : Xm → X such that f1, . . . , fn ∈ LConm(X).

Earlier several authors tried to coin the name generalized IFS. One should notethe paper [19] in which notion tightly corresponds to contractive multivalued IFS from [2].

3. The Existence of the Attractor of a GIFS for Lipschitz Contractions

In this section m is a fixed natural number, (X, d) will be a fixed complete metric space, andall the GIFSs are of order m and have the form S = (X, (fk)k=1,n), where n is a natural number.

We prove the existence of the attractor of S (Theorem 3.9) and study its properties(among them we give an upper bound for the Hausdorff-Pompeiu distance betweenthe attractors of two such GIFSs (Theorem 3.12), an upper bound for the Hausdorff-Pompeiu distance between the attractor of such a GIFS, and an arbitrary compact set of X(Theorem 3.17) and we prove its continuous dependence in the fk’s (Theorem 3.15)).

Definition 3.1. Let f : Xm → X be a function. The function Ff : K∗(X)m → K∗(X) defined by

Ff(K1, K2, . . . , Km) = f(K1, K2, . . . , Km) ={f(x1, x2, . . . , xm) : xj ∈ Kj ∀j ∈ {1, . . . , m}},

(3.1)

for all K1, K2, . . . , Km ∈ K∗(X), is called the set function associated to the function f .


The function FS : K∗(X)m → K∗(X) defined by

FS(K1, K2, . . . , Km) =n⋃k=1

Ffk(K1, K2, . . . , Km), (3.2)

for all K1, K2, . . . , Km ∈ K∗(X), is called the set function associated to the GIFS S.

Lemma 3.2. For a sequence (fn)n of elements of C(Xm,X) and f ∈ C(Xm,X) such that fnu→ f ,

one has

fn(K1, K2, . . . , Km) −→ f(K1, K2, . . . , Km), (3.3)

in(K∗(X), h), for all K1, K2, . . . , Km ∈ K∗(X).

Proposition 3.3. Let (X, dX) and (Y, dY ) be two complete metric spaces and let fn, f ∈ C(X,Y ) besuch that supn≥1Lip(fn) < +∞ and fn

s→ f on a dense set in X. Then

Lip(f) ≤ sup

n≥1Lip

(fn),

fnu.c→ f.

(3.4)

Proof. In this proof, by M we mean supn≥1Lip(fn).Let us consider A = {x ∈ X | fn(x) → f(x)}, which is a dense set in X, let K be a

compact set in X, and let ε > 0.Since f is uniformly continuous on K, there exists δ ∈ (0, ε/3(M + 1)) such that if

x, y ∈ K and dX(x, y) < δ, then

dY(f(x), f

(y))

<ε

3. (3.5)

Since K is compact, there exist x1, x2, . . . , xp ∈ K such that

K ⊆p⋃i=1

B

(xi,

δ

2

). (3.6)

Taking into account the fact that A is dense in X, we can choose y1, y2, . . . , yp ∈ A suchthat

y1 ∈ B(x1,

δ

2

), . . . , yp ∈ B

(xp,

δ

2

). (3.7)

Since, for all i ∈ {1, . . . , p}, limn→∞fn(yi) = f(yi), there exists nε ∈ N such that forevery n ∈ N, n ≥ nε, we have

dY(fm

(yi), f

(yi))

<ε

3, (3.8)

for every i ∈ {1, . . . , p}.


For x ∈ K, there exists i ∈ {1, . . . , p}, such that x ∈ B(xi, δ/2) and therefore

dX(x, yi

) ≤ dX(x, xi) + dX(xi, yi) < δ

2+δ

2< δ <

ε

3(M + 1), (3.9)

and so

dY(f(yi), f(x)

)<ε

3. (3.10)

Hence, for n ≥ nε, we have

dY(fn(x), f(x)

) ≤ dY(fn(x), fn(yi)) + dY(fn(yi), f(yi)) + dY(f(yi), f(x))

≤MdX(x, yi

)+ε

3+ε

3≤M ε

3(M + 1)+

2ε3< ε.

(3.11)

Consequently, as x was arbitrary chosen in K, we infer that fnu→ f on K, and so

fnu.c→ f. (3.12)

The inequality

Lip(f) ≤ sup

n≥1Lip

(fn)

(3.13)

is obvious.

Lemma 3.4. Let A1, A2, . . . , Am be subsets of R.Then

(1) infa1∈A1,...,am∈Am max{a1, . . . , am} = max{infA1, . . . , infAm};(2) supa1∈A1,...,am∈Am

max{a1, . . . , am} = max{supA1, . . . , supAm}.

Lemma 3.5. If f : Xm → X is a Lipschitz function, then

Lip(Ff

)= Lip

(f). (3.14)

Lemma 3.6. In the framework of this section, one has

Lip(FS) ≤ max{Lip

(f1), . . . ,Lip

(fn)}. (3.15)

The proofs of the above lemmas are almost obvious.


Theorem 3.7 (Banach contraction principle for LConm(X)). For every f ∈ LConm(X), thereexists a unique α ∈ X, such that

f(α, α, . . . , α) = α. (3.16)

For every x0, x1, . . . , xm−1 ∈ X, the sequence (xn)n≥1, defined by xk+m = f(xk+m−1,xk+m−2, . . . , xk), for all k ∈ N, has the property that

limn→∞

xn = α. (3.17)

Concerning the speed of the convergence, one has the following estimation:

d(xn, α) ≤m(Lip

(f))[n/m] max{d(x0, x1), d(x1, x2), . . . , d(xn−1, xn)}

1 − Lip(f) (3.18)

for every n ∈ N.

Proof. See [20, Remark 5.1].

Remark 3.8. The point α from the above theorem is called the fixed point of f .

From Theorem 3.7 and Lemma 3.6 we have the following.

Theorem 3.9. In the framework of this section, there exists a unique A(S) ∈ K∗(X) such that

FS(A(S), A(S), . . . , A(S)) = A(S). (3.19)

Moreover, for any H0,H1, . . . ,Hm−1 ∈ K∗(X), the sequence (Hn)n≥1 defined by Hk+m =FS(Hk+m−1,Hk+m−2, . . . ,Hk), for all k ∈ N, has the property that

limn→∞

Hn = A(S). (3.20)

Concerning the speed of the convergence, one has the following estimation:

h(Hn,A(S)) ≤ m(max

{Lip

(f1), . . . ,Lip

(fn)})[n/m] max{h(H0,H1), . . . , h(Hm−1,Hm)}

1 −max{Lip

(f1), . . . ,Lip

(fn)}

(3.21)

for all n ∈ N.


Definition 3.10. The unique set A(S) given by the previous theorem is called the attractor ofthe GIFS S.

Theorem 3.11. If f, g ∈ LConm(X) have the fixed points α and β, then

d(α, β

) ≤ min

{1

1 − Lip(f)d(f(β, β, . . . , β

), β),

11 − Lip(g)d

(α, g(α, α, . . . , α)

)}

≤ 11 −min

{Lip

(f),Lip

(g)}d(f, g).

(3.22)

Proof. We have

d(α, β

)= d

(f(α, . . . , α), g

(β, . . . , β

))

≤ d(f(α, . . . , α), f(β, . . . , β)) + d(f(β, . . . , β), g(β, . . . , β))

= d(f(α, . . . , α), f

(β, . . . , β

))+ d

(f(β, . . . , β

), β)

≤ Lip(f)d(α, β

)+ d

(f(β, . . . , β

), β),

(3.23)

so

d(α, β

) ≤ 11 − Lip(f)d

(f(β, β, . . . , β

), β), (3.24)

and in a similar manner we get

d(α, β

) ≤ 11 − Lip(g)d

(α, g(α, α, . . . , α)

). (3.25)

Therefore

d(α, β

) ≤ min

{1

1 − Lip(f)d(f(β, β, . . . , β

), β),

11 − Lip(g)d

(α, g(α, α, . . . , α)

)}

= min

{1

1 − Lip(f)d(f(β, . . . , β

), g

(β, . . . , β

)),

11 − Lip(g)d

(f(α, . . . , α), g(α, . . . , α)

)}

≤ min

{1

1 − Lip(f)d(f, g

),

11 − Lip(g)d

(f, g

)}=

11 −min

{Lip

(f),Lip

(g)}d(f, g).

(3.26)

From Theorem 3.11 and Lemma 3.6, we have the following.


Theorem 3.12. In the framework of this section, if S = (X, (fk)k=1,n) and S′ = (X, (gk)k=1,n) aretwom dimensional GIFSs, then

h(A(S), A(S′)) ≤ 1

1 − μ max{d(f1, g1

), . . . , d

(fn, gn

)}, (3.27)

where μ = min(max{Lip(f1), . . . ,Lip(fn)},max{Lip(g1), . . . ,Lip(gn)}).

Theorem 3.13. Let fn, f ∈ LConm(X) with fixed points αn and α, respectively, such that

supn≥1

Lip(fn)< 1,

fns→ f

(3.28)

on a dense set in Xm.Then

αn −→ α. (3.29)

Proof. From the fact that supn≥1Lip(fn) < 1 and fns→ f on a dense set in Xm, it follows, using

Proposition 3.3, that

fnu.c.→ f (3.30)

on Xm and

Lip(f) ≤ sup

n≥1Lip

(fn). (3.31)

From Theorem 3.11, we have

d(α, αn) ≤ 11 − Lip(fn)d

(α, fn(α, α, . . . , α)

), (3.32)

and hence

d(α, αn) ≤ 11 − supn≥1Lip

(fn)d(α, fn(α, α, . . . , α)) (3.33)

for all n ∈ N.Since fn

u.c→ f on Xm, we obtain that

limn→∞

fn(α, α, . . . , α) = f(α, α, . . . , α), (3.34)


and consequently, using the above inequality, we obtain that

limn→∞

αn = α. (3.35)

Proposition 3.14. Let Sj = (X, (fjk)k=1,n), where j ∈ N∗, and let S = (X, (fk)k=1,n) be m-

dimensional generalized iterated function systems such that

supj≥1

max{Lip

(fj

1

), . . . ,Lip

(fjn

)}< 1,

fj

k

s→ fk

(3.36)

on a dense set in Xm, for every k ∈ {1, . . . , n}.Then

FSju.c→ FS. (3.37)

Proof. Using Proposition 3.3, we obtain that

fj

k

u.c→ fk (3.38)

on Xm and

max{Lip

(f1), . . . ,Lip

(fn)} ≤ sup

j≥1max

{Lip

(fj

1

), . . . ,Lip

(fjn

)}. (3.39)

Then, using Lemma 3.2 and Proposition 2.6(ii), we get

FSjs→ FS. (3.40)

Since, according to Lemma 3.6, we have

Lip(FSj

)≤ max

{Lip

(fj

1

), . . . ,Lip

(fjn

)}≤ sup

j≥1max

{Lip

(fj

1

), . . . ,Lip

(fjn

)}< 1

(3.41)

for all j ∈ N, we obtain, using again the arguments from Proposition 3.3, that

FSju.c→ FS. (3.42)

From Theorem 3.13, Proposition 3.14, and Lemma 3.6, we have the following.


Theorem 3.15. Let Sj = (X, (fjk)k=1,n), where j ∈ N∗, and let S = (X, (fk)k=1,n) be m-dimensional

generalized iterated function systems having the property that

supj≥1

max{Lip

(fj

1

), . . . ,Lip

(fjn

)}< 1,

fj

k

s→ fk

(3.43)

on a dense set in Xm, for every k ∈ {1, . . . , n}.Then

A(Sj) −→ A(S). (3.44)

Theorem 3.16. Forf ∈ LConm(X) having the unique fixed point α and for every x ∈ X, one has

dX(x, α) ≤d(f(x, x, . . . , x), x

)1 − Lip(f) . (3.45)

Proof. We can use the Banach contraction principle for g ∈ LCon1(X), where

g(x) = f(x, x, . . . , x) (3.46)

for all x ∈ X.

Theorem 3.17. For a generalized iterated function system S = (X, (fk)k=1,n) and H ∈ K∗(X), thefollowing inequality is valid:

h(A(S),H) ≤ h(f(H,H, . . . ,H),H

)1 −max

{Lip

(f1), . . . ,Lip

(fn)} . (3.47)

Proof. The function GS : K∗(X) → K∗(X), defined by

GS(K) = FS(K,K, . . . , K) =n⋃k=1

fk(K,K, . . . , K), (3.48)

for all K ∈ K∗(X), is a contraction and

Lip(GS) ≤ Lip(FS) ≤ max{Lip

(f1), . . . ,Lip

(fn)}. (3.49)

4. Examples

In this section we present some examples of attractors of GIFSs. Example 4.3 shows that thenotion of GIFS is a natural generalization of the notion of IFS.


Example 4.1. Let A1, A2, . . . , Am ∈ B(X) and α ∈ X, where X is a Banach space and B(X) is theset of linear and continuous operators from X to X.

Let us consider the function f : Xm → X, given by

f(x1, x2, . . . , xm) = A1x1 +A2x2 + · · · +Amxm + α, (4.1)

for every x1, x2, . . . , xm ∈ X.Then

∥∥f(x1, x2, . . . , xm) − f(y1, y2, . . . , ym

)∥∥=∥∥A1

(x1 − y1

)+A2

(x2 − y2

)+ · · · +Am

(xm − ym

)∥∥

≤m∑k=1

‖Ak‖∥∥xk − yk∥∥ ≤

(m∑k=1

‖Ak‖)

max{∥∥x1 − y1

∥∥, . . . ,∥∥xm − ym∥∥},(4.2)

for every x1, x2, . . . , xm, y1, y2, . . . , ym ∈ X, and so

Lip(f) ≤ m∑

k=1

‖Ak‖. (4.3)

In particular, if X = R and Ak = akIR, for every k ∈ {1, . . . , m}, then

Lip(f) ≤ m∑

k=1

|ak|. (4.4)

Let us consider fm0 , fm1 : R

m → R given by

f0(x1, x2, . . . , xm) =m∑k=1

18xk,

f1(x1, x2, . . . , xm) =8 −m

8+

m∑k=1

18xk

(4.5)

for every x1, x2, . . . , xm ∈ R.Then

Lip(f0)= Lip

(f1) ≤ m

8. (4.6)

If m < 8, then f0, f1 are contractions.We consider the GIFS Sm = (R, (fm0 , f

m1 )), wherem ∈ {1, 2, . . . , 7}.

If m ≥ 4, then

A(Sm) = [0, 1]. (4.7)


Indeed, fm0 ([0, 1], [0, 1], . . . , [0, 1]) = [0, m/8], fm1 ([0, 1], [0, 1], . . . , [0, 1]) = [1 −m/8, 1]and so [0, 1] = fm0 ([0, 1], [0, 1], . . . , [0, 1]) ∪ fm1 ([0, 1], [0, 1], . . . , [0, 1]), that is, [0, 1] =FSm([0, 1], . . . , [0, 1]). This proves that A(Sm) = [0, 1].

If m = 3, then

A(S3

)=[

0,38

]∪[

58, 1]. (4.8)

Indeed, if A = [0, 3/8] ∪ [5/8, 1], then

f30 (A,A,A) = f3

0

([0,

38

],

[0,

38

],

[0,

38

])∪ f3

0

([0,

38

],

[0,

38

],

[58, 1])

∪ f30

([0,

38

],

[58, 1],

[58, 1])∪ f3

0

([58, 1],

[58, 1],

[58, 1])

=[

0,9

64

]∪[

564,

1464

]∪[

1064,

2164

]∪[

1564,

38

]=[

0,38

](4.9)

and f31 (A,A,A) = [5/8, 1]. Hence A = f3

0 (A,A,A) ∪ f31 (A,A,A). This proves that A(S3) =

A = [0, 3/8] ∪ [5/8, 1].If m = 2, then

A(S2

)=[

0,2

32

]∪[

332,

532

]∪[

632,

832

]∪[

2432,

2632

]∪[

2732,

2932

]∪[

3032, 1]. (4.10)

If m = 1, then A(S1) is a Cantor type set (more precisely A(S1) consists of those elements of[0, 1] for which one can use the digits 0 and 7 in order to write them in base 8).

Remark 4.2. Finally let us note that

A(S1

)⊆ A

(S2

)⊆ A

(S3

)⊆ A

(S4

)= A

(S5

)= A

(S6

)= A

(S7

). (4.11)

Example 4.3. Let X be one of the spaces lp, l∞, or c0, where p ≥ 1.

Let j : X → X, im : Rm → X and π1 : X → R be given by

j((xn)n≥1

)= (0, x1, x2, . . . , xm, . . .),

im((xn)n≥1

)= (x1, x2, . . . , xm, 0, 0, 0, . . .) ,

π1((xn)n≥1

)= x1

(4.12)

for all (xn)n≥1 ∈ X.


We consider the GIFS S = (X, (f0, f1)), where f0 : X × X → X and f1 : X × X → X aregiven by

f0(x, y

)= i1

(π1(x)

2

)+j(y)

2,

f1(x, y

)= i1

(π1(x)

2+

12

)+j(y)

2

(4.13)

for all x, y ∈ X.Then

A(S) = ∞×k=0

[0,

12k

]. (4.14)

Indeed, ifA = ×∞k=0[0, 1/2k], then j(A) = {0}×(×∞

k=0[0, 1/2k]) and π1(A) = [0, 1]. Hence

f0(A,A) = im1

(π1(A)

2

)+j(A)

2=[

0,12

]× {(0, 0, 0, . . .)} + {0} ×

(∞×k=0

[0,

12k+1

])

=[

0,12

]×(∞×k=0

[0,

12k+1

]),

f0(A,A) = im1

(π1(A)

2+

12

)+j(A)

2=[

12, 1]× {(0, 0, 0, . . .)} + {0} ×

(∞×k=0

[0,

12k+1

])

=[

12, 1]×(∞×k=0

[0,

12k+1

]),

(4.15)

and therefore A = f0(A,A) ∪ f1(A,A). This, together with the fact that A is compact, provesthat A(S) = A = ×∞k=0[0, 1/2k].

On one hand it is obvious that A(S) has infinite Hausdorff dimension. On the otherhand, for every finite IFS S, with contraction constant less then 1, we have dimH(A(S)) <∞.Indeed, the proof of the above claim is similar with the one of Proposition 9.6, page 135, from[18].

Therefore there exists no finite IFS consisting of Lipschitz contractions having as attractor theset A(S) = ×∞

k=0[0, 1/2k].

Acknowledgment

The authors want to thank the referees whose generous and valuable remarks and commentsbrought improvements to the paper and enhanced clarity.

References

[1] M. F. Barnsley, Fractals Everywhere, Academic Press, Boston, Mass, USA, 2nd edition, 1993.[2] J. Andres, J. Fiser, G. Gabor, and K. Lesniak, “Multivalued fractals,” Chaos, Solitons and Fractals, vol.

24, no. 3, pp. 665–700, 2005.


[3] G. Gwozdz-Łukawska and J. Jachymski, “The Hutchinson-Barnsley theory for infinite iteratedfunction systems,” Bulletin of the Australian Mathematical Society, vol. 72, no. 3, pp. 441–454, 2005.

[4] A. Kaenmaki, “On natural invariant measures on generalised iterated function systems,” AnnalesAcademiæ Scientiarium Fennicæ. Mathematica, vol. 29, no. 2, pp. 419–458, 2004.

[5] K. Lesniak, “Infinite iterated function systems: a multivalued approach,” Bulletin of the Polish Academyof Sciences. Mathematics, vol. 52, no. 1, pp. 1–8, 2004.

[6] A. Łozinski, K. Zyczkowski, and W. Słomczynski, “Quantum iterated function systems,” PhysicalReview E, vol. 68, no. 4, Article ID 046110, 9 pages, 2003.

[7] R. Miculescu and A. Mihail, “Lipscomb’s space ωA is the attractor of an infinite IFS containing affinetransformations of l2(A),” Proceedings of the American Mathematical Society, vol. 136, no. 2, pp. 587–592,2008.

[8] J. H. Elton and M. Piccioni, “Iterated function systems arising from recursive estimation problems,”Probability Theory and Related Fields, vol. 91, no. 1, pp. 103–114, 1992.

[9] B. Forte and E. R. Vrscay, “Solving the inverse problem for function/image approximation usingiterated function systems. I. Theoretical basis,” Fractals, vol. 2, no. 3, pp. 325–334, 1994.

[10] L. Montrucchio and F. Privileggi, “Fractal steady states in stochastic optimal control models,” Annalsof Operations Research, vol. 88, pp. 183–197, 1999.

[11] M. S. El Naschie, “Iterated function systems and the two-slit experiment of quantum mechanics,”Chaos, Solitons and Fractals, vol. 4, no. 10, pp. 1965–1968, 1994.

[12] W. Słomczynski, “From quantum entropy to iterated function systems,” Chaos, Solitons and Fractals,vol. 8, no. 11, pp. 1861–1864, 1997.

[13] S. Bahar, “Chaotic orbits and bifurcation from a fixed point generated by an iterated function system,”Chaos, Solitons and Fractals, vol. 5, no. 6, pp. 1001–1006, 1995.

[14] S. Bahar, “Further studies of bifurcations and chaotic orbits generated by iterated function systems,”Chaos, Solitons and Fractals, vol. 7, no. 1, pp. 41–47, 1996.

[15] S. Bahar, “Chaotic attractors generated by iterated function systems: “harmonic decompositions” andthe onset of chaos,” Chaos, Solitons and Fractals, vol. 8, no. 3, pp. 303–312, 1997.

[16] H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, Springer, NewYork, NY, USA, 2nd edition, 2004.

[17] K. J. Falconer, The Geometry of Fractal Sets, vol. 85 of Cambridge Tracts in Mathematics, CambridgeUniversity Press, Cambridge, UK, 1986.

[18] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons,Chichester, UK, 1990.

[19] J. J. P. Veerman and L. B. Jonker, “Rigidity propertiesof locally scaling fractals,” http://arxiv.org/abs/math.DS/9701216.

[20] M.-A. Serban, “Fixed point theorems for operators on Cartesian product spaces and applications,”Seminar on Fixed Point Theory Cluj-Napoca, vol. 3, pp. 163–172, 2002.


Research ArticleCommon Fixed Point of Multivalued Generalizedϕ-Weak Contractive Mappings

Behzad Djafari Rouhani1 and Sirous Moradi2

1 Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, USA2 Department of Mathematics, Faculty of Science, University of Arak, Arak 38156-879, Iran

Correspondence should be addressed to Behzad Djafari Rouhani, [email protected]



Copyright q 2010 B. Djafari Rouhani and S. Moradi. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Fixed point and coincidence results are presented for multivalued generalized ϕ-weak contractivemappings on complete metric spaces, where ϕ : [0,+∞) −→ [0,+∞) is a lower semicontinuousfunction with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0. Our results extend previous results by Zhang andSong (2009), as well as by Rhoades (2001), Nadler (1969), and Daffer and Kaneko (1995).

1. Introduction

Let (X, d) be a metric space. We denote the family of all nonempty closed and boundedsubsets of X by CB(X).

A mapping T : X → X is said to be ϕ-weak contractive if there exists a map ϕ :[0,+∞) → [0,+∞) with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0 such that

d(Tx, Ty

) ≤ d(x, y) − ϕ(d(x, y)) (1.1)

for all x, y ∈ X.Also two mappings T, S : X → X are called generalized ϕ-weak contractions if there

exists a map ϕ : [0,+∞) → [0,+∞) with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0 such that

d(Tx, Sy

) ≤M(x, y

) − ϕ(M(x, y

))(1.2)


for all x, y ∈ X, where

M(x, y

):= max

{d(x, y

), d(x, Tx), d

(y, Sy

),d(x, Sy

)+ d

(y, Tx

)2

}. (1.3)

A mapping T : X → CB(X) is said to be a weak contraction if there exists 0 ≤ α < 1 such that

H(Tx, Ty

) ≤ αN(x, y

), (1.4)

for all x, y ∈ X, where H denotes the Hausdorff metric on CB(X) induced by d, that is,

H(A,B) := max

{supx∈A

d(x, B), supy∈B

d(y,A

)}, (1.5)

for all A,B ∈ CB(X), and where

N(x, y

):= max

{d(x, y

), d(x, Tx), d

(y, Ty

),d(x, Ty

)+ d

(y, Tx

)2

}. (1.6)

A mapping T : X → CB(X) is said to be ϕ-weak contractive if there exists a mapϕ : [0,+∞) → [0,+∞) with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0 such that

H(Tx, Ty

) ≤ d(x, y) − ϕ(d(x, y)), (1.7)

for all x, y ∈ X.The concepts of weak and ϕ-weak contractive mappings were defined by Daffer and

Kaneko [1] in 1995.Many authors have studied fixed points for multivalued mappings. Among many

others, see, for example, [1–4], and the references therein.In the following theorem, Nadler [3] extended the Banach Contraction Principle to

multivalued mappings.

Theorem 1.1. Let (X, d) be a complete metric space. Suppose T : X → CB(X) is a contractionmapping in the sense that for some 0 ≤ α < 1,

H(Tx, Ty

) ≤ αd(x, y), (1.8)

for all x, y ∈ X. Then there exists a point x ∈ X such that x ∈ Tx (i.e., x is a fixed point of T ).

Daffer and Kaneko [1] proved the existence of a fixed point for a multivalued weakcontraction mapping of a complete metric space X into CB(X).

Theorem 1.2. Let (X, d) be a complete metric space and T : X → CB(X) be such that

H(Tx, Ty

) ≤ αN(x, y

), (1.9)


for some 0 ≤ α < 1 and for all x, y ∈ X (i.e., weak contraction). If x �→ d(x, Tx) is lowersemicontinuous (l.s.c.), then there exists x0 ∈ X such that x0 ∈ Tx0.

In Section 3 we extend Nadler and Daffer-Kaneko’s theorems to multivaluedgeneralized weak contraction mappings (see Definition 2.1).

Rhoades [5, Theorem 2] proved the following fixed point theorem for ϕ-weakcontractive single valued mappings, giving another generalization of the Banach ContractionPrinciple.

Theorem 1.3. Let (X, d) be a complete metric space, and let T : X → X be a mapping such that

d(Tx, Ty

) ≤ d(x, y) − ϕ(d(x, y)), (1.10)

for every x, y ∈ X (i.e., ϕ-weak contractive), where ϕ : [0,+∞) → [0,+∞) is a continuous andnondecreasing function with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0. Then T has a unique fixed point.

By choosing ψ(t) = t−ϕ(t), ϕ-weak contractions become mappings of Boyd and Wongtype [6], and by defining k(t) = (1 − ϕ(t))/t for t > 0 and k(0) = 0, then ϕ-weak contractionsbecome mappings of Reich type [7].

Recently Zhang and Song [8] proved the following theorem on the existence of acommon fixed point for two single valued generalized ϕ-weak contraction mappings.

Theorem 1.4. Let (X, d) be a complete metric space, and let T, S : X → X be two mappings suchthat for all x, y ∈ X

d(Tx, Sy

) ≤M(x, y

) − ϕ(M(x, y

)), (1.11)

(i.e., generalized ϕ-weak contractions), where ϕ : [0,+∞) → [0,+∞) is an l.s.c. function withϕ(0) = 0 and ϕ(t) > 0 for all t > 0. Then there exists a unique point x ∈ X such that x = Tx = Sx.

In Section 4, we extend Theorem 1.3 by assuming ϕ to be only l.s.c., and extendTheorem 1.4 to multivalued mappings.

2. Preliminaries

In this paper, (X, d) denotes a complete metric space and H denotes the Hausdorff metric onCB(X).

Definition 2.1. Two mappings T, S : X → CB(X) are called generalized weak contractions ifthere exists 0 ≤ α < 1 such that

H(Tx, Sy

) ≤ αM(x, y

), (2.1)

for all x, y ∈ X.


Definition 2.2. Two mappings T, S : X → CB(X) are called generalized ϕ-weak contractive ifthere exists a map ϕ : [0,+∞) → [0,+∞) with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0 such that

H(Tx, Sy

) ≤M(x, y

) − ϕ(M(x, y

))(2.2)

for all x, y ∈ X.

In the proof of our main results, we will use the following well-known lemma, andrefer to Nadler [3] or Assad and Kirk [9] for its proof.

Lemma 2.3. If A,B ∈ CB(X) and a ∈ A, then for each ε > 0, there exists b ∈ B such that

d(a, b) ≤ H(A,B) + ε. (2.3)

3. Extension of Nadler and Daffer-Kaneko’s Theorems

The following theorem extends Nadler and Daffer-Kaneko’s Theorems to a coincidencetheorem, without assuming x �→ d(x, Tx) to be l.s.c.

Theorem 3.1. Let (X, d) be a complete metric space, and let T, S : X → CB(X) be two multivaluedmappings such that for all x, y ∈ X,

H(Tx, Sy

) ≤ αM(x, y

), (3.1)

where 0 ≤ α < 1 (i.e., multivalued generalized weak contractions). Then there exists a point x ∈ Xsuch that x ∈ Tx and x ∈ Sx (i.e., T and S have a common fixed point). Moreover, if either T or S issingle valued, then this common fixed point is unique.

Proof. Obviously M(x, y) = 0 if and only if x = y is a common fixed point of T and S.Let ε > 0 be such that β = α + ε < 1. Let x0 ∈ X and x1 ∈ Sx0. By Lemma 2.3, there

exists x2 ∈ Tx1 such that d(x2, x1) ≤ H(Tx1, Sx0) + εM(x1, x0). Again by using Lemma 2.3,there exists x3 ∈ Sx2 such that d(x3, x2) ≤ H(Sx2, Tx1) + εM(x2, x1). By induction and usingLemma 2.3, we can find in this way a sequence {xn} in X such that x2k+1 ∈ Sx2k and

d(x2k+1, x2k) ≤ H(Sx2k, Tx2k−1) + εM(x2k, x2k−1) (3.2)

and x2k+2 ∈ Tx2k+1 and

d(x2k+2, x2k+1) ≤ H(Tx2k+1, Sx2k) + εM(x2k+1, x2k). (3.3)


It follows that

d(x2n+1, x2n)

≤ H(Tx2n−1, Sx2n) + εM(x2n−1, x2n)

≤ βM(x2n−1, x2n)

= βmax{d(x2n−1, x2n), d(x2n−1, Tx2n−1), d(x2n, Sx2n),

d(x2n−1, Sx2n) + d(x2n, Tx2n−1)2

}

≤ βmax{d(x2n−1, x2n), d(x2n−1, x2n), d(x2n, x2n+1),

d(x2n−1, x2n+1) + 02

}

= βmax{d(x2n−1, x2n), d(x2n, x2n+1)}= βd(x2n−1, x2n),

(3.4)

since if otherwise d(x2n, x2n+1) > d(x2n, x2n−1), then d(x2n, x2n+1) ≤ βd(x2n, x2n+1) and sod(x2n, x2n+1) = 0. Hence 0 = d(x2n, x2n+1) > d(x2n, x2n−1) and this is a contradiction.

Similarly,

d(x2n+2, x2n+1) ≤ βd(x2n+1, x2n). (3.5)

From (3.4) and (3.5), we conclude that

d(xk+1, xk) ≤ βd(xk, xk−1), (3.6)

for all k ∈ N. Since β < 1 and (3.6) holds, {xn} is a Cauchy sequence. Since (X, d) is complete,there exists x ∈ X such that limn→∞xn = x.

We have

d(x2n+2, Sx) ≤ H(Tx2n+1, Sx)

≤ αM(x2n+1, x)

= αmax{d(x2n+1, x), d(x2n+1, Tx2n+1), d(x, Sx),

d(x2n+1, Sx) + d(x, Tx2n+1)2

}

≤ αmax{d(x2n+1, x), d(x2n+1, x2n+2), d(x, Sx),

d(x2n+1, Sx) + d(x, x2n+2)2

}.

(3.7)

Letting n → ∞ in the above inequality, we conclude that d(x, Sx) ≤ αd(x, Sx). So d(x, Sx) =0. Since Sx ∈ CB(X), we have x ∈ Sx.

Similarly, x ∈ Tx. Therefore, T and S have a common fixed point.


Furthermore, if T is single valued, then this common fixed point is unique. In fact, if xand y are two common fixed points for T and S, then

d(x, y

) ≤ H({x}, Sy)= H

({Tx}, Sy)≤ αM(

x, y)

= αmax

{d(x, y

), d(x, Tx), d

(y, Sy

),d(x, Sy

)+ d

(y, Tx

)2

}

≤ αmax

{d(x, y

), 0, 0,

d(x, y

)+ d

(y, x

)2

}

= αd(x, y

).

(3.8)

Since 0 ≤ α < 1, d(x, y) = 0, and so x = y.

Remark 3.2. The last part of the proof of Theorem 3.1 shows that if S, T : X → CB(X) aremultivalued and x0 is a common fixed point, and Tx0 or Sx0 is a singleton, then the commonfixed point of T and S is unique.

By taking T = S in Theorem 3.1, we get the following corollary that extends the Dafferand Kaneko theorem (Theorem 1.2).

Corollary 3.3. Let (X, d) be a complete metric space and let T : X → CB(X) be such that

H(Tx, Ty

) ≤ αN(x, y

), (3.9)

for some 0 ≤ α < 1 and for all x, y ∈ X (i.e., weak contraction). Then there exists x0 ∈ X such thatx0 ∈ Tx0.

Example 3.4. Let X = [0, 1] be endowed with the Euclidean metric. Let S, T : X → CB(X) bedefined by Tx = [0, x/4] and Sy = {y/4}. Obviously,

H(Tx, Sy

)= max

{∣∣∣y4− x

4

∣∣∣, y4

}

≤ 12

max{∣∣y − x∣∣, ∣∣∣y − y

4

∣∣∣}

=12

max{d(x, y

), d

(y, Sy

)}

≤ 12M

(x, y

).

(3.10)


So T and S have a common fixed point (x = 0), and since S is single valued, this fixed pointis unique.

4. Extension of Rhoades and Zhang-Song’s Theorems

First we extend Zhang and Song’s theorem (Theorem 1.4) to the case where one of themappings is multivalued.

Theorem 4.1. Let (X, d) be a complete metric space and let T : X → X and S : X → CB(X) betwo mappings such that for all x, y ∈ X,

H({Tx}, Sy) ≤M(

x, y) − ϕ(M(

x, y)), (4.1)

(i.e., generalized ϕ-weak contractive) where ϕ : [0,+∞) → [0,+∞) is l.s.c. with ϕ(0) = 0 andϕ(t) > 0 for all t > 0. Then there exists a unique point x ∈ X such that Tx = x ∈ Sx.

Proof. Unicity of the common fixed point follows from (4.1).Obviously M(x, y) = 0 if and only if x = y is a common fixed point of T and S.Let x0 ∈ X and x1 ∈ Sx0. Let x2 := Tx1. By Lemma 2.3, there exists x3 ∈ Sx2 such that

d(x3, x2) ≤ H(Sx2, {Tx1}) + 12ϕ(M(x2, x1)). (4.2)

We let x4 := Tx3. Inductively, we let x2n := Tx2n−1, and by Lemma 2.3, we choose x2n+1 ∈ Sx2n

such that

d(x2n+1, x2n) ≤ H(Sx2n, {Tx2n−1}) + 12ϕ(M(x2n, x2n−1)). (4.3)

We break the argument into four steps.

Step 1. limn→∞d(xn+1, xn) = 0.

Proof. Using (4.1) and (4.3),

d(x2n+1, x2n) ≤ H({Tx2n−1}, Sx2n) +12ϕ(M(x2n−1, x2n))

≤M(x2n−1, x2n) − 12ϕ(M(x2n−1, x2n)),

(4.4)


where

d(x2n−1, x2n)

≤M(x2n−1, x2n)

= max{d(x2n−1, x2n), d(x2n−1, Tx2n−1), d(x2n, Sx2n),

d(x2n−1, Sx2n) + d(x2n, Tx2n−1)2

}

≤ max{d(x2n−1, x2n), d(x2n−1, x2n), d(x2n, x2n+1),

d(x2n−1, x2n+1) + 02

}

= max{d(x2n−1, x2n), d(x2n, x2n+1)}

= d(x2n−1, x2n)(by (4.4)

).

(4.5)

So M(x2n−1, x2n) = d(x2n−1, x2n). Hence by (4.4),

d(x2n+1, x2n) ≤ d(x2n, x2n−1). (4.6)

Also

d(x2n+2, x2n+1) = d(Tx2n+1, x2n+1)

≤ H({Tx2n+1}, Sx2n)

≤M(x2n+1, x2n) − ϕ(M(x2n+1, x2n)),

(4.7)

where

d(x2n+1, x2n)

≤M(x2n+1, x2n)

= max{d(x2n+1, x2n), d(x2n+1, Tx2n+1), d(x2n, Sx2n),

d(x2n+1, Sx2n) + d(x2n, Tx2n+1)2

}

≤ max{d(x2n+1, x2n), d(x2n+1, x2n+2), d(x2n, x2n+1),

0 + d(x2n, x2n+2)2

}

= max{d(x2n+1, x2n), d(x2n+1, x2n+2)}= d(x2n+1, x2n)

(by (4.7)

).

(4.8)

So M(x2n+1, x2n) = d(x2n+1, x2n). Hence by (4.7),

d(x2n+2, x2n+1) ≤ d(x2n+1, x2n). (4.9)

Therefore, by (4.6) and (4.9), we conclude that


d(xk+1, xk) ≤ d(xk, xk−1), (4.10)

for all k ∈ N.Therefore, the sequence {d(xk+1, xk)} is monotone nonincreasing and bounded below.

So there exists r ≥ 0 such that

limn→∞

d(xn+1, xn) = limn→∞

M(xn+1, xn) = r. (4.11)

Since ϕ is l.s.c.,

ϕ(r) ≤ lim infn→∞

ϕ(M(xn, xn−1)) ≤ lim infn→∞

ϕ(M(x2n−1, x2n)). (4.12)

By (4.4), we conclude that

r ≤ r − 12ϕ(r), (4.13)

and so ϕ(r) = 0. Hence r = 0.

Step 2. {xn} is a bounded sequence.

Proof. If {xn} were unbounded, then by Step 1, {x2n} and {x2n−1} are unbounded. We choosethe sequence {n(k)}∞k=1 such that n(1) = 1, n(2) > n(1) is even and minimal in the sense thatd(xn(2), xn(1)) > 1, and d(xn(2)−2, xn(1)) ≤ 1, and similarly n(3) > n(2) is odd and minimal in thesense that d(xn(3), xn(2)) > 1, and d(xn(3)−2, xn(2)) ≤ 1, . . . , n(2k) > n(2k−1) is even and minimalin the sense that d(xn(2k), xn(2k−1)) > 1 and d(xn(2k)−2, xn(2k−1)) ≤ 1, and n(2k + 1) > n(2k) isodd and minimal in the sense that d(xn(2k+1), xn(2k)) > 1 and d(xn(2k+1)−2, xn(2k)) ≤ 1.

Obviously n(k) ≥ k for every k ∈ N. By Step 1, there exists N0 ∈ N such that for allk ≥N0 we have d(xk+1, xk) < 1/4. So for every k ≥N0, we have n(k + 1) − n(k) ≥ 2 and

1 < d(xn(k+1), xn(k)

)≤ d(xn(k+1), xn(k+1)−1

)+ d

(xn(k+1)−1, xn(k+1)−2

)+ d

(xn(k+1)−2, xn(k)

)≤ d(xn(k+1), xn(k+1)−1

)+ d

(xn(k+1)−1, xn(k+1)−2

)+ 1.

(4.14)

Hence limk→∞d(xn(k+1), xn(k)) = 1. Also

1 < d(xn(k+1), xn(k)

)≤ d(xn(k+1), xn(k+1)+1

)+ d

(xn(k+1)+1, xn(k)+1

)+ d

(xn(k)+1, xn(k)

)≤ d(xn(k+1), xn(k+1)+1

)+ d

(xn(k+1)+1, xn(k+1)

)+ d

(xn(k+1), xn(k)

)+ d

(xn(k), xn(k+1)

)+ d

(xn(k)+1, xn(k)

)≤ 2d

(xn(k+1), xn(k+1)+1

)+ d

(xn(k+1), xn(k)

)+ 2d

(xn(k)+1, xn(k)

),

(4.15)

and this shows that limk→∞d(xn(k+1)+1, xn(k)+1) = 1.


So if n(k + 1) is odd, then

d(xn(k+1)+1, xn(k)+1

) ≤M(xn(k+1), xn(k)

) − ϕ(M(xn(k+1), xn(k)

)), (4.16)

where

1 < d(xn(k+1), xn(k)

) ≤M(xn(k+1), xn(k)

)

= max{d(xn(k+1), xn(k)

), d

(xn(k+1), Txn(k+1)

), d

(xn(k), Sxn(k)

),

d(xn(k+1), Sxn(k)

)+ d

(xn(k), Txn(k+1)

)2

}

≤ max{d(xn(k+1), xn(k)

), d

(xn(k+1), xn(k+1)+1

), d

(xn(k), xn(k)+1

),

d(xn(k+1), xn(k)+1

)+ d

(xn(k), xn(k+1)+1

)2

}

≤ max{d(xn(k+1), xn(k)

), d

(xn(k+1), xn(k+1)+1

), d

(xn(k), xn(k)+1

),

2d(xn(k+1), xn(k)

)+ d

(xn(k)+1, xn(k)

)+ d

(xn(k+1)+1, xn(k+1)

)2

},

(4.17)

and this shows that limk→∞M(xn(k+1), xn(k)) = 1. Since ϕ is l.s.c. and (4.16) holds, we have1 ≤ 1 − ϕ(1). So ϕ(1) = 0 and this is a contradiction.

Step 3. {xn} is Cauchy.

Proof. Let Cn = sup{d(xi, xj) : i, j ≥ n}. Since {xn} is bounded, Cn < +∞ for all n ∈ N.Obviously {Cn} is decreasing. So there exists C ≥ 0 such that limn→∞ Cn = C. We need toshow that C = 0.

For every k ∈ N, there exists n(k), m(k) ∈ N such that m(k) > n(k) ≥ k and

Ck − 1k≤ d(xm(k), xn(k)

) ≤ Ck. (4.18)

By (4.18), we conclude that

limk→∞

d(xm(k), xn(k)

)= C. (4.19)

From Step 1 and (4.19), we have

limk→∞

d(xm(k)+1, xn(k)+1

)= lim

k→∞d(xm(k)+1, xn(k)

)

= limk→∞

d(xm(k), xn(k)+1

)= lim

k→∞d(xm(k), xn(k)

)= C.

(4.20)


So we may assume that for every k ∈ N, m(k) is odd and n(k) is even. Hence

d(xm(k)+1, xn(k)+1

)= d

(Txm(k), xn(k)+1

)≤ H({

Txm(k)}, Sxn(k)

)≤M(

xm(k), xn(k)) − ϕ(M(

xm(k), xn(k))),

(4.21)

where

d(xm(k), xn(k)

) ≤M(xm(k), xn(k)

)

= max{d(xm(k), xn(k)

), d

(xm(k), Txm(k)

), d

(xn(k), Sxn(k)

),

d(xm(k), Sxn(k)

)+ d

(xn(k), Txm(k)

)2

}

≤ max{d(xm(k), xn(k)

), d

(xm(k), xm(k)+1

), d

(xn(k), xn(k)+1

),

d(xm(k), xn(k)+1

)+ d

(xn(k), xm(k)+1

)2

}.

(4.22)

This inequality shows that limk→∞M(xm(k), xn(k)) = C. Since ϕ is l.s.c. and (4.21) holds, wehave C ≤ C − ϕ(C). Hence ϕ(C) = 0 and so C = 0. Therefore, {xn} is a Cauchy sequence.

Step 4. T and S have a common fixed point.

Proof. Since (X, d) is complete and {xn} is Cauchy, there exists x ∈ X such that limn→∞xn = x.For every n ∈ N

d(x2n+2, Sx) = d(Tx2n+1, Sx) ≤ H({Tx2n+1}, Sx)≤M(x2n+1, x) − ϕ(M(x2n+1, x)),

(4.23)

where

M(x2n+1, x)

= max{d(x2n+1, x), d(x2n+1, Tx2n+1), d(x, Sx),

d(x2n+1, Sx) + d(x, Tx2n+1)2

}

= max{d(x2n+1, x), d(x2n+1, x2n+2), d(x, Sx),

d(x2n+1, Sx) + d(x, x2n+2)2

},

(4.24)

and this shows that limn→∞M(x2n+1, x) = d(x, Sx).Since ϕ is l.s.c. and (4.23) holds, letting n → ∞ in (4.23) we get

d(x, Sx) ≤ d(x, Sx) − ϕ(d(x, Sx)). (4.25)


So ϕ(d(x, Sx)) = 0 and hence d(x, Sx) = 0. Since Sx ∈ CB(X), then x ∈ Sx.Also

d(Tx, x) ≤ H({Tx}, Sx) ≤M(x, x) − ϕ(M(x, x)), (4.26)

where

M(x, x) = max{d(x, x), d(x, Tx), d(x, Sx),

d(x, Sx) + d(x, Tx)2

}= d(x, Tx). (4.27)

So from (4.26), we have

d(Tx, x) ≤ d(Tx, x) − ϕ(d(Tx, x)). (4.28)

Thus ϕ(d(Tx, x)) = 0, and hence d(Tx, x) = 0. Therefore, x = Tx.

Remark 4.2. In the proof of Theorem 2.1 in Zhang and Song [8], the boundedness of thesequence {Cn} is used, but not proved. Also, for the proof that {xn} is a Cauchy sequence, themonotonicity of ϕ is used, without being explicitly mentioned.

In our proof of Theorem 4.1, which is different from [8, Theorem 2.1], ϕ is not assumedto be nondecreasing.

The following theorem extends Rhoades’ theorem by assuming ϕ to be only l.s.c..

Theorem 4.3. Let (X, d) be a complete metric space, and let T : X → X be a mapping such that

d(Tx, Ty

) ≤ d(x, y) − ϕ(d(x, y)), (4.29)

for every x, y ∈ X (i.e., ϕ-weak contractive), where ϕ : [0,+∞) → [0,+∞) is an l.s.c. function withϕ(0) = 0 and ϕ(t) > 0 for all t > 0. Then T has a unique fixed point.

Proof. The proof is similar to the proof of Theorem 4.1, by taking S = T , and replacingM(x, y)with d(x, y).

Remark 4.4. With a similar proof as in Theorem 4.1, in Theorem 4.3 we can replace theinequality (4.29) by the following inequality (4.30) for two single valued mappings T, S :X → X.

d(Tx, Sy

) ≤M(x, y

) − ϕ(d(x, y)). (4.30)

5. Conclusion and Future Directions

We have extended Nadler and Daffer-Kaneko’s theorems to a coincidence theorem withoutassuming the lower semicontinuity of the mapping x �→ d(x, Tx).

We have also extended Rhoades’ theorem by assuming ϕ to be only l.s.c., as well asZhang and Song’s theorem to the case where one of the mappings is multivalued. Futuredirections to be pursued in the context of this research include the investigation of the casewhere both mappings in Zhang and Song’s theorem are multivalued.


Acknowledgment

This work is dedicated to Professor W. A. Kirk for his 70th birthday

References

[1] P. Z. Daffer and H. Kaneko, “Fixed points of generalized contractive multi-valued mappings,” Journalof Mathematical Analysis and Applications, vol. 192, no. 2, pp. 655–666, 1995.

[2] C. Chifu and G. Petrusel, “Existence and data dependence of fixed points and strict fixed points forcontractive-type multivalued operators,” Fixed Point Theory and Applications, vol. 2007, Article ID 34248,8 pages, 2007.

[3] S. B. Nadler, “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488,1969.

[4] N. Shahzad and A. Lone, “Fixed points of multimaps which are not necessarily nonexpansive,” FixedPoint Theory and Applications, no. 2, pp. 169–176, 2005.

[5] B. E. Rhoades, “Some theorems on weakly contractive maps,” Nonlinear Analysis: Theory, Methods &Applications, vol. 47, no. 4, pp. 2683–2693, 2001.

[6] D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American MathematicalSociety, vol. 20, pp. 458–464, 1969.

[7] S. Reich, “Some fixed point problems,” Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe diScienze Fisiche, Matematiche e Naturali, vol. 57, no. 3-4, pp. 194–198, 1974.

[8] Q. Zhang and Y. Song, “Fixed point theory for generalized ϕ-weak contractions,” Applied MathematicsLetters, vol. 22, no. 1, pp. 75–78, 2009.

[9] N. A. Assad and W. A. Kirk, “Fixed point theorems for set-valued mappings of contractive type,”Pacific Journal of Mathematics, vol. 43, pp. 553–562, 1972.


Research ArticleConvergence of Inexact Iterative Schemes forNonexpansive Set-Valued Mappings

Simeon Reich and Alexander J. Zaslavski

Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel

Correspondence should be addressed to Alexander J. Zaslavski, [email protected]

Received 23 October 2009; Accepted 10 January 2010


Copyright q 2010 S. Reich and A. J. Zaslavski. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Taking into account possibly inexact data, we study iterative schemes for approximating fixedpoints and attractors of contractive and nonexpansive set-valued mappings, respectively. Moreprecisely, we are concerned with the existence of convergent trajectories of nonstationarydynamical systems induced by approximations of a given set-valued mapping.

1. Introduction

The study of iterative schemes for various classes of nonexpansive mappings is a centraltopic in Nonlinear Functional Analysis. It began with the classical Banach theorem [1] onthe existence of a unique fixed point for a strict contraction. This celebrated result alsoyields convergence of iterates to the unique fixed point. Since Banach’s seminal result,many developments have taken place in this area. We mention, in particular, existence andapproximation results regarding fixed points of those nonexpansive mappings which are notnecessarily strictly contractive [2, 3]. Such results were obtained for general nonexpansivemappings in special Banach spaces, while for self-mappings of general complete metricspaces most of the results were established for several classes of contractive mappings[4]. More recently, interesting developments have occurred for nonexpansive set-valuedmappings, where the situation is more difficult and less understood. See, for instance, [5–8] and the references cited therein. As we have already mentioned, one of the methodsfor proving the classical Banach result is to show the convergence of Picard iterations,which holds for any initial point. In the case of set-valued mappings, not all the trajectoriesof the dynamical system induced by the given mapping converge. Therefore, convergenttrajectories have to be constructed in a special way. For example, in the setting of [9], if at themoment t = 0, 1, . . . we reach a point xt, then the next iterate xt+1 is an element of T(xt), where


T is the given mapping, which approximates the best approximation of xt in T(xt). Since T isassumed to act on a general complete metric space, we cannot, in general, choose xt+1 to be thebest approximation of xt by elements of T(xt). Instead, we choose xt+1 so that it provides anapproximation up to a positive number εt, such that the sequence {εt}∞t=0 is summable. Thismethod allowed Nadler [9] to obtain the existence of a fixed point of a strictly contractiveset-valued mapping and the authors of [10] to obtain more general results.

In view of the above discussion, it is obviously important to study convergenceproperties of the iterates of (set-valued) nonexpansive mappings in the presence of errors andpossibly inaccurate data. The present paper is a contribution in this direction. More precisely,we are concerned with the existence of convergent trajectories of nonstationary dynamicalsystems induced by approximations of a given set-valued mapping. In the second section ofthe paper, we consider an iterative scheme for approximating fixed points of closed-valuedstrict contractions in metric spaces and prove our first convergence theorem (see Theorem 2.1below). Our second convergence theorem (Theorem 3.1) is established in the third sectionof our paper. We show there that if for any initial point, there exists a trajectory of thedynamical system induced by a nonexpansive set-valued mapping T , which converges toa given invariant set F, then a convergent trajectory also exists for a nonstationary dynamicalsystem induced by approximations of T .

2. Convergence to a Fixed Point of a Contractive Mapping

In this section we consider iterative schemes for approximating fixed points of closed-valuedstrict contractions in metric spaces.

We begin with a few notations.Throughout this paper, (X, ρ) is a complete metric space.For x ∈ X and a nonempty subset A of X, set

ρ(x,A) = inf{ρ(x, y)

: y ∈ A}. (2.1)

For each pair of nonempty A,B ⊂ X, put

H(A,B) = max

{supx∈A

ρ(x, B), supy∈B

ρ(y,A

)}. (2.2)

Let T : X → 2X \ {∅} be such that T(x) is a closed subset of X for each x ∈ X and

H(T(x), T

(y)) ≤ cρ(x, y), ∀x, y ∈ X, (2.3)

where c ∈ [0, 1) is a constant.

Theorem 2.1. Let {εi}∞i=0 ⊂ (0,∞) and {δi}∞i=0 ⊂ (0,∞) satisfy

∞∑i=0

εi <∞,∞∑i=0

δi <∞. (2.4)


Let Ti : X → 2X \ {∅} satisfy, for each integer i ≥ 0,

H(T(x), Ti(x)) ≤ εi, ∀x ∈ X. (2.5)

Assume that x0 ∈ X and that for each integer i ≥ 0,

xi+1 ∈ Ti(xi), ρ(xi, xi+1) ≤ ρ(xi, Ti(xi)) + δi. (2.6)

Then {xi}∞i=0 converges to a fixed point of T .

Proof. We first show that {xi}∞i=0 is a Cauchy sequence. To this end, let i ≥ 0 be an integer. Thenby (2.6) and (2.5),

ρ(xi+1, xi+2) ≤ ρ(xi+1, Ti+1(xi+1)) + δi+1

≤ ρ(xi+1, T(xi+1)) + sup{ρ(z, Ti+1(xi+1)) : z ∈ T(xi+1)

}+ δi+1

≤ ρ(xi+1, T(xi+1)) + εi+1 + δi+1

≤ H(Ti(xi), T(xi+1)) + εi+1 + δi+1

≤ H(T(xi), T(xi+1)) + εi + εi+1 + δi+1

≤ cρ(xi, xi+1) + εi + εi+1 + δi+1.

(2.7)

By (2.7),

ρ(x1, x2) ≤ cρ(x0, x1) + ε1 + δ1 + ε0, (2.8)

ρ(x2, x3) ≤ cρ(x1, x2) + ε1 + ε2 + δ2 (2.9)

≤ c2ρ(x0, x1) + c(ε1 + ε0 + δ1) + ε1 + ε2 + δ2. (2.10)

Now we show by induction that for each integer n ≥ 1,

ρ(xn, xn+1) ≤ cnρ(x0, x1) +n−1∑i=0

ci(εn−i + δn−i + εn−i−1). (2.11)

In view of (2.8) and (2.10), inequality (2.11) holds for n = 1, 2.


Assume that k ≥ 1 is an integer and that (2.11) holds for n = k. When combined with(2.7), this implies that

ρ(xk+1, xk+2) ≤ cρ(xk, xk+1) + εk+1 + δk+1 + εk

≤ ck+1ρ(x0, x1) +k−1∑i=0

ci+1(εk−i + δk−i + εk−1−i) + εk+1 + δk+1 + εk

= ck+1ρ(x0, x1) +k∑i=0

ci(εk+1−i + δk+1−i + εk−i).

(2.12)

Thus (2.11) holds for n = k + 1. Therefore, we have shown by induction that (2.11) holds forall integers n ≥ 1. By (2.11),

∞∑n=1

ρ(xn, xn+1) ≤∞∑n=1

(cnρ(x0, x1) +

n∑i=1

cn−i(εi + δi + εi−1)

)

≤ ρ(x0, x1)∞∑n=1

cn +∞∑i=1

⎛⎝ ∞∑

j=0

cj

⎞⎠(εi + δi + εi−1)

≤( ∞∑

n=0

cn)[

ρ(x0, x1) +∞∑n=1

(εn + δn + εn−1)

]<∞.

(2.13)

Thus {xn}∞n=0 is a Cauchy sequence and there exists

x∗ = limn→∞

xn. (2.14)

We claim that

x∗ ∈ T(x∗). (2.15)

Indeed, by (2.14), there is an integer n0 ≥ 1 such that for each integer n ≥ n0,

ρ(xn, x∗) ≤ ε8 . (2.16)


Let n ≥ n0 be an integer. By (2.3), (2.16) and (2.5),

ρ(x∗, T(x∗)) ≤ ρ(x∗, xn+1) + ρ(xn+1, T(x∗))

≤ ρ(x∗, xn+1) + ρ(xn+1, T(xn)) +H(T(xn), T(x∗))

≤ ρ(x∗, xn+1) + ρ(xn+1, T(xn)) + ρ(xn, x∗)

≤ ρ(xn+1, T(xn)) +ε

4

≤ ρ(xn+1, Tn(xn)) +H(Tn(xn), T(xn)) +ε

4

≤ εn + ε

4−→ ε

4

(2.17)

as n → ∞. Since ε is an arbitrary positive number, we conclude that

x∗ ∈ T(x∗), (2.18)

as claimed. Theorem 2.1 is proved.

3. Convergence to an Attractor of a Nonexpansive Mapping

In this section we show that if for any initial point, there exists a trajectory of the dynamicalsystem induced by a nonexpansive set-valued mapping T , which converges to an invariantset F, then a convergent trajectory also exists for a nonstationary dynamical system inducedby approximations of T .

Let T : X → 2X \ {∅} be such that T(x) is a closed set for each x ∈ X and

H(T(x), T

(y)) ≤ ρ(x, y), ∀x, y ∈ X. (3.1)

Theorem 3.1. Let {εi}∞i=0 ⊂ (0,∞),∑∞

i=0 εi <∞, F a nonempty closed subset of X,

T(F) ⊂ F, (3.2)

and for each integer i ≥ 0, let Ti : X → 2X \ {∅} satisfy

H(T(x), Ti(x)) ≤ εi, ∀x ∈ X. (3.3)

Assume that for each x ∈ X, there exists a sequence {xi}∞i=0 ⊂ X such that

x0 = x, xi+1 ∈ T(xi), i = 0, 1, . . . ,

limi→∞

ρ(xi, F) = 0.(3.4)


Then for each x ∈ X, there is a sequence {xi}∞i=0 ⊂ X such that

x0 = x, xi+1 ∈ Ti(xi), i = 0, 1, . . . ,

limi→∞

ρ(xi, F) = 0.(3.5)

We begin the proof of Theorem 3.1 with two lemmata.

Lemma 3.2. Let x ∈ X, p a natural number, {xi}pi=0 ⊂ X,

x0 = x, xi+1 ∈ Ti(xi), i = 0, . . . , p − 1, (3.6)

and let δ > 0. Then there is a natural number q > p and a sequence {xi}qi=p ⊂ X such that

xi+1 ∈ Ti(xi), i = p, . . . , q − 1,

ρ(xq, F

) ≤ δ. (3.7)

Proof. Choose a natural number p1 > p such that

∞∑i=p1

εi <δ

8(3.8)

and a sequence {xi}p1

i=p ⊂ X such that

xi+1 ∈ Ti(xi), i = p, . . . , p1 − 1. (3.9)

There is a sequence {yi}∞i=p1⊂ X such that

yp1 = xp1 , limi→∞

ρ(yi, F

)= 0, (3.10)

yi+1 ∈ T(yi), for all integers i ≥ p1. (3.11)

We are now going to define by induction a sequence {xi}∞i=p1⊂ X.

To this end, assume that k ≥ p1 is an integer and that we have already defined xi ∈ X,i = p1, . . . , k, such that

xi+1 ∈ Ti(xi), i = p1, . . . , k − 1, (3.12)

ρ(xk, yk

) ≤ 3

⎛⎝ k∑

i=p1

εi − εk⎞⎠. (3.13)

(Clearly, this assumption holds for k = p1.)


By (3.11) and (3.1),

yk+1 ∈ T(yk), (3.14)

H(T(yk), T(xk)

) ≤ ρ(xk, yk). (3.15)

By (3.15), there is yk+1 ∈ X such that

yk+1 ∈ T(xk), ρ(yk+1, yk+1

) ≤ ρ(xk, yk) + εk. (3.16)

Together with (3.3), this implies that

ρ(yk+1, Tk(xk)

) ≤ εk, (3.17)

and there is

xk+1 ∈ Tk(xk) (3.18)

such that

ρ(yk+1, xk+1

) ≤ 2εk. (3.19)

When combined with (3.16) and (3.13), this implies that

ρ(xk+1, yk+1

) ≤ ρ(xk+1, yk+1)+ ρ(yk+1, yk+1

) ≤ ρ(xk, yk) + 3εk ≤ 3k∑

i=p1

εi. (3.20)

Thus, by (3.18) and (3.20), the assumption we have made concerning k also holds for k + 1.Therefore, we have indeed defined by induction a sequence {xi}∞i=p1

such that

xi+1 ∈ Ti(xi), i = p1, . . . , (3.21)

and (3.13) holds for all integers k ≥ p1. By (3.11), there is an integer q > p1 + 2 such that

ρ(yq, F

)<δ

4. (3.22)

Together with (3.8) and (3.13), this inequality implies that

ρ(xq, F

) ≤ ρ(xq, yq) + ρ(yq, F) ≤∞∑i=p1

εi +δ

4<δ

2+δ

4. (3.23)

Lemma 3.2 is proved.


Lemma 3.3. Let {xi}∞i=0 ⊂ X,

xi+1 ∈ Ti(xi), i = 0, 1, . . . , (3.24)

δ > 0, p a natural number,

ρ(xp, F

) ≤ δ, (3.25)

∞∑i=p

εi < δ. (3.26)

Then ρ(xi, F) ≤ 3δ for all integers i ≥ p.

Proof. We intend to show by induction that for all integers n ≥ p,

ρ(xn, F) ≤ δ +n∑i=p

(2εi) − 2εn. (3.27)

Clearly, for n = p inequality (3.27) does hold. Assume now that n ≥ p is an integer and (3.27)holds. Then there is

yn ∈ F (3.28)

such that

ρ(xn, yn

) ≤ δ +n∑i=p

2εi − εn. (3.29)

By (3.24) and (3.3), there is

xn+1 ∈ T(xn) (3.30)

such that

ρ(xn+1, xn+1) ≤ 2εn. (3.31)

By (3.29) and (3.1),

H(T(xn), T

(yn)) ≤ ρ(xn, yn), (3.32)

and, in view of (3.30), there is

yn+1 ∈ T(yn)

(3.33)


such that

ρ(yn+1, xn+1

) ≤ ρ(xn, yn) + εn. (3.34)

By (3.33), (3.28), and (3.2),

yn+1 ∈ F. (3.35)

By (3.35), (3.31), (3.34), and (3.27),

ρ(xn+1, F) ≤ ρ(xn+1, yn+1

) ≤ ρ(xn+1, xn+1) + ρ(xn+1, yn+1

)

≤ 2εn + εn + ρ(xn, yn

) ≤ δ + 2εn +n∑i=p

2εi.(3.36)

Thus, the assumption we have made concerning n also holds for n + 1. Therefore, we mayconclude that inequality (3.27) indeed holds for all integers n ≥ p. Together with (3.26), thisimplies that for all integers n ≥ p,

ρ(xn, F) ≤ δ + 2δ = 3δ. (3.37)

Lemma 3.3 is proved.

Completion of the Proof of Theorem 3.1

Let x ∈ X. Since∑∞

i=0 εi < ∞, it follows from Lemma 3.2 that there exist a sequence {xi}∞i=0and a strictly increasing sequence of natural numbers {nk}∞k=1, constructed by induction, suchthat

xi+1 ∈ Ti(xi), i = 0, 1, . . . , (3.38)

and for each integer k ≥ 1,

ρ(xnk , F) ≤ 2−k,∞∑i=nk

εi < 2−k. (3.39)

It now follows from (3.39) and Lemma 3.3 that

limn→∞

ρ(xn, F) = 0. (3.40)

Theorem 3.1 is proved.


Acknowledgment

This research was supported by the Israel Science Foundation (Grant no. 647/07), the Fundfor the Promotion of Research at the Technion, and by the Technion President’s ResearchFund.

References

[1] S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equationsintegrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.

[2] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in AdvancedMathematics, Cambridge University Press, Cambridge, UK, 1990.

[3] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 ofMonographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.

[4] W. A. Kirk, “Contraction mappings and extensions,” in Handbook of Metric Fixed Point Theory, pp. 1–34,Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

[5] S. Reich and A. J. Zaslavski, “Convergence of iterates of nonexpansive set-valued mappings,” in SetValuedMappings with Applications in Nonlinear Analysis, vol. 4 of Mathematical Analysis and Applications,pp. 411–420, Taylor & Francis, London, UK, 2002.

[6] S. Reich and A. J. Zaslavski, “Generic existence of fixed points for set-valued mappings,” Set-ValuedAnalysis, vol. 10, no. 4, pp. 287–296, 2002.

[7] S. Reich and A. J. Zaslavski, “Two results on fixed points of set-valued nonexpansive mappings,”Revue Roumaine de Mathematiques Pures et Appliques, vol. 51, no. 1, pp. 89–94, 2006.

[8] B. Ricceri, “Une propriete topologique de l’ensemble des points fixes d’une contraction multivoque avaleurs convexes,” Atti della Accademia Nazionale dei Lincei, vol. 81, no. 3, pp. 283–286, 1987.


[10] F. S. de Blasi, J. Myjak, S. Reich, and A. J. Zaslavski, “Generic existence and approximation of fixedpoints for nonexpansive set-valued maps,” Set-Valued and Variational Analysis, vol. 17, no. 1, pp. 97–112, 2009.


Research ArticleDoes Kirk’s Theorem Hold for MultivaluedNonexpansive Mappings?

T. Domınguez Benavides and B. Gavira

Facultad de Matematicas, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain

Correspondence should be addressed to T. Domınguez Benavides, [email protected]

Received 25 September 2009; Accepted 29 December 2009


Copyright q 2010 T. Domınguez Benavides and B. Gavira. This is an open access articledistributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fixed Point Theory for multivalued mappings has many useful applications in Applied Sciences,in particular, in Game Theory and Mathematical Economics. Thus, it is natural to try of extendingthe known fixed point results for single-valued mappings to the setting of multivalued mappings.Some theorems of existence of fixed points of single-valued mappings have already been extendedto the multivalued case. However, many other questions remain still open, for instance, thepossibility of extending the well-known Kirk’s Theorem, that is: do Banach spaces with weaknormal structure have the fixed point property (FPP) for multivalued nonexpansive mappings?There are many properties of Banach spaces which imply weak normal structure and consequentlythe FPP for single-valued mappings (for example, uniform convexity, nearly uniform convexity,uniform smoothness,. . .). Thus, it is natural to consider the following problem: do these propertiesalso imply the FPP for multivalued mappings? In this way, some partial answers to the problemof extending Kirk’s Theorem have appeared, proving that those properties imply the existenceof fixed point for multivalued nonexpansive mappings. Here we present the main known resultsand current research directions in this subject. This paper can be considered as a survey, but somenew results are also shown.

1. Introduction

The presence or absence of a fixed point (i.e., a point which remains invariant under a map)is an intrinsic property of a map. However, many necessary or sufficient conditions for theexistence of such points involve a mixture of algebraic, topological, or metric properties ofthe mapping or its domain. By Metric Fixed Point Theory, we understand the branch ofFixed Point Theory concerning those results which depend on a metric and which are notpreserved when this metric is replaced by another equivalent metric. The first metric fixedpoint theorem was given by Banach in 1922.


Theorem 1.1 (Banach Contraction Principle, [1]). Let X be a complete metric space and T : X →X a contractive mapping, that is, there exists k ∈ [0, 1) such that d(Tx, Ty) ≤ kd(x, y) for everyx, y ∈ X. Then T has a (unique) fixed point x0. Moreover, x0 = limnT

nx for every x ∈ X.

Banach Theorem is a basic tool in Functional Analysis, Nonlinear Analysis andDifferential Equations. Thus, it is natural to look for some generalizations under weakerassumptions.

For many years Metric Fixed Point Theory just studied some extensions of BanachTheorem relaxing the contractiveness condition, and the extension of this result formultivalued mappings. In the 1960s, Metric Fixed Point Theory received a strong boost whenKirk [2] proved that every (singlevalued) nonexpansive mapping T : C → C, defined froma convex closed bounded subset C of a reflexive Banach space with normal structure, has afixed point.

The celebrated Kirk’s theorem had a profound impact in the development of FixedPoint Theory and iniciated the search of more general conditions for a Banach space and fora subset C which assure the existence of fixed points.

The result obtained by Kirk is, in some sense, surprising because it uses geometricproperties of Banach spaces (commonly used in Linear Functional Analysis, but rarelyconsidered in Nonlinear Analysis until then). Thus, it is the starting point for a newmathematical field: the application of the Geometric Theory of Banach Spaces to FixedPoint Theory. From that moment on, many researchers have tried to exploit this connection,essentially considering some other geometric properties of Banach spaces which can beapplied to prove the existence of fixed points for different types of nonlinear operators (e.g.,uniform smoothness, Opial property, nearly uniform convexity, nearly uniform smoothness,etc.).

Fixed Point Theory for multivalued mappings has useful applications in AppliedSciences, in particular, in Game Theory and Mathematical Economics. Thus, it is naturalto study the problem of the extension of the known fixed point results for singlevaluedmappings to the setting of multivalued mappings.

Some theorems of existence of fixed points of single-valued mappings have alreadybeen extended to the multivalued case. For example, in 1969 Nadler [3] extended the BanachContraction Principle to multivalued contractive mappings in complete metric spaces.However, many other questions remain open, for instance, the possibility of extending thewell-known Kirk’s Theorem [2], that is, do Banach spaces with weak normal structure havethe fixed point property (FPP) for multivalued nonexpansive mappings?

There are many properties of Banach spaces which imply weak normal structure andconsequently the FPP for singlevalued mappings (e.g., uniform convexity, nearly uniformconvexity, uniform smoothness, . . .). Thus, it is natural to consider the following problem:Do these properties also imply the FPP for multivalued mappings? As a consequence, somepartial answers to the problem of extending Kirk’s Theorem have appeared, which aredirected to prove that those properties imply the existence of fixed point for multivaluednonexpansive mappings.

Here we present the main known results and current research directions in this subject.This paper can be considered as a survey, but some new results are also included.

2. PreliminariesIn this section we recall the notion of normal structure and some properties of Banach spaceswhich imply normal structure.


Normal structure plays an essential role in some problems of Metric Fixed PointTheory, especially those concerning nonexpansive mappings. The notion of normal structurewas introduced by Brodskiı and Mil’man [4] in 1948 in order to study fixed points ofisometries. Later, the notion of normal structure was generalized for the weak topology.

Definition 2.1. A Banach space X is said to have normal structure (NS) (resp., weak normalstructure (w-NS)) if for every bounded closed (resp., weakly compact) convex subset C of Xwith diam(C) := sup{‖x − y‖ : x, y ∈ C} > 0, there exists x ∈ C such that sup{‖x − y‖ : y ∈C} < diam(C).

In 1965 Kirk [2] obtained a strong connection between normal structure and the FPPfor nonexpansive mappings.

Theorem 2.2. Let C be a bounded closed (resp., weakly compact) convex subset of a Banach space Xand let T : C → C be a nonexpansive mapping (i.e., ‖Tx − Ty‖ ≤ ‖x −y‖ for every x, y ∈ C). If X isa reflexive Banach space with normal structure (resp., a Banach space with w-NS), then T has a fixedpoint.

Bynum [5] defined two coefficients related to normal structure and weak normalstructure.

Definition 2.3. The normal structure coefficient of a Banach space X is defined by

N(X) = inf{

diam(A)r(A)

: A ⊂ X convex closed and bounded with diam(A) > 0}, (2.1)

where diam(A) denotes the diameter of A defined by diam(A) = sup{‖x −y‖ : x, y ∈ A} andr(A) denotes the Chebyshev radius of A defined by r(A) = inf{sup{‖x−y‖ : y ∈ A} : x ∈ A}.

The weakly convergent sequence coefficient of X is defined by

WCS(X) = inf{

diama({xn})ra({xn})

}, (2.2)

where the infimum is taken over all weakly convergent sequences {xn} which are not normconvergent, where,

diama({xn}) = limk→∞

sup{‖xn − xm‖ : n,m ≥ k},

ra({xn}) = inf{

lim supn‖xn − x‖ : x ∈ co({xn})

} (2.3)

denote the asymptotic diameter and radius of {xn}, respectively.

We recall that X is said to have uniform normal structure (UNS) (resp., weak uniformnormal structure (w-UNS)) if N(X) > 1 (resp., WCS(X) > 1). Notice that this is notthe common definition of weak uniform normal structure and is often known as Bynum’scondition. It is known that if X has uniform normal structure, then X is reflexive [6].


In the latest fifty years, some geometrical properties implying normal structure havebeen studied. Here we are going to recall some of these properties and some results whichprove that these properties imply the existence of fixed point for multivalued mappings.

First we consider the Opial property. Opial [7] was the first who studied such aproperty giving applications to Fixed Point Theory. The uniform Opial property was definedin [8] by Prus, and the Opial modulus was introduced in [9] by Lin et al.

Definition 2.4. We will say that a Banach space X satisfies the Opial property if for everyweakly null sequence {xn} and every x /= 0 in X, we have

lim infn→∞

‖xn‖ < lim infn→∞

‖xn + x‖. (2.4)

We will say that X satisfies the nonstrict Opial property if

lim infn→∞

‖xn‖ ≤ lim infn→∞

‖xn + x‖ (2.5)

under the same conditions.The Opial modulus of X is defined for c ≥ 0 as

rX(c) = inf{

lim infn‖xn + x‖ − 1

}, (2.6)

where the infimum is taken over all x ∈ X with ‖x‖ ≥ c and all weakly null sequences {xn}in X with lim infn‖xn‖ ≥ 1.

We will say that X satisfies the uniform Opial property if rX(c) > 0 for all c > 0.

There are some relationships between the notions of Opial property and normalstructure. If X is a Banach space which satisfies the Opial property, then X has w-NS [10].On the other hand, WCS(X) ≥ 1 + rX(1) [9, Theorem 3.2]. Consequently, X has w-UNS ifrX(1) > 0.

Next we study the uniform convexity of the space, which is another geometricalproperty related with normal structure. We recall that a Banach space X is uniformly convex(UC) if and only if

δX(ε) := inf{

1 −∥∥∥∥x + y

2

∥∥∥∥ : x, y ∈ Bx,∥∥x − y∥∥ ≥ ε

}> 0 (2.7)

for each ε ∈ [0, 2], or equivalently

ε0(X) := sup{ε ≥ 0 : δX(ε) = 0} = 0. (2.8)

The Clarkson modulus δX(ε) and the coefficient of normal structure N(X) are relatedby the following inequality: N(X) ≥ (1 − δX(1))−1. Consequently, the condition δX(1) > 0implies that X is reflexive and has uniform normal structure. In particular, notice that notonly do uniformly convex spaces have normal structure, but so do all those spaces which donot have segments of length greater than or equal to 1 near the unit sphere.


In 1980 Huff [11] initiated the study of nearly uniform convexity which is aninfinite-dimensional generalization of uniform convexity. Independently of Huff, Goebel andSekowski [12] also introduced a property which is equivalent to nearly uniform convexityunder the name of noncompact uniform convexity. It is known that a Banach space X isnearly uniformly convex (NUC) if and only if

ΔX,φ(ε) := inf{

1 − d(0, A) : A ⊂ BX convex, φ(A) > ε}> 0 (2.9)

for each ε > 0, or equivalently

εφ(X) := sup{ε ≥ 0 : ΔX,φ(ε) = 0

}= 0, (2.10)

where φ is a measure of noncompactness. Also we are going to use the following equivalentdefinition: X is NUC if and only if X is reflexive and

ΔX(ε) := inf{

1 − ‖x‖ : {xn} ⊂ BX, xn ⇀ x, lim infn‖xn − x‖ ≥ ε

}> 0 (2.11)

for each ε > 0, or equivalently

Δ0(X) := sup{ε > 0 : ΔX(ε) = 0} = 0. (2.12)

WhenX is a reflexive Banach space, β is the separation measure and χ is the Hausdorffmeasure (for definitions see, for instance, [13] or [14]), we have the following relationshipsamong the different moduli:

ΔX,β(ε) ≤ ΔX(ε) ≤ ΔX,χ(ε), (2.13)

and consequently,

εβ(X) ≥ Δ0(X) ≥ εχ(X). (2.14)

If the space X satisfies the nonstrict Opial property, then Δ0(X) coincides with εχ(X).On the other hand, if εβ(X) < 1 (in particular, if X is NUC), then X is reflexive and has

weak uniform normal structure (see [13, page 125]).The dual concept of uniform convexity is uniform smoothness which is also related to

normal structure. A Banach space X is said to be uniformly smooth (US) if

ρ′X(0) = limt→ 0+

ρX(t)t

= 0, (2.15)


where ρX is the modulus of smoothness of X, defined by

ρX(t) = sup{

12(∥∥x + ty

∥∥ + ∥∥x − ty∥∥) − 1 : ‖x‖ ≤ 1,∥∥y∥∥ ≤ 1

}(2.16)

for t ≥ 0.It is known that ρ′X(0) < 1/2 implies that X is reflexive and has uniform normal

structure [15–17]. However, the infinite-dimensional generalization of uniform smoothness,nearly uniform smoothness, does not imply normal structure [13, Example VI.2].

3. Some Properties Implying Weak Normal Structure andthe FPP for Multivalued Mappings

In this section we are going to show some results which prove that some properties implyingweak normal structure also imply the existence of fixed point for multivalued nonexpansivemappings. As a consequence these results give some partial answers to the problem ofextending Kirk’s Theorem.

Throughout this section K(X) (resp., KC(X)) will denote the family of all nonemptycompact (resp., compact convex) subsets of X. We recall that a multivalued mapping T :X → K(X) is said to be nonexpansive if H(Tx, Ty) ≤ ‖x − y‖ for every x, y ∈ X, whereH(·, ·) denotes the Hausdorff metric given by

H(A,B) := max

{supa∈A

d(a, B), supb∈B

d(b,A)

}(3.1)

for every bounded subsets A and B of X.In 1973 Lami Dozo gave the following result of existence of fixed point for those spaces

which satisfy the Opial property.

Theorem 3.1 (Lami Dozo [18, Theorem 3.2]). Let X be a Banach space which satisfies the Opialproperty, let C be a weakly compact convex subset of X, and let T : C → K(C) be a nonexpansivemapping. Then T has a fixed point, that is, there exists x ∈ C such that x ∈ Tx.

In 1974 Lim [19] gave a similar result for uniformly convex spaces using Edelstein’smethod of asymptotic centers [20].

Theorem 3.2 (Lim [19]). Let X be a uniformly convex Banach space, let C be a closed boundedconvex subset of X and T : C → K(C) be a nonexpansive mapping. Then T has a fixed point.

In 1990 Kirk and Massa proved the following partial generalization of Lim’s Theoremusing asymptotic centers of sequences and nets. We recall that, given a bounded sequence{xn} in a Banach space X and a subset C of X, the asymptotic center of {xn} with respect toC is defined by

A(C, {xn}) :={x ∈ C : lim sup

n‖xn − x‖ = r(C, {xn})

}, (3.2)


where r(C, {xn}) denotes the asymptotic radius of {xn}with respect to C defined by

r(C, {xn}) := inf{

lim supn‖xn − x‖ : x ∈ C

}. (3.3)

Theorem 3.3 (Kirk and Massa [21]). Let C be a closed bounded convex subset of a Banach spaceX and T : C → KC(C) a nonexpansive mapping. If the asymptotic center in C of each boundedsequence of X is nonempty and compact, then T has a fixed point.

We do not know a complete characterization of those spaces in which asymptoticcenters of bounded sequences are compact. Nevertheless, there are some partial answers,for example, k-uniformly convex Banach spaces satisfy that condition [22]. However, anexample given by Kuczumov and Prus [23] shows that in nearly uniformly convex spaces,the asymptotic center of a bounded sequence with respect to a closed bounded convex subsetis not necessarily compact. Therefore, the problem of obtaining fixed point results in nearlyuniformly convex spaces remained open. This question (together with the same question foruniformly smooth spaces) explicitly appeared in a survey about Metric Fixed Point Theoryfor multivalued mappings published by Xu [24] in 2000.

The analysis of the importance of the asymptotic center in Kirk-Massa Theorem ledDomınguez Benavides and Lorenzo to study some connections between asymptotic centersand the geometry of certain spaces, including nearly uniformly convex spaces. Thus, in [25]Domınguez and Lorenzo obtained the following relationship between the Chebyshev radiusof the asymptotic center of a bounded sequence and the modulus of noncompact convexitywith respect to the measures β and χ.

Theorem 3.4 (see [25, Theorem 3.4]). Let C be a closed convex subset of a reflexive Banach spaceX and {xn} a bounded sequence in C which is regular with respect to C (i.e., the asymptotic radius isinvariant for all subsequences of {xn}). Then

rC(A(C, {xn})) ≤(1 −ΔX,β

(1−))r(C, {xn}), (3.4)

where the Chebyshev radius of a bounded subset D of X relative to C is defined by

rC(D) := inf{

sup{∥∥x − y∥∥ : y ∈ D} : x ∈ C}. (3.5)

Moreover, if X satisfies the nonstrict Opial property, then

rC(A(C, {xn})) ≤(1 −ΔX,χ

(1−))r(C, {xn}). (3.6)

The previous inequalities give an iterative method which reduces at each step thevalue of the Chebyshev radius for a chain of asymptotic centers. Consequently, Domınguezand Lorenzo deduced in [26] the following partial extension of Kirk’s Theorem which, inparticular, assures that nearly uniformly convex spaces have the fixed point property formultivalued nonexpansive mappings.


Theorem 3.5 (see [26, Theorem 3.5]). Let C be a nonempty closed bounded convex subset of aBanach space X such that εβ(X) < 1. Let T : C → KC(C) be a nonexpansive mapping. Then T hasa fixed point.

This result guarantees, in particular, the existence of fixed points in nearly uniformlyconvex spaces (because εβ(X) = 0 if X is NUC), giving a positive answer to one of theprevious open problems proposed by Xu.

Dhompongsa et al. [27] observed that the main tool used in the proofs in [25, 26], inorder to obtain fixed point results for multivalued nonexpansive mappings, is a relationshipbetween the Chebyshev radius of the asymptotic center of a bounded sequence andthe asymptotic radius of the sequence. This relationship also gives an iterative methodwhich reduces at each step the value of the Chebyshev radius for a chain of asymptoticcenters. Consequently, in [27, 28] they introduced the Domınguez-Lorenzo condition ((DL)-condition, in short) and property (D) in the following way.

We recall that a sequence {xn} is regular with respect to C if r(C, {xn}) = r(C, {xni})for all subsequences {xni} of {xn}, and {xn} is asymptotically uniform with respect to C ifA(C, {xn}) = A(C, {xni}) for all subsequences {xni} of {xn}.

Definition 3.6. A Banach space X is said to satisfy the (DL)-condition if there exists λ ∈ [0, 1)such that for every weakly compact convex subset C of X and for every bounded sequence{xn} in C which is regular with respect to C

rC(A(C, {xn})) ≤ λr(C, {xn}). (3.7)

A Banach space X is said to satisfy property (D) if there exists λ ∈ [0, 1) such that forany nonempty weakly compact convex subsetC ofX, any bounded sequence {xn} inCwhichis regular and asymptotically uniform with respect to C, and any sequence {yn} ⊂ A(C, {xn})which is regular and asymptotically uniform with respect to C, we have

r(C,{yn}) ≤ λr(C, {xn}). (3.8)

From the definition it is easy to deduce that property (D) is weaker than the (DL)-condition. Dhompongsa et al. proved in [28, Theorem 3.2] and [28, Theorem 3.5] thatproperty (D) implies w-NS and the FPP for multivalued nonexpansive mappings.

Theorem 3.7 (see [28, Theorem 3.3]). LetX be a Banach space satisfying property (D). ThenX hasw-NS.

Theorem 3.8 (see [28, Theorem 3.6]). Let C be a nonempty weakly compact convex subset of aBanach space X which satisfies property (D). Let T : C → KC(C) be a nonexpansive mapping. ThenT has a fixed point.

From Theorem 3.5 every Banach space with εβ(X) < 1 satisfies the (DL)-condition.In this paper we present some other properties concerning geometrical constants of Banachspaces which also imply the (DL)-condition or property (D).

Since our goal is to study if properties implying w-NS also imply the FPP formultivalued mappings, a possible approach to that problem is to study if these propertiesimply either the (DL)-condition or property (D). These results will give only partial answers


to the problem of extending Kirk’s Theorem for multivalued mappings because we knowthat uniform normal structure does not imply property (D) ([29, Proposition 5]); therefore,the problem of extending Kirk’s Theorem cannot be fully solved by this approach. In thissetting the following results have been obtained.

Theorem 3.9 (Dhompongsa et al. [27, Theorem 3.4]). Let X be a uniformly nonsquare Banachspace with property WORTH. Then X satisfies the (DL)-condition.

We recall that a Banach space X is uniformly nonsquare if there exists δ > 0 such that ‖x +y‖ ∧ ‖x − y‖ ≤ 2 − δ for every x, y ∈ BX or equivalently J(X) < 2, where J(X) denotes the Jamesconstant of X defined by

J(X) = sup{∥∥x + y

∥∥ ∧ ∥∥x − y∥∥ : x, y ∈ BX}. (3.9)

X is said to satisfy property WORTH if

lim supn‖xn + x‖ = lim sup

n‖xn − x‖ (3.10)

for any x ∈ X and any weakly null sequence {xn} in X.

Theorem 3.10 (Dhompongsa et al. [28, Theorem 3.7]). Let X be Banach space such that

CNJ(X) < 1 +WCS(X)2

4, (3.11)

where CNJ(X) denotes the Jordan-von Neumann constant of X defined by

CNJ(X) = sup

{∥∥x + y∥∥2 +

∥∥x − y∥∥2

2‖x‖2 + 2∥∥y∥∥2

: x, y ∈ X not both zero

}. (3.12)

Then X satisfies property (D).

Theorem 3.11 (Domınguez Benavides and Gavira [29, Corollary 1]). Let X be a Banach spacesuch that

ρ′X(0) <12. (3.13)

Then X satisfies the (DL)-condition. In particular, uniformly smooth Banach spaces (ρ′X(0) = 0)satisfy the (DL)-condition.

Theorem 3.12 (Domınguez Benavides and Gavira [29, Corollary 2]). Let X be a Banach spacesuch that one of the following two equivalent conditions is satisfied:

(1) rX(1) > 0,

(2) Δ0(X) < 1.

Then X satisfies the (DL)-condition.


Theorem 3.13 (Saejung [30, Theorem 3]). A Banach space X has property (D) whenever ε0(X) <WCS(X).

This result improves Theorem 3.10 because it is easy to see that CNJ(X) ≥ 1 +(1/4)(ε0(X))2.

Theorem 3.14 (Kaewkhao [31, Corollary 3.2]). Let X be a Banach space such that

J(X) < 1 +1

μ(X), (3.14)

where J(X) denotes the James constant of X defined by

J(X) := sup{

min(∥∥x + y

∥∥,∥∥x − y∥∥) : x, y ∈ BX}, (3.15)

and μ(X) denotes the coefficient of worthwhileness of X defined as the infimum of the set of realnumbers r > 0 such that

lim supn→∞

‖x + xn‖ ≤ r lim supn→∞

‖x − xn‖ (3.16)

for all x ∈ X and all weakly null sequences {xn} in X. Then X satisfies the (DL)-condition.

Remark 3.15. This result improves Theorem 3.9 because if X is a uniformly nonsquare Banachspace with property WORTH, then

J(X) < 2 = 1 +1

μ(X). (3.17)

Theorem 3.16 (Kaewkhao [31, Theorem 4.1]). Let X be a Banach space such that

CNJ(X) < 1 +1

μ(X)2. (3.18)


Theorem 3.17 (Gavira [32, Theorem 4]). Let X be a Banach space such that

ρ′X(0) <1

μ(X). (3.19)


Remarks 3.18. (i) This result is a strict generalization of Theorem 3.16 (see [32]).(ii) Theorem 3.17 applies to the Bynum space �2,1 while Theorem 3.11 does not (see

[32]). However, we do not know if ρ′X(0) < 1/2 implies ρ′X(0) < 1/μ(X).


Finally we show a new result which gives a property implying the (DL)-condition interms of Clarkson modulus and the Garcıa-Falset coefficient.

Theorem 3.19. Let X be a Banach space such that

δX

(1

R(X)+

√1 − 1

R(X)+

1

(R(X))2

)>

12

(1 − 1

R(X)

), (3.20)

where R(X) denotes the Garcıa-Falset coefficient of X defined by

R(X) = sup{

lim infn→∞

‖xn + x‖ : x ∈ BX, {xn} ⊂ BX, xn ⇀ 0}. (3.21)


Proof. Let C be a nonempty weakly compact convex subset of X. Let {xn} be a boundedsequence in C which is regular with respect to C. Denote A = A(C, {xn}), r = r(C, {xn}),and R = R(X). By translation and multiplication we can assume that {xn} is weakly null andlimn‖xn‖ = 1. Let z ∈ A, then lim supn‖xn − z‖ = r ≤ 1. Denote ‖z‖ by a. By the definition ofR, we have

lim infn

∥∥∥xn + z

a

∥∥∥ ≤ R. (3.22)

For every ε > 0, there exists N ∈ N such that

(1) ‖xN − z‖ < r + ε,

(2) ‖xN + z/a‖ < R + ε,

(3) ‖(1/(r+ε)−1/(R+ε))xN −(1/(r+ε)+1/a(R+ε))z‖ > (1/(r+ε)+1/a(R+ε) )a((r−ε)/r),

(4) ‖xN − (1/(r + ε) − 1/a(R + ε))/(1/(r + ε) + 1/(R + ε))z‖ > r − ε.

Consider u = (1/(r + ε))(xN − z) ∈ BX and v = (1/(R + ε))(xN + z/a) ∈ BX . Using theabove estimates we obtain

‖u − v‖ =∥∥∥∥(

1r + ε

− 1R + ε

)xN −

(1

r + ε+

1a(R + ε)

)z

∥∥∥∥

>

(1

r + ε+

1a(R + ε)

)a

(r − εr

)=(

a

r + ε+

1R + ε

)(r − εr

)

>

(a

r+

1R

)− o(ε),

(3.23)


where o(ε) tends to 0 as ε → 0+. Furthermore,

‖u + v‖ =∥∥∥∥(

1r + ε

− 1R + ε

)xN −

(1

r + ε+

1a(R + ε)

)z

∥∥∥∥

=(

1r + ε

+1

R + ε

)∥∥∥∥xN − 1/(r + ε) + 1/a(R + ε)1/(r + ε) + 1/(R + ε)

z

∥∥∥∥ >(

1r + ε

+1

R + ε

)(r − ε)

>

(1r+

1R

)r − o(ε).

(3.24)

Also we have

‖u + v‖ ≥ 1r + ε

+1

R + ε−(

1r + ε

− 1a(R + ε)

)a ≥ 2

R + ε+

1r + ε

− 1 >2R

+1r− 1 − o(ε).

(3.25)

Define f(r) = 2/R+ 1/r − 1 and g(r) = 1+ r/R. Thus, ‖u+v‖ ≥ max{f(r), g(r)}−o(ε).Since f(r) = g(r) for r = r0 = 1 − R +

√R2 + 1 − R, we obtain

‖u + v‖ ≥ g(r0) =1R

+

√1 − 1

R+

1R2. (3.26)

Consequently, we have

12

(a

r+

1R

)− o(ε) ≤ ‖u − v‖

2≤ 1 − δX

⎛⎝ 1R

+

√1 − 1

R+

1R2− o(ε)

⎞⎠. (3.27)

Since the last inequality is true for every ε > 0 and every z ∈ A, letting ε → 0 andusing the continuity of δ(·), we obtain

rC(A) ≤⎛⎝2 − 1

R− 2δX

⎛⎝ 1R

+

√1 − 1

R+

1R2

⎞⎠⎞⎠r. (3.28)

In [33] it is proved that X has normal structure under the slightly weaker condition

δX

(1 +

1R(X)

)>

12

(1 − 1

R(X)

). (3.29)

It is an open question if this condition implies the (DL)-condition.

Corollary 3.20. Let X be a uniformly nonsquare Banach space such that R(X) = 1. Then X satisfiesthe (DL)-condition.


4. Fixed Point Results for Multivalued Nonexpansive Mappings inModular Function Spaces

The theory of modular spaces was initiated by Nakano [34] in 1950 in connection withthe theory of order spaces and redefined and generalized by Musielak and Orlicz [35] in1959. Even though a metric is not defined, many problems in metric fixed point theory canbe reformulated and solved in modular spaces (see, for instance, [36–39]). In particular,Dhompongsa et al. [40] have obtained some fixed point results for multivalued mappingsin modular functions spaces.

Let us recall some basic concepts about modular function spaces (for more details thereader is referred to [41, 42]).

Let Ω be a nonempty set and Σ a nontrivial σ-algebra of subsets of Ω. Let P be a δ-ringof subsets of Ω, such that E ∩ A ∈ P for any E ∈ P and A ∈ Σ. Let us assume that thereexists an increasing sequence of sets Kn ∈ P such that Ω =

⋃Kn (for instance, P can be the

class of sets of finite measure in a σ-finite measure space). By Ewe denote the linear space ofall simple functions with supports from P. ByM we will denote the space of all measurablefunctions, that is, all functions f : Ω → R such that there exist a sequence {gn} ∈ E, |gn| ≤ |f |and gn(ω) → f(ω) for all ω ∈ Ω.

Let us recall that a set function μ : Σ → [0,∞] is called a σ-subadditive measureif μ(∅) = 0, μ(A) ≤ μ(B) for any A ⊂ B and μ(

⋃An) ≤

∑μ(An) for any sequence of sets

{An} ⊂ Σ. By 1A, we denote the characteristic function of the set A.

Definition 4.1. A functional ρ : E × Σ → [0,∞] is called a function modular if:

(1) ρ(0, E) = 0 for any E ∈ Σ;

(2) ρ(f, E) ≤ ρ(g, E) whenever |f(ω)| ≤ |g(ω)| for any ω ∈ Ω, f, g ∈ E and E ∈ Σ;

(3) ρ(f, ·) : Σ → [0,∞] is a σ-subadditive measure for every f ∈ E;

(4) ρ(α,A) → 0 as α decreases to 0 for every A ∈ P, where ρ(α,A) = ρ(α1A,A);

(5) if there exists α > 0 such that ρ(α,A) = 0, then ρ(β,A) = 0 for every β > 0;

(6) for any α > 0, ρ(α, ·) is order continuous on P, that is, ρ(α,An) → 0 if {An} ⊂ Pand decreases to ∅.

A σ-subadditive measure ρ is said to be additive if ρ(f,A ∪ B) = ρ(f,A) + ρ(f, B),whenever A,B ∈ Σ such that A ∩ B = ∅ and f ∈ M.

The definition of ρ is then extended to f ∈ M by

ρ(f, E)= sup

{ρ(g, E)

: g ∈ E, ∣∣g(ω)∣∣ ≤ ∣∣f(ω)∣∣ for every ω ∈ Ω}. (4.1)

Definition 4.2. A set E is said to be ρ-null if ρ(α, E) = 0 for every α > 0. A property p(ω) issaid to hold ρ-almost everywhere (ρ-a.e.) if the set {ω ∈ Ω : p(ω) does not hold} is ρ-null.For example, we will say frequently fn → fρ-a.e.

Note that a countable union of ρ-null sets is still ρ-null. In the sequel we willidentify sets A and B whose symmetric difference AΔB is ρ-null, similarly we will identifymeasurable functions which differ only on a ρ-null set.


Under the above conditions, we define the function ρ :M → [0,∞] by ρ(f) = ρ(f,Ω).We know from [41] that ρ satisfies the following properties:

(i) ρ(f) = 0 if and only if f = 0 ρ-a.e.

(ii) ρ(αf) = ρ(f) for every scalar α with |α| = 1 and f ∈ M.

(iii) ρ(αf + βg) ≤ ρ(f) + ρ(g) if α + β = 1, α, β ≥ 0 and f, g ∈ M.

In addition, if the following property is satisfied

(iii)′ ρ(αf + βg) ≤ αρ(f) + βρ(g) if α + β = 1, α, β ≥ 0 and f, g ∈ M,

we say that ρ is a convex modular.A function modular ρ is called σ-finite if there exists an increasing sequence of sets

Kn ∈ P such that 0 < ρ(1Kn) <∞ and Ω =⋃Kn.

The modular ρ defines a corresponding modular space Lρ, which is given by

Lρ ={f ∈ M : ρ

(λf) −→ 0 as λ −→ 0

}. (4.2)

In general the modular ρ is not subadditive and therefore does not behave as a normor a distance. But one can associate to a modular an F-norm. In fact, when ρ is convex, theformula

∥∥f∥∥l = inf{α > 0 : ρ

(f

α

)≤ 1}

(4.3)

defines a norm which is frequently called the Luxemburg norm. The formula

∥∥f∥∥a = inf{

1k

(1 + ρ

(kf))

: k > 0}

(4.4)

defines a different norm which is called the Amemiya norm. Moreover, ‖ · ‖l and ‖ · ‖a areequivalent norms. We can also consider the space

Eρ ={f ∈ M : ρ

(αf, ·) is order continuous for all α > 0

}. (4.5)

Definition 4.3. A function modular ρ is said to satisfy the Δ2-condition if

supn≥1

ρ(2fn,Dk

) −→ 0 as k −→ ∞ whenever{fn} ⊂ M, Dk ∈ Σ

decreases to ∅ and supn≥1

ρ(fn,Dk

) −→ 0 as k −→ ∞.(4.6)

It is known that the Δ2-condition is equivalent to Eρ = Lρ.

Definition 4.4. A function modular ρ is said to satisfy the Δ2-type condition if there existsK > 0 such that for any f ∈ Lρ we have ρ(2f) ≤ Kρ(f).


In general, the Δ2-type condition and Δ2-condition are not equivalent, even though itis obvious that the Δ2-type condition implies the Δ2-condition.

Definition 4.5. Let Lρ be a modular space.

(1) The sequence {fn} ⊂ Lρ is said to be ρ-convergent to f ∈ Lρ if ρ(fn − f) → 0 asn → ∞.

(2) The sequence {fn} ⊂ Lρ is said to be ρ-a.e. convergent to f ∈ Lρ if the set {ω ∈ Ω :fn(ω) � f(ω)} is ρ-null.

(3) A subset C of Lρ is called ρ-a.e. closed if the ρ-a.e. limit of a ρ-a.e. convergentsequence of C always belongs to C.

(4) A subset C of Lρ is called ρ-a.e. compact if every sequence in C has a ρ-a.e.convergent subsequence in C.

(5) A subset C of Lρ is called ρ-bounded if

diamρ(C) = sup{ρ(f − g) : f, g ∈ C} <∞. (4.7)

We know by [41] that under the Δ2-condition the norm convergence and modularconvergence are equivalent, which implies that the norm and modular convergence are alsothe same when we deal with the Δ2-type condition. In the sequel we will assume that themodular function ρ is convex and satisfies the Δ2-type condition. Hence, the ρ-convergencedefines a topology which is identical to the norm topology.

In the same way as the Hausdorff distance defined on the family of bounded closedsubsets of a metric space, we can define the analogue to the Hausdorff distance for modularfunction spaces. We will speak of ρ-Hausdorff distance even though it is not a metric.

Definition 4.6. Let C be a nonempty subset of Lρ. We will denote by Fρ(C) the family ofnonempty ρ-closed subsets of C and by Kρ(C) the family of nonempty ρ-compact subsetsof C. Let Hρ(·, ·) be the ρ-Hausdorff distance on Fρ(Lρ), that is,

Hρ(A,B) = max

{supf∈A

distρ(f, B), supg∈B

distρ(g,A

)}, A, B ∈ Fρ

(Lρ), (4.8)

where distρ(f, B) = inf{ρ(f − g) : g ∈ B} is the ρ-distance between f and B. A multivaluedmapping T : C → Fρ(Lρ) is said to be a ρ-contraction if there exists a constant k ∈ [0, 1) suchthat

Hρ

(Tf, Tg

) ≤ kρ(f − g), f, g ∈ C. (4.9)

If it is valid when k = 1, then T is called ρ-nonexpansive.A function f ∈ C is called a fixed point for a multivalued mapping T if f ∈ TF.

Dhompongsa et al. [40] stated the Banach Contraction Principle for multivaluedmappings in modular function spaces.


Theorem 4.7 (see [40, Theorem 3.1]). Let ρ be a convex function modular satisfying the Δ2-typecondition, C a nonempty ρ-bounded ρ-closed subset of Lρ, and T : C → Fρ(C) a ρ-contractionmapping, that is, there exists a constant k ∈ [0, 1) such that

Hρ

(Tf, Tg

) ≤ kρ(f − g), f, g ∈ C. (4.10)

Then T has a fixed point.

By using that result, they proved the existence of fixed points for multivalued ρ-nonexpansive mappings.

Theorem 4.8 (see [40, Theorem 3.4]). Let ρ be a convex function modular satisfying the Δ2-typecondition, C a nonempty ρ-a.e. compact ρ-bounded convex subset of Lρ, and T : C → Kρ(C) aρ-nonexpansive mapping. Then T has a fixed point.

They also applied the above theorem to obtain fixed point results in the Banach spaceL1 (resp., �1) for multivalued mappings whose domains are compact in the topology of theconvergence locally in measure (resp., w∗-topology).

Consider the space Lp(Ω, μ) for a σ-finite measure μ with the usual norm. Let C bea bounded closed convex subset of Lp for 1 < p < ∞ and T : C → K(C) a multivaluednonexpansive mapping. Because of uniform convexity of Lp, it is known that T has afixed point. For p = 1, T can fail to have a fixed point even in the singlevalued case fora weakly compact convex set C (see [43]). However, since L1 is a modular space whereρ(f) =

∫Ω|f |dμ = ‖f‖ for all f ∈ L1, Theorem 4.8 implies the existence of a fixed point when

we define mappings on a ρ-a.e. compact ρ-bounded convex subset of L1. Thus the followingcan be stated.

Corollary 4.9 (see [40, Corollary 3.5]). Let (Ω, μ) be as above, C ⊂ L1(Ω, μ) a nonempty boundedconvex set which is compact for the topology of the convergence locally in measure, and T : C → K(C)a nonexpansive mapping. Then T has a fixed point.

In the case of the space �1, we also can obtain a bounded closed convex set C and anonexpansive mapping T : C → C which is fixed point free. Indeed, consider the followingeasy and well-known example.

Let

C =

{{xn} ∈ �1 : 0 ≤ xn ≤ 1,

∞∑n=1

xn = 1

}. (4.11)

Define a nonexpansive mapping T : C → C by

T(x) = (0, x1, x2, x3, . . .), where x = {xn}, (4.12)

then T is a fixed point free map. However, if we consider Lρ = �1, where ρ(x) = ‖x‖, for allx ∈ �1, then ρ-a.e. convergence and ω∗-convergence are identical on bounded subsets of �1

(see [36]). This fact leads to the following corollary.


Corollary 4.10 (see [40, Corollary 3.6]). Let C be a nonempty ω∗-compact convex subset of �1 andT : C → K(C) a nonexpansive mapping. Then T has a fixed point.

Next we will give a property of closed convex bounded subsets of �1 more general thanweak star compactness which implies the fixed point property for nonexpansive mappings.

Domınguez et al. introduced in [44] some compactness conditions concerningproximinal subsets called Property (P). Following this idea we will use the following similarnotion for modular function spaces.

Definition 4.11. Let C be a nonempty ρ-closed convex ρ-bounded subset of Lρ. It is said that Chas Property (Pρ) if for every f ∈ Lρ,which is the ρ-a.e. limit of a sequence inC, the set Pρ,C(f)is a nonempty and ρ-compact subset of C, where Pρ,C(f) = {g ∈ C : ρ(g − f) = distρ(f, C)}.

Using that notion and the following two lemmas, we obtain a new fixed point resultfor multivalued ρ-nonexpansive mappings.

Lemma 4.12 (see [40, Lemma 3.3]). Let ρ be a convex function modular satisfying the Δ2-typecondition, f ∈ Lρ, and K a nonempty ρ-compact subset of Lρ. Then there exists g0 ∈ K such that

ρ(f − g0

)= distρ

(f,K

). (4.13)

Lemma 4.13 (see [37, Lemma 1.3]). Let ρ be a function modular satisfying the Δ2-type condition,

and {fn}n be a sequence in Lρ such that fnρ-a.e.→ f ∈ Lρ and there exists k > 1 such that supnρ(k(fn−

f)) <∞. Then,

lim infn→∞

ρ(fn − g

)= lim inf

n→∞ρ(fn − f

)+ ρ(f − g) ∀g ∈ Lρ. (4.14)

Theorem 4.14. Let ρ be a convex function modular satisfying the Δ2-type condition, C a nonemptyρ-closed ρ-bounded convex subset of Lρ satisfying Property (Pρ) such that every sequence in C has aρ-a.e. convergent subsequence in Lρ, and T : C → KρC(C) a ρ-nonexpansive mapping. Then T hasa fixed point.

Proof. Fix f0 ∈ C. For each n ∈ N, the ρ-contraction Tn : C → Fρ(C) is defined by

Tn(f)=

1nf0 +

(1 − 1

n

)Tf, f ∈ C. (4.15)

By Theorem 4.7, we can conclude that Tn has a fixed point, say fn. It is easy to see that

distρ(fn, Tfn

) ≤ 1n

diamρ(C) −→ 0. (4.16)

By our assumptions, we can assume, by passing through a subsequence, that fnρ-a.e.→ f for

some f ∈ Lρ. By Lemma 4.12, for each n ∈ N there exists gn ∈ Tfn such that

ρ(fn − gn

)= distρ

(fn, Tfn

). (4.17)


Now we are going to show that Pρ,C(f) ∩ Th/= ∅ for each h ∈ Pρ,C(f). Taking any h ∈ Pρ,C(f),from the ρ-compactness of Th and Lemma 4.12, we can find hn ∈ Th such that

ρ(gn − hn

)= distρ

(gn, Th

) ≤ Hρ

(Tfn, Th

) ≤ ρ(fn − h), (4.18)

and we can assume, by passing through a subsequence, that hnρ→ h0 for some h0 ∈ Th. From

above and using Lemma 4.13, it follows that

lim infn

ρ(fn − h0

)= lim inf

nρ(gn − h0

)= lim inf

nρ(gn − hn

) ≤ lim infn

ρ(fn − h

)

= lim infn

ρ(fn − f

)+ ρ(f − h). (4.19)

On the other hand, by Lemma 4.13 we also have

lim infn

ρ(fn − h0

)= lim inf

nρ(fn − f

)+ ρ(f − h0

). (4.20)

Thus, we deduce ρ(f −h0) ≤ ρ(f −h), which implies that h0 ∈ Pρ,C(f) and so Pρ,C(f)∩ Th/= ∅.Now we define the mapping T : Pρ,C(f) → KC(Pρ,C(f)) by T(h) = Pρ,C(f) ∩ Th.

From [45, Proposition 2.45] we know that the mapping T is upper semicontinuous. SincePρ,C(f) ∩ Th is a nonempty ρ-compact convex set and the ρ-topology is a norm-topology, wecan apply the Kakutani-Bohnenblust-Karlin Theorem (see [14]) to obtain a fixed point for Tand hence for T .

If we apply the previous theorem in the particular case of the space L1(Ω, μ) for aσ-finite measure μ with the usual norm, we obtain the following result, which can be alsodeduced from [44, Theorem 4.9].

Corollary 4.15. Let (Ω, μ) be as above, C ⊂ L1(Ω, μ) a nonempty closed bounded convex set whichsatisfies Property (P). Suppose, in addition, that every sequence in C has a convergent locally inmeasure subsequence in L1. If T : C → KC(C) is a nonexpansive mapping, then T has a fixedpoint.

If we consider now the space �1, then the assumption of existence of a w∗-convergentsubsequence for every sequence in C can be removed and we can state the following result.

Corollary 4.16. Let C be a nonempty closed bounded convex subset of �1 which satisfies Property (P).If T : C → KC(C) is a nonexpansive mapping, then T has a fixed point.

Notice that in �1 there exists a subset with Property (P) which is not w∗-compact.

Example 4.17 (see [44, Example 4.8]). Let (an) be a bounded sequence of nonnegative realnumbers and let (en) be the standard Schauder basis of �1. It is clear that the setC := conv(xn),where xn := (1 + an)en, is never weakly star compact. Nevertheless, by using [46, Example1] it is easy to show that C has Property (P) if and only if N0 := {n ∈ N : an = infm∈Nam} isnonempty and finite.


Acknowledgments

The authors are very grateful to the anonymous referee for some useful suggestions toimprove the presentation of this paper. This research was partially supported by DGESGrant no.BFM2006-13997-C02-01 and Junta de Andalucıa Grant no.FQM-127. This researchis dedicated to W. A. Kirk celebrating his wide and deep contribution in Metric Fixed PointTheory.

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Research ArticleConvergence of the Sequence ofSuccessive Approximations to a Fixed Point

Tomonari Suzuki

Department of Basic Sciences, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan

Correspondence should be addressed to Tomonari Suzuki, [email protected]

Received 29 September 2009; Accepted 21 December 2009


Copyright q 2010 Tomonari Suzuki. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

If (X, d) is a complete metric space and T is a contraction on X, then the conclusion of the Banach-Caccioppoli contraction principle is that the sequence of successive approximations {Tnx} of Tstarting from any point x ∈ X converges to a unique fixed point. In this paper, using the conceptof τ-distance, we obtain simple, sufficient, and necessary conditions of the above conclusion.

1. Introduction

The following famous theorem is referred to as the Banach-Caccioppoli contraction principle.This theorem is very forceful and simple, and it became a classical tool in nonlinear analysis.

Theorem 1.1 (see Banach [1] and Caccioppoli [2]). Let (X, d) be a complete metric space and letT be a self contraction on X, that is, there exists r ∈ [0, 1) such that d(Tx, Ty) ≤ rd(x, y) for allx, y ∈ X. Then the following holds.

(A) T has a unique fixed point z, and {Tnx} converges to z for any x ∈ X.

We note that the conclusion of Kannan’s fixed point theorem [3] is also (A). See Kirk’ssurvey [4]. Recently, we obtained that (A) holds if and only if T is a strong Leader mapping[5, 6].

Theorem 1.2 (see [6]). Let T be a mapping on a complete metric space (X, d). Then the followingare equivalent.

(i) (A) holds.

(ii) T is a strong Leader mapping, that is, the following hold.


(a) For x, y ∈ X and ε > 0, there exist δ > 0 and ν ∈ N such that

d(Tix, Tjy

)< ε + δ =⇒ d

(Ti+νx, Tj+νy

)< ε, (1.1)

for all i, j ∈ N ∪ {0}, where T0 is the identity mapping on X.

(b) For x, y ∈ X, there exist ν ∈ N and a sequence {αn} in (0,∞) such that

d(Tix, Tjy

)< αn =⇒ d

(Ti+νx, Tj+νy

)<

1n, (1.2)

for all i, j ∈ N ∪ {0} and n ∈ N.

The following theorem is proved in [7, 8].

Theorem 1.3 (see Rus [7] and Subrahmanyam [8]). Let (X, d) be a complete metric space andlet T be a continuous mapping on X. Assume that there exists r ∈ [0, 1) satisfying d(Tx, T2x) ≤rd(x, Tx) for all x ∈ X. Then the following holds.

(B) {Tnx} converges to a fixed point for every x ∈ X.

We obtained a condition equivalent to (B) in [9].

Theorem 1.4 (see [9]). Let T be a mapping on a complete metric space (X, d). Then the followingare equivalent.

(i) (B) holds.

(ii) The following hold.

(a) For x ∈ X and ε > 0, there exist δ > 0 and ν ∈ N such that

d(Tix, Tjx

)< ε + δ =⇒ d

(Ti+νx, Tj+νx

)< ε, (1.3)

for all i, j ∈ N ∪ {0}.(b) For x, y ∈ X, there exist ν ∈ N and a sequence {αn} in (0,∞) such that

d(Tix, Tjy

)< αn =⇒ d

(Ti+νx, Tj+νy

)<

1n, (1.4)

for all i, j ∈ N ∪ {0} and n ∈ N.

We sometimes call a mapping satisfying (A) a Picard operator [10]. We also call amapping satisfying (B) a weakly Picard operator [11–13].

We cannot tell that the conditions (ii) of Theorems 1.2 and 1.4 are simple. Motivatedby this, we obtain simpler conditions which are equivalent to Conditions (A) and (B).


2. Preliminaries

Throughout this paper, we denote by N, Z, and R the sets of positive integers, integers andreal numbers, respectively.

In 2001, Suzuki introduced the concept of τ-distance in order to improve results inTataru [14], Zhong [15, 16], and others. See also [17].

Definition 2.1 (see [18]). Let (X, d) be a metric space. Then a function p from X×X into [0,∞)is called a τ-distance on X if there exists a function η from X × [0,∞) into [0,∞) and thefollowing are satisfied:

(τ1) p(x, z) ≤ p(x, y) + p(y, z) for all x, y, z ∈ X,

(τ2) η(x, 0) = 0 and η(x, t) ≥ t for all x ∈ X and t ∈ [0,∞), and η is concave andcontinuous in its second variable,

(τ3) limnxn = x and limn sup{η(zn, p(zn, xm)) : m ≥ n} = 0 imply p(w,x) ≤lim infnp(w,xn) for all w ∈ X,

(τ4) limn sup{p(xn, ym) : m ≥ n} = 0 and limnη(xn, tn) = 0 imply that limnη(yn, tn) = 0,

(τ5) limnη(zn, p(zn, xn)) = 0 and limnη(zn, p(zn, yn)) = 0 imply that limnd(xn, yn) = 0.

The metric d is a τ-distance on X. Many useful examples and propositions are statedin [9, 18–23] and references therein. The following fixed point theorems are proved in [18].

Theorem 2.2 (see [18]). LetX be a complete metric space and let T be a mapping onX. Assume thatthere exist a τ-distance p and r ∈ [0, 1) such that p(Tx, T2x) ≤ rp(x, Tx) for all x ∈ X. Assume thefollowing.

(i) If limn sup{p(xn, xm) : m > n} = 0, limnp(xn, Txn) = 0, and limnp(xn, y) = 0, thenTy = y.

Then (B) holds. Moreover, if Tz = z, then p(z, z) = 0.

Theorem 2.3 (see[18]). Let X be a complete metric space and let T be a mapping on X. Assume thatT is a contraction with respect to some τ-distance p, that is, there exist a τ-distance p and r ∈ [0, 1)such that

p(Tx, Ty

) ≤ rp(x, y), (2.1)

for all x, y ∈ X. Then (A) and p(z, z) = 0 hold.

The following lemmas are useful in our proofs.

Lemma 2.4 (see [18]). Let (X, d) be a metric space and let p be a τ-distance on X. If sequences {xn}and {yn} in X satisfy limnp(z, xn) = 0 and limnp(z, yn) = 0 for some z ∈ X, then limnd(xn, yn) =0. In particular for x, y, z ∈ X, p(z, x) = 0 and p(z, y) = 0 imply that x = y.

Lemma 2.5 (see [18]). Let (X, d) be a metric space and let p be a τ-distance on X. If a sequence{xn} in X satisfies limn sup{p(xn, xm) : m > n} = 0, then {xn} is a Cauchy sequence. Moreover if asequence {yn} in X satisfies limnp(xn, yn) = 0, then limnd(xn, yn) = 0.

The following lemmas are easily deduced from Lemmas 2.4 and 2.5.


Lemma 2.6. Let (X, d) be a metric space and let p be a τ-distance on X. Then for every z ∈ X andε > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply that d(x, y) ≤ ε.

Lemma 2.7. Let X be a metric space and let p be a τ-distance on X. Assume that a sequence {xn} inX satisfies limn sup{p(xn, xm) : m > n} = 0, limnp(xn, y) = 0, and limnp(xn, z) = 0. Then y = z.

The following is proved at Page 442 of [18]. However we give a proof because we usereductio ad absurdum in [18].

Lemma 2.8 (see [18]). Let g be a nondecreasing function from [0,∞) into itself satisfying inf{g(t) :t > 0} = 0. Define a function f from [0,∞) into itself by

f(t) = t + sup

{n∑i=1

αi min{g(si), 1

}: t =

n∑i=1

αisi, si ≥ 0, αi > 0,n∑i=1

αi = 1

}. (2.2)

Then f(0) = 0, f(t) ≥ t + g(t) for all t ∈ [0,∞); and f is concave and continuous.

Proof. It is clear that f(0) = 0, f(t) ≥ t + g(t), and f is concave. We shall prove that f iscontinuous at 0. Fix ε > 0. Then there exists δ > 0 such that g(δ) ≤ ε. Choose τ > 0 withτ + τ/δ ≤ ε. Fix t ∈ (0, τ). Let α1, α2, . . . , αn > 0 and s1, s2, . . . , sn ≥ 0 such that t =

∑ni=1 αisi and∑n

i=1 αi = 1. Since δ∑{αi : si ≥ δ} ≤ t, we have

t +n∑i=1

αi min{g(si), 1

} ≤ t +∑si<δ

αig(si) +∑si≥δ

αi ≤ t +∑si<δ

αiε +∑si≥δ

αi

≤ t + ε + t

δ≤ τ + ε +

τ

δ≤ 2ε.

(2.3)

Since α1, α2, . . . , αn > 0 and s1, s2, . . . , sn ≥ 0 are arbitrary, we obtain f(t) ≤ 2ε. Thus,limt→+0f(t) = 0 = f(0).

The following is obvious.

Lemma 2.9. Let T be a mapping on a set X. Let A0 be a subset of X such that T(A0) ⊂ A0. Define asequence {An} of subsets of X by

A1 = T−1(A0) \A0, An+1 = T−1(An). (2.4)

Then the following hold.

(i) For every n ∈ N and x ∈ X, x ∈ An if and only if Tjx /∈A0 for j = 0, 1, . . . , n − 1 andTnx ∈ A0.

(ii) Am ∩An = ∅ form,n ∈ N ∪ {0} withm/=n.

(iii) T(An+1) = An for every n ∈ N.


3. Condition (B)

In this section, we discuss Condition (B).

Theorem 3.1. Let X be a complete metric space and let T be a mapping on X. Assume that there exista τ-distance p, r ∈ [0, 1), andM ∈ [0,∞) such that

p(Tx, T2x

)≤ rp(x, Tx), p

(Tx, Ty

) ≤Mp(x, y), (3.1)

for all x, y ∈ X. Then (B) holds. Moreover, if Tz = z, then p(z, z) = 0.

Proof. Assume that limn sup{p(xn, xm) : m > n} = 0, limnp(xn, Txn) = 0, and limnp(xn, y) = 0.Then we have

p(xn, Ty

) ≤ p(xn, Txn) + p(Txn, Ty) ≤ p(xn, Txn) +Mp(xn, y

), (3.2)

and hence, limnp(xn, Ty) = 0. By Lemma 2.7, we obtain Ty = y. By Theorem 2.2, we obtainthe desired result.

As a direct consequence of Theorem 3.1, we obtain the following.

Corollary 3.2. LetX be a complete metric space and let T be a mapping onX. Assume that there exista τ-distance p and r ∈ (0, 1) such that

p(Tx, T2x

)≤ rp(x, Tx), p

(Tx, Ty

) ≤ p(x, y), (3.3)

for all x, y ∈ X. Then (B) holds.

Corollary 3.2 characterizes Condition (B).

Theorem 3.3. Let T be a mapping on a metric space (X, d) such that (B) holds. Then there exist aτ-distance p and r ∈ (0, 1) satisfying (3.3).

Proof. Let r ∈ (0, 1) be fixed. We note that every periodic point is a fixed point. That is, ifx ∈ X satisfies Tnx = x for some n ∈ N, then Tx = x. Define a mapping T∞ from X onto F(T)by T∞x = limnT

nx for x ∈ X, where F(T) is the set of all fixed points of T . Define a mappingC from X into the set of subsets of X by

Cx ={Tx, T2x, T3x, . . . , T∞x

}. (3.4)

Since T∞x is a fixed point of T , we have

y ∈ Cx =⇒ Cy ⊂ Cx. (3.5)


Next, we define a function f from X into Z ∪ {∞} satisfying

f(Tx) ≥ f(x) + 1, f(x) =∞⇐⇒ Tx = x, (3.6)

for all x ∈ X. We put f(x) = ∞ for x ∈ F(T). It is obvious that f(Tx) = f(x) = ∞ = f(x) + 1for x ∈ F(T). Define a sequence {An} of subsets of X by

A1 = T−1(F(T)) \ F(T), An+1 = T−1(An). (3.7)

Then by Lemma 2.9,

F(T) ∩An = ∅, Am ∩An = ∅, (3.8)

for m,n ∈ N with m/=n. We put f(x) = −n for x ∈ An. We note that

f(Tx) =

⎧⎪⎪⎨⎪⎪⎩∞ if x ∈ A1,

f(x) + 1 if x ∈∞⊔n=2

An.(3.9)

Put

Y = X \(F(T)

(⊔n∈N

An

)). (3.10)

It is obvious that T(Y ) ⊂ Y , T−1(Y ) = Y , and Y ∩ F(T) = ∅. So,

Tmx = Tnx ⇐⇒ m = n, (3.11)

for x ∈ Y and m,n ∈ N ∪ {0}. Define an equivalence relation ∼ on Y as follows: x ∼ y if andonly if there exist m,n ∈ N ∪ {0} such that Tmx = Tny. By Axiom of Choice, there exists amapping B on Y such that

Bx ∼ x, x ∼ y ⇐⇒ Bx = By. (3.12)

Let u ∈ Y with Bu = u. Then we put f(Tnu) = n for n ∈ N ∪ {0}. Define a sequence {Dn} ofsubsets of Y by

D0 ={u, Tu, T2u, T3u, . . .

}, D1 = T−1(D0) \D0, Dn+1 = T−1(Dn). (3.13)

Then we have Dm ∩Dn = ∅ for m,n ∈ N ∪ {0}with m/=n; and

{x ∈ Y : x ∼ u} =⊔

n∈N∪{0}Dn. (3.14)


We put f(x) = −n for x ∈ Y with n ∈ N and x ∈ Dn. We have defined f . We note that f(x) ∈ N

implies that x ∈ Y .Next, we define a τ-distance p by

p(x, y)=

⎧⎨⎩rf(x) + rf(y) if y ∈ Cx,rf(x) + rf(y) + 1 if y /∈Cx,

(3.15)

where r∞ = 0. We note that p(x, y) < 1 implies either of the following.

(i) Tx = x = y.

(ii) There exist u ∈ Y , k ∈ N, and � ∈ N ∪ {∞} such that Bu = u, k < l, x = Tku, andy = T�u. (In this case, x ∈ Y , u = Bx, f(x) = k, and f(y) = � hold.)

We shall show that p is a τ-distance. Let x, y, z ∈ X. If y ∈ Cx and z ∈ Cy, then z ∈ Cx. So wehave

p(x, z) = rf(x) + rf(z) ≤ rf(x) + rf(y) + rf(y) + rf(z) = p(x, y) + p(y, z). (3.16)

If y /∈Cx or z/∈Cy, then

p(x, z) ≤ rf(x) + rf(z) + 1 ≤ rf(x) + rf(y) + rf(y) + rf(z) + 1 ≤ p(x, y) + p(y, z). (3.17)

These imply (τ1). We shall define a function η from X × [0,∞) into [0,∞). For x ∈ X \ Y ,we put η(x, t) = t. For x ∈ Y , we put u = Bx. Since {Tnu} converges to T∞u, there exists astrictly increasing sequence {hu(n)} in N such that j ≥ hu(n) implies that d(Tju, T∞u) ≤ 1/nfor j ∈ N ∪ {∞}. Since limnhu(n) =∞, we can define a nondecreasing function gu from [0,∞)into [0, 1] such that gu(rhu(n)) = 1/n. It is obvious that gu(0) = limt→+0gu(t) = 0. Put

η(x, t) = t + sup

{n∑i=1

αigu(si) : t =n∑i=1

αisi, si ≥ 0, αi > 0,n∑i=1

αi = 1

}. (3.18)

Then η(x, t) satisfies (τ2) and η(x, t) ≥ t + gu(t) by Lemma 2.8. In order to show (τ3), weassume that limnxn = x and limn sup{η(zn, p(zn, xm)) : m ≥ n} = 0. Then without loss ofgenerality, we may assume that sup{η(zn, p(zn, xm)) : m ≥ n} < 1. Thus sup{p(zn, xm) : m ≥n} < 1 for n ∈ N. It is obvious that xm ∈ Czn for m,n ∈ N with m ≥ n. We consider thefollowing two cases.

(i) There exists ν ∈ N such that xn ∈ F(T) for n ≥ ν.

(ii) There exists a subsequence {xnj} of {xn} such that xnj /∈F(T).In the first case, since F(T) ∩ Czν exactly consists of one element and xn ∈ F(T) ∩ Czν forn ≥ ν, xn = xν holds for all n ≥ ν. So x = xν. Thus, p(w,x) = limnp(w,xn) holds for everyw ∈ X. In the second case, we note that zn /∈F(T) for all n ∈ N. Hence zn ∈ Y . Put u = Pz1.Since xn ∈ Cz1, there exists a sequence {�n} in N ∪ {∞} such that xn = T�nu. Since �nj ∈ N forall j ∈ N, there exists a sequence {kn} in N such that zn = Tknu. Since limnp(zn, xn) = 0, we


have limnkn = ∞ and limn�n = ∞. So we obtain x = T∞u. We note that x ∈ Cxn for all n ∈ N.Let w ∈ X. In the case where x ∈ Cw, we have

p(w,x) = rf(w) = limn→∞

(rf(w) + rf(xn)

)≤ lim inf

n→∞p(w,xn). (3.19)

In the other case, where x /∈Cw, we have xn /∈Cw, and hence,

p(w,x) = rf(w) + 1 = limn→∞

(rf(w) + rf(xn) + 1

)= lim

n→∞p(w,xn). (3.20)

Therefore we have shown (τ3). Let us prove (τ4). We assume that limn sup{p(xn, ym) : m ≥n} = 0 and limnη(xn, tn) = 0. Without loss of generality, we may assume that sup{p(xn, ym) :m ≥ n} < 1. We consider the following two cases.

(i) There exists ν ∈ N such that yn ∈ F(T) for n ≥ ν.

(ii) There exists a subsequence {ynj} of {yn} such that ynj /∈F(T).

In the first case, we have

limn→∞

η(yn, tn

)= lim

n→∞tn ≤ lim

n→∞η(xn, tn) = 0. (3.21)

In the second case, as in the proof of (τ3), there exist u ∈ Y , a sequence {kn} in N, anda sequence {�n} in N ∪ {∞} such that Bu = u, xn = Tknu, and yn = T�nu. We note thatη(xn, t) = η(u, t). If yn ∈ F(T), then η(yn, t) = t ≤ η(u, t). If yn /∈F(T), then η(yn, t) = η(u, t).Therefore

limn→∞

η(yn, tn

) ≤ limn→∞

η(u, tn) = limn→∞

η(xn, tn) = 0. (3.22)

Let us prove (τ5). We assume that η(z, p(z, x)) < 1/n. We note that p(z, x) < 1. In the casewhere Tz = z = x, we have d(z, x) = 0 < 1/n. In the other case, where there exist u ∈ Y ,k ∈ N, and � ∈ N ∪ {∞} such that Bu = u, k < l, z = Tku, and x = T�u, we have

η(z, p(z, x)

)<

1n= gu

(rhu(n)

)≤ η(z, rhu(n)

). (3.23)

Hence

rk + r� = p(z, x) < rhu(n). (3.24)

Thus, we obtain k > hu(n) and � > hu(n). So we have

d(z, x) = d(Tku, T�u

)≤ d(Tku, T∞u

)+ d(T�u, T∞u

)≤ 1n+

1n=

2n. (3.25)


Therefore

η(z, p(z, x)

)<

1n, η

(z, p(z, y))

<1n=⇒ d

(x, y) ≤ 4

n, (3.26)

which imply (τ5). Therefore we have shown that p is a τ-distance on X.We shall show (3.3). Let x, y ∈ X. Since Tx ∈ Cx, T2x ∈ C(Tx), f(Tx) ≥ f(x) + 1, and

f(T2x) ≥ f(Tx) + 1, we have

p(Tx, T2x

)= rf(Tx) + rf(T

2x) ≤ rf(x)+1 + rf(Tx)+1 = rp(x, Tx). (3.27)

If y ∈ Cx, then Ty ∈ C(Tx) holds. So we have

p(Tx, Ty

)= rf(Tx) + rf(Ty) ≤ rf(x)+1 + rf(y)+1 = rp

(x, y) ≤ p(x, y). (3.28)

If y /∈Cx, then we have

p(Tx, Ty

) ≤ rf(Tx) + rf(Ty) + 1 ≤ rf(x) + rf(y) + 1 = p(x, y). (3.29)

Therefore (3.3) holds.

Remark 3.4. We have proved that, for every r ∈ (0, 1), there exists a τ-distance p satisfying(3.3).

Combining Theorem 6 in [9], we obtain the following.

Corollary 3.5. Let T be a mapping on a complete metric space (X, d). Then the following areequivalent.

(i) (B) holds.

(ii) There exists a τ-distance p on X satisfying the following.

(a) For x ∈ X and ε > 0, there exist δ > 0 and ν ∈ N such that

p(Tix, Tjx

)< ε + δ =⇒ p

(Ti+νx, Tj+νx

)< ε, (3.30)

for all i, j ∈ N ∪ {0} with i < j.(b) For x, y ∈ X, there exist ν ∈ N and a sequence {αn} in (0,∞) such that

p(Tix, Tjy

)< αn =⇒ p

(Ti+νx, Tj+νy

)<

1n, (3.31)

for all n ∈ N and i, j ∈ N ∪ {0} with i > j.

(iii) There exist a τ-distance p and r ∈ (0, 1) such that p(Tx, T2x) ≤ rp(x, Tx) andp(Tx, Ty) ≤ p(x, y) for all x, y ∈ X.


4. Condition (A)

In this section, we discuss Condition (A).Define a relation <

0on [0,∞) as follows: s <

0t if and only if either s = t = 0 or s < t

holds.

Theorem 4.1. Let X be a complete metric space and let T be a mapping on X. Assume that there exista τ-distance p and r ∈ (0, 1) such that

p(Tx, T2x

)≤ rp(x, Tx), p

(Tx, Ty

)<0p(x, y), (4.1)

for all x, y ∈ X. Then (A) holds.

Proof. By Theorem 3.1, (B) holds. Moreover, if Tx = x, then p(x, x) = 0. Let z,w ∈ X be fixedpoints of T . Then

p(z,w) = p(Tz, Tw) <0p(z,w), (4.2)

which implies that p(z,w) = 0. Since p(z, z) = 0, we obtain z = w by Lemma 2.4. Thus thefixed point is unique.

Theorem 4.2. Let X be a complete metric space and let T be a mapping on X. Assume that there exista τ-distance p and r ∈ (0, 1) such that

p(Tx, T2x

)≤ rp(x, Tx), p

(Tx, Ty

)< p(x, y), (4.3)

for all x, y ∈ X with x /=y. Then (A) holds.

Proof. In the case where X consists of one element, the conclusion obviously holds. So weconsider the other case. Assume that limn sup{p(xn, xm) : m > n} = 0, limnp(xn, Txn) = 0, andlimnp(xn, y) = 0. We consider the following two cases:

(i) xn /=y for sufficiently large n ∈ N,

(ii) there exists a sequence {xnj} of {xn} such that xnj = y.

In the first case, we have

p(xn, Ty

) ≤ p(xn, Txn) + p(Txn, Ty) < p(xn, Txn) + p(xn, y) (4.4)

for sufficiently large n, and hence, limnp(xn, Ty) = 0. By Lemma 2.7, we obtain Ty = y. In thesecond case, we have

p(y, Ty

)= lim

j→∞p(xnj , Txnj

)= 0, p

(y, y)= lim

j→∞p(xnj , y

)= 0. (4.5)


By Lemma 2.4, we obtain Ty = y. By Theorem 2.2, (B) holds. Let z,w ∈ X be distinct fixedpoints of T . Then

p(z,w) = p(Tz, Tw) < p(z,w), (4.6)

which implies a contradiction. Thus the fixed point is unique.

We shall show that Theorems 4.1 and 4.2 characterize Condition (A).

Theorem 4.3. Let T be a mapping on a metric space (X, d) such that (A) holds. Then there exist aτ-distance p and r ∈ (0, 1) satisfying (4.1).

Proof. Let p, r, f , and C be as in the proof of Theorem 3.3. Then p(Tx, T2x) ≤ rp(x, Tx) holds.Fix x, y ∈ X. We consider the following two cases:

(i) Tx = x and Ty = y,

(ii) either Tx /=x or Ty /=y.

In the first case, x = y holds by (A). Since

p(x, x) = p(Tx, T2x

)≤ rp(x, Tx) = p(x, x), (4.7)

we obtain p(x, x) = 0. Thus, p(Tx, Ty) = p(x, y) = 0. In the second case, we note that eitherf(x) ∈ Z or f(y) ∈ Z holds. Thus

rf(x)+1 + rf(y)+1 < rf(x) + rf(y). (4.8)

If y ∈ Cx, then Ty ∈ C(Tx) holds. So we have

p(Tx, Ty

)= rf(Tx) + rf(Ty) ≤ rf(x)+1 + rf(y)+1 < rf(x) + rf(y) = p

(x, y). (4.9)

If y /∈Cx, then we have

p(Tx, Ty

) ≤ rf(Tx) + rf(Ty) + 1 < rf(x) + rf(y) + 1 = p(x, y). (4.10)

Therefore (4.1) holds.

Theorem 4.4. Let T be a mapping on a metric space (X, d) such that (A) holds. Then there exist aτ-distance p and r ∈ (0, 1) satisfying (4.3) for all x, y ∈ X with x /=y.

Proof. The proof of Theorem 4.3 works.


Combining Theorem 7 in [9], we obtain the following.

Corollary 4.5. Let T be a mapping on a complete metric space (X, d). Then the following areequivalent.

(i) (A) holds.

(ii) There exists a τ-distance p on X satisfying the following.

(a) For x, y ∈ X and ε > 0, there exist δ > 0 and ν ∈ N such that

p(Tix, Tjy

)< ε + δ =⇒ p

(Ti+νx, Tj+νy

)< ε, (4.11)

for all i, j ∈ N ∪ {0} with i < j.(b) For x, y ∈ X, there exist ν ∈ N and a sequence {αn} in (0,∞) such that

p(Tix, Tjy

)< αn =⇒ p

(Ti+νx, Tj+νy

)<

1n, (4.12)

for all n ∈ N and i, j ∈ N ∪ {0} with i > j.

(iii) There exist a τ-distance p and r ∈ (0, 1) such that p(Tx, T2x) ≤ rp(x, Tx) andp(Tx, Ty) <

0p(x, y) for all x, y ∈ X.

(iv) There exist a τ-distance p and r ∈ (0, 1) such that p(Tx, T2x) ≤ rp(x, Tx) andp(Tx, Ty) < p(x, y) for all x, y ∈ X with x /=y.

5. Additional Result

Since Theorem 2.2 deduces Corollary 3.2, we can tell that Theorem 2.2 characterizesCondition (B). However, the following example tells that Theorem 2.3 does not characterizeCondition (A).

Example 5.1. Let A be the set of all real sequences {an} such that an ∈ (0,∞) for n ∈ N,{an} is strictly decreasing, and {an} converges to 0. Let H be a Hilbert space consisting ofall the functions x from A into R satisfying

∑a∈A |x(a)|2 < ∞ with inner product 〈x, y〉 =∑

a∈A x(a)y(a) for all x, y ∈ H. Define a subset X of H by

X = {0} ∪(⋃

a∈A{anea : n ∈ N}

), (5.1)

where ea ∈ H is defined by ea(a) = 1 and ea(b) = 0 for b ∈ A \ {a}. Define a mapping T on Xby

T0 = 0, T(anea) = an+1ea. (5.2)

Then (A) holds. However, T is not a contraction with respect to any τ-distance p.


Proof. It is obvious that (A) holds. Arguing by contradiction, we assume that T is a contractionwith respect to some τ-distance p. That is, there exist a τ-distance p and r ∈ [0, 1) such thatp(Tx, Ty) ≤ rp(x, y) for all x, y ∈ X. Since

p(0, 0) = p(T0, T0) ≤ rp(0, 0), (5.3)

we have p(0, 0) = 0. By Lemma 2.6, there exists a strictly increasing sequence {κn} in N suchthat

p(0, x) ≤ rκn =⇒ d(0, x) ≤ 1n. (5.4)

We choose α ∈ A such that α2κn+1 > 1/n. Fix ν ∈ N with rκνp(0, α1eα) ≤ 1. Then we have

p(0, α2κν+1eα) = p(T2κν0, T2κν(α1eα)

)≤ r2κνp(0, α1eα) ≤ rκν , (5.5)

and hence,

1ν< α2κν+1 = d(0, α2κν+1eα) ≤ 1

ν. (5.6)

This is a contradiction.

Acknowledgments

The author is supported in part by Grant-in-Aid for Scientific Research from Japan Societyfor the Promotion of Science. The author wishes to express his gratitude to the referees forcareful reading and giving a historical comment.

References

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[2] R. Caccioppoli, “Un teorema generale sull’esistenza di elementi uniti in una transformazionefunzionale,” Rendiconti dell’Accademia Nazionale dei Lincei, vol. 11, pp. 794–799, 1930.

[3] R. Kannan, “Some results on fixed points. II,” The AmericanMathematical Monthly, vol. 76, pp. 405–408,1969.

[4] W. A. Kirk, “Contraction mappings and extensions,” in Handbook of Metric Fixed Point Theory, W. A.Kirk and B. Sims, Eds., pp. 1–34, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

[5] S. Leader, “Equivalent Cauchy sequences and contractive fixed points in metric spaces,” StudiaMathematica, vol. 76, no. 1, pp. 63–67, 1983.

[6] T. Suzuki, “A sufficient and necessary condition for the convergence of the sequence of successiveapproximations to a unique fixed point,” Proceedings of the American Mathematical Society, vol. 136, no.11, pp. 4089–4093, 2008.

[7] I. A. Rus, “The method of successive approximations,” Revue Roumaine de Mathematiques Pures etAppliquees, vol. 17, pp. 1433–1437, 1972.

[8] P. V. Subrahmanyam, “Remarks on some fixed-point theorems related to Banach’s contractionprinciple,” Journal of Mathematical and Physical Sciences, vol. 8, pp. 445–457, 1974.


[9] T. Suzuki, “Subrahmanyam’s fixed point theorem,” Nonlinear Analysis: Theory, Methods & Applications,vol. 71, no. 5-6, pp. 1678–1683, 2009.

[10] I. A. Rus, “Picard operators and applications,” Scientiae Mathematicae Japonicae, vol. 58, no. 1, pp. 191–219, 2003.

[11] I. A. Rus, “The theory of a metrical fixed point theorem: theoretical and applicative relevances,” FixedPoint Theory, vol. 9, no. 2, pp. 541–559, 2008.

[12] I. A. Rus, A. S. Muresan, and V. Muresan, “Weakly Picard operators on a set with two metrics,” FixedPoint Theory, vol. 6, no. 2, pp. 323–331, 2005.

[13] I. A. Rus, A. Petrusel, and M. A. Serban, “Weakly Picard operators: equivalent definitions,applications and open problems,” Fixed Point Theory, vol. 7, no. 1, pp. 3–22, 2006.

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[23] T. Suzuki, “On the relation between the weak Palais-Smale condition and coercivity given by Zhong,”Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2471–2478, 2008.


Research ArticleFixed Point Theorems on Spaces Endowed withVector-Valued Metrics

Alexandru-Darius Filip and Adrian Petrusel

Department of Applied Mathematics, Babes-Bolyai University, Kogalniceanu Street, No. 1,400084 Cluj-Napoca, Romania

Correspondence should be addressed to Adrian Petrusel, [email protected]

Received 2 July 2009; Revised 4 October 2009; Accepted 21 December 2009

Academic Editor: Tomas Dominguez Benavides

Copyright q 2010 A.-D. Filip and A. Petrusel. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The purpose of this work is to present some (local and global) fixed point results for singlevaluedand multivalued generalized contractions on spaces endowed with vector-valued metrics. Theresults are extensions of some theorems given by Perov (1964), Bucur et al. (2009), M. Berinde andV. Berinde (2007), O’Regan et al. (2007), and so forth.

1. Introduction

The classical Banach contraction principle was extended for contraction mappings on spacesendowed with vector-valued metrics by Perov in 1964 (see [1]).

Let X be a nonempty set. A mapping d : X ×X → Rm is called a vector-valued metric

on X if the following properties are satisfied:

(d1) d(x, y) ≥ 0 for all x, y ∈ X; if d(x, y) = 0, then x = y;

(d2) d(x, y) = d(y, x) for all x, y ∈ X;

(d3) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X.

If α, β ∈ Rm, α = (α1, α2, . . . , αm), β = (β1, β2, . . . , βm), and c ∈ R, by α ≤ β (resp., α < β)

we mean that αi ≤ βi (resp., αi < βi) for i ∈ {1, 2, . . . , m} and by α ≤ c we mean that αi ≤ c fori ∈ {1, 2, . . . , m}.

A set X equipped with a vector-valued metric d is called a generalized metric space.We will denote such a space with (X, d). For the generalized metric spaces, the notions ofconvergent sequence, Cauchy sequence, completeness, open subset, and closed subset aresimilar to those for usual metric spaces.


If (X, d) is a generalized metric space, x0 ∈ X and r = (ri)mi=1 ∈ R

m, with ri > 0 for eachi ∈ {1, 2, . . . , m}, then we will denote by

B(x0, r) := {x ∈ X | d(x0, x) < r} (1.1)

the open ball centered in x0 with radius r, by B(x0, r) the closure (in (X, d)) of the open ball,and by

B(x0, r) := {x ∈ X | d(x0, x) ≤ r} (1.2)

the closed ball centered in x0 with radius r.If f : X → X is a singlevalued operator, then we denote by Fix(f) the set of all fixed

points of f ; that is, Fix(f) := {x ∈ X | x = f(x)}.For the multivalued operators we use the following notations:

P(X) := {Y ⊂ X | Y /= ∅};Pb(X) := {Y ∈ P(X) | Y is bounded};Pcl(X) := {Y ∈ P(X) | Y is closed}.

(1.3)

Now, if F : X → P(X) is a multivalued operator, then we denote by Fix(F) the fixed pointsset of F, that is, Fix(F) := {x ∈ X | x ∈ F(x)}.

The set Graph(F) = {(x, y) ∈ X ×X | y ∈ F(x)} is called the graph of the multivaluedoperator F.

In the context of a metric space (X, d), if A,B ∈ P(X), then we will use the followingnotations:

(a) the gap functional D : P(X) × P(X) → R+:

D(A,B) := inf{d(a, b) | a ∈ A, b ∈ B}; (1.4)

(b) the generalized excess functional ρ : Pcl(X) × Pcl(X) → R+ ∪ {+∞}:

ρ(A,B) := sup{D(a, B) | a ∈ A}; (1.5)

(c) the generalized Pompeiu-Hausdorff functional H : Pcl(X) × Pcl(X) → R+ ∪ {+∞}:

H(A,B) := max{ρ(A,B), ρ(B,A)

}. (1.6)

It is well known that H is a generalized metric, in the sense that if A,B ∈ Pcl(X), thenH(A,B) ∈ R+ ∪ {+∞}.

Throughout this paper we denote by Mm,m(R+) the set of all m × m matrices withpositive elements, by Θ the zero m × m matrix, and by I the identity m × m matrix. If


A ∈Mm,m(R+), then the symbol Aτ stands for the transpose matrix of A. Notice also that,for the sake of simplicity, we will make an identification between row and column vectors inRm.

Recall that a matrix A is said to be convergent to zero if and only if An → 0 as n → ∞(see Varga [2]).

Notice that, for the proof of the main results, we need the following theorem, part ofwhich being a classical result in matrix analysis; see, for example, [3, Lemma 3.3.1, page 55],[4, page 37], and [2, page 12]. For the assertion (iv) see [5].

Theorem 1.1. Let A ∈Mm,m(R+). The following are equivalents.

(i) A is convergent towards zero.

(ii) An → 0 as n → ∞.

(iii) The eigenvalues of A are in the open unit disc, that is, |λ| < 1, for every λ ∈ C withdet(A − λI) = 0.

(iv) The matrix I −A is nonsingular and

(I −A)−1 = I +A + · · · +An + · · · . (1.7)

(v) The matrix I −A is nonsingular and (I −A)−1 has nonnenegative elements.

(vi) Anq → 0 and qAn → 0 as n → ∞, for each q ∈ Rm.

Remark 1.2. Some examples of matrix convergent to zero are

(a) any matrix A :=(a a

b b

), where a, b ∈ R+ and a + b < 1;

(b) any matrix A :=(a b

a b

), where a, b ∈ R+ and a + b < 1;

(c) any matrix A :=(a b

0 c

), where a, b, c ∈ R+ and max{a, c} < 1.

For other examples and considerations on matrices which converge to zero, see Rus[4], Turinici [6], and so forth.

Main result for self contractions on generalized metric spaces is Perov’s fixed pointtheorem; see [1].

Theorem 1.3 (Perov [3]). Let (X, d) be a complete generalized metric space and the mapping f :X → X with the property that there exists a matrix A ∈ Mm,m(R) such that d(f(x), f(y)) ≤Ad(x, y) for all x, y ∈ X.

If A is a matrix convergent towards zero, then

(1) Fix(f) = {x∗};(2) the sequence of successive approximations (xn)n∈N, xn = fn(x0) is convergent and it has

the limit x∗, for all x0 ∈ X;

(3) one has the following estimation:

d(xn, x∗) ≤ An(I −A)−1d(x0, x1); (1.8)


(4) if g : X → X satisfies the condition d(f(x), g(x)) ≤ η, for all x ∈ X, η ∈ Rm and

considering the sequence yn = gn(x0) one has

d(yn, x

∗) ≤ (I −A)−1η +An(I −A)−1d(x0, x1). (1.9)

On the other hand, notice that the evolution of macrosystems under uncertainty or lackof precision, from control theory, biology, economics, artificial intelligence, or other fields ofknowledge, is often modeled by semilinear inclusion systems:

x1 ∈ T1(x1, x2),

x2 ∈ T1(x1, x2),(1.10)

(where Ti : X × X → P(X) for i ∈ {1, 2} are multivalued operators; here P(X) stands for thefamily of all nonempty subsets of a Banach space X). The system above can be representedas a fixed point problem of the form

x ∈ T(x)(

where T := (T1, T2) : X2 −→ P(X2

), x = (x1, x2)

). (1.11)

Hence, it is of great interest to give fixed point results for multivalued operators on a setendowed with vector-valued metrics or norms. However, some advantages of a vector-valued norm with respect to the usual scalar norms were already pointed out by Precupin [5]. The purpose of this work is to present some new fixed point results for generalized(singlevalued and multivalued) contractions on spaces endowed with vector-valued metrics.The results are extensions of the theorems given by Perov [1], O’Regan et al. [7], M. Berindeand V. Berinde [8], and by Bucur et al. [9].

2. Main Results

We start our considerations by a local fixed point theorem for a class of generalizedsinglevalued contractions.

Theorem 2.1. Let (X, d) be a complete generalized metric space, x0 ∈ X, r := (ri)mi=1 ∈ R

m+ with

ri > 0 for each i ∈ {1, 2, . . . , m} and let f : B(x0, r) → X having the property that there existA,B ∈Mm,m(R+) such that

d(f(x), f

(y)) ≤ Ad(x, y) + Bd(y, f(x)) (2.1)

for all x, y ∈ B(x0, r). We suppose that

(1) A is a matrix that converges toward zero;

(2) if u ∈ Rm+ is such that u(I −A)−1 ≤ (I −A)−1r, then u ≤ r;

(3) d(x0, f(x0))(I −A)−1 ≤ r.Then Fix(f)/= ∅.

In addition, if the matrix A + B converges to zero, then Fix(f) = {x∗}.


Proof. We consider (xn)n∈N the sequence of successive approximations for the mapping f ,defined by

xn+1 = f(xn), ∀n ∈ N,

x0 ∈ X, be arbitrary.(2.2)

Using (3), we have d(x0, x1)(I −A)−1 = d(x0, f(x0))(I −A)−1 ≤ r ≤ (I −A)−1r.Thus, by (2) we get that d(x0, x1) ≤ r and hence x1 ∈ B(x0, r). Similarly, d(x1, x2)(I −

A)−1 = d(f(x0), f(x1))(I −A)−1 ≤ Ad(x0, x1)(I −A)−1 + Bd(x1, f(x0))(I −A)−1 ≤ Ar.Since d(x0, x2) ≤ d(x0, x1) + d(x1, x2), by (2) we get

d(x0, x2)(I −A)−1 ≤ d(x0, x1)(I −A)−1 + d(x1, x2)(I −A)−1

≤ Ir +Ar ≤(I +A +A2 + · · ·

)r = (I −A)−1r.

(2.3)

Thus d(x0, x2) ≤ r and hence x2 ∈ B(x0, r).Inductively, we construct the sequence (xn)n∈N in B(x0, r) satisfying, for all n ∈ N, the

following conditions:

(i) xn+1 = f(xn);

(ii) d(x0, xn)(I −A)−1 ≤ (I −A)−1r;

(iii) d(xn, xn+1)(I −A)−1 ≤ Anr.

From (iii) we get, for all n ∈ N and p ∈ N, p > 0, that

d(xn, xn+p

)(I −A)−1 = d(xn, xn+1)(I −A)−1 + d(xn+1, xn+2)(I −A)−1

+ · · · + d(xn+p−1, xn+p)(I −A)−1

≤ Anr +An+1r + · · · +An+p−1r

≤ An(I +A +A2 + · · · +Ap−1 + · · ·

)r

≤ An(I −A)−1r −→ 0, as n −→ ∞.

(2.4)

Hence (xn)n∈N is a Cauchy sequence. Using the fact that (B(x0, r), d) is a completemetric space, we get that (xn)n∈N is convergent in the closed set B(x0, r). Thus, there existsx∗ ∈ B(x0, r) such that x∗ = limn→∞xn.

Next, we show that x∗ ∈ Fix(f).Indeed, we have the following estimation:

d(x∗, f(x∗)

) ≤ d(x∗, xn) + d(xn, f(x∗)) = d(x∗, xn) + d(f(xn−1), f(x∗)

)≤ d(x∗, xn) +Ad(xn−1, x

∗) + Bd(x∗, xn) −→ 0, as n −→ ∞.(2.5)


Hence x∗ ∈ Fix(f). In addition, letting p → ∞ in the estimation of d(xn, xn+p), we get

d(xn, x∗) ≤ An(I −A)−1d(x0, x1). (2.6)

We show now the uniqueness of the fixed point.Let x∗, y∗ ∈ Fix(f) with x∗ /=y∗. Then

d(x∗, y∗

)= d

(f(x∗), f

(y∗

)) ≤ Ad(x∗, y∗) + Bd(y∗, f(x∗)) = (A + B)d(x∗, y∗

), (2.7)

which implies (I −A−B)d(x∗, y∗) ≤ 0 ∈ Rm. Taking into account that I −A−B is nonsingular

and (I −A − B)−1 ∈Mm,m(R+) we deduce that d(x∗, y∗) ≤ 0 and thus x∗ = y∗.

Remark 2.2. By similitude to [10], a mapping f : Y ⊆ X → X satisfying the condition

d(f(x), f

(y)) ≤ Ad(x, y) + Bd(y, f(x)), ∀x, y ∈ Y, (2.8)

for some matrices A,B ∈ Mm,m(R+) with A a matrix that converges toward zero, could becalled an almost contraction of Perov type.

We have also a global version of Theorem 2.1, expressed by the following result.

Corollary 2.3. Let (X, d) be a complete generalized metric space. Let f : X → X be a mappinghaving the property that there exist A,B ∈Mm,m(R+) such that

d(f(x), f

(y)) ≤ Ad(x, y) + Bd(y, f(x)), ∀x, y ∈ X. (2.9)

If A is a matrix that converges towards zero, then

(1) Fix(f)/= ∅;(2) the sequence (xn)n∈N given by xn := fn(x0) converges towards a fixed point of f , for all

x0 ∈ X;

(3) one has the estimation

d(xn, x∗) ≤ An(I −A)−1d(x0, x1), (2.10)

where x∗ ∈ Fix(f).


Remark 2.4. Any matrix A =(a 0

0 c

), where a, c ∈ R+ and max{a, c} < 1, satisfies the

assumptions (1)-(2) in Theorem 2.1.

Remark 2.5. Let us notice here that some advantages of a vector-valued norm with respectto the usual scalar norms were very nice pointed out, by several examples, in Precup in [5].More precisely, one can show that, in general, the condition that A is a matrix convergent


to zero is weaker than the contraction conditions for operators given in terms of the scalarnorms on X of the following type:

‖x‖M := ‖x1‖ + ‖x2‖‖x‖C := max{‖x1‖, ‖x2‖} or

‖x‖E := (‖x1‖2 + ‖x2‖2)1/2.

As an application of the previous results we present an existence theorem for a systemof operatorial equations.

Theorem 2.6. Let (X, | · |) be a Banach space and let f1, f2 : X ×X → X be two operators. Supposethat there exist aij , bij ∈ R+, i, j ∈ {1, 2} such that, for each x := (x1, x2), y := (y1, y2) ∈ X ×X, onehas:

(1) |f1(x1, x2)−f1(y1, y2)| ≤ a11|x1−y1|+a12|x2−y2|+b11|x1−f1(y1, y2)|+b12|x2−f2(y1, y2)|,(2) |f2(x1, x2)−f2(y1, y2)| ≤ a21|x1−y1|+a22|x2−y2|+b21|x1−f1(y1, y2)|+b22|x2−f2(y1, y2)|.

In addition, assume that the matrix A :=(a11 a12

a21 a22

)converges to 0.

Then, the system

u1 = f1(u1, u2), u2 = f1(u1, u2) (2.11)

has at least one solution x∗ ∈ X × X. Moreover, if, in addition, the matrix A + B converges to zero,then the above solution is unique.

Proof. Consider E := X × X and the operator f : E → Pcl(E) given by the expressionf(x1, x2) := (f1(x1, x2), f2(x1, x2)). Then our system is now represented as a fixed pointequation of the following form: x = f(x), x ∈ E. Notice also that the conditions (1) + (2)can be jointly represented as follows:

∥∥f(x) − f(y)∥∥ ≤ A · ∥∥x − y∥∥ + B · ∥∥x − f(y)∥∥, for each x, y ∈ E := X ×X. (2.12)

Hence, Corollary 2.3 applies in (E, d), with d(u, v) := ‖u − v‖ :=(|u1−v1||u2−v2|

).

We present another result in the case of a generalized metric space but endowed withtwo metrics.

Theorem 2.7. Let X be a nonempty set and let d, ρ be two generalized metrics on X. Let f : X → Xbe an operator. We assume that

(1) there exists C ∈Mm,m(R+) such that d(f(x), f(y)) ≤ ρ(x, y) · C;

(2) (X, d) is a complete generalized metric space;

(3) f : (X, d) → (X, d) is continuous;

(4) there exists A,B ∈Mm,m(R+) such that for all x, y ∈ X one has

ρ(f(x), f

(y)) ≤ Aρ(x, y) + Bρ(y, f(x)). (2.13)

If the matrix A converges towards zero, then Fix(f)/= ∅.



Proof. We consider the sequence of successive approximations (xn)n∈N defined recurrently byxn+1 = f(xn), x0 ∈ X being arbitrary. The following statements hold:

ρ(x1, x2) = ρ(f(x0), f(x1)

) ≤ Aρ(x0, x1) + Bρ(x1, f(x0)

)= Aρ(x0, x1),

ρ(x2, x3) = ρ(f(x1), f(x2)

) ≤ Aρ(x1, x2) + Bρ(x2, f(x1)

) ≤ A2ρ(x0, x1),

...

ρ(xn, xn+1) ≤ Anρ(x0, x1), ∀n ∈ N, n ≥ 1.

(2.14)

Now, let p ∈ N, p > 0. We estimate

ρ(xn, xn+p

) ≤ ρ(xn, xn+1) + ρ(xn+1, xn+2) + · · · + ρ(xn+p−1, xn+p

)

≤ Anρ(x0, x1) +An+1ρ(x0, x1) + · · · +An+p−1ρ(x0, x1)

≤ An(I +A +A2 + · · · +Ap−1 + · · ·

)ρ(x0, x1)

= An(I −A)−1ρ(x0, x1).

(2.15)

Letting n → ∞ we obtain that ρ(xn, xn+p) → 0 ∈ Rm. Thus (xn)n∈N is a Cauchy sequence

with respect to ρ.On the other hand, using the statement (1), we get

d(xn, xn+p

)= d

(f(xn−1), f

(xn+p−1

)) ≤ ρ(xn−1, xn+p−1) · C

≤ An−1(I −A)−1ρ(x0, x1)C −→ 0, as n −→ ∞.(2.16)

Hence, (xn)n∈N is a Cauchy sequence with respect to d. Since (X, d) is complete, oneobtains the existence of an element x∗ ∈ X such that x∗ = limn→∞xn with respect to d.

We prove next that x∗ = f(x∗), that is, Fix(f)/= ∅. Indeed, since xn+1 = f(xn), for alln ∈ N, letting n → ∞ and taking into account that f is continuous with respect to d, we getthat x∗ = f(x∗).

The uniqueness of the fixed point x∗ is proved below.Let x∗, y∗ ∈ Fix(f) such that x∗ /=y∗. We estimate

ρ(x∗, y∗

)= ρ

(f(x∗), f

(y∗

)) ≤ Aρ(x∗, y∗) + Bρ(y∗, f(x∗)) = (A + B)ρ(x∗, y∗

). (2.17)

Thus, using the additional assumption on the matrix A + B, we have that

(I −A − B)ρ(x∗, y∗) ≤ 0 =⇒ ρ(x∗, y∗

) ≤ 0 =⇒ x∗ = y∗. (2.18)

In what follows, we will present some results for the case of multivalued operators.


Theorem 2.8. Let (X, d) be a complete generalized metric space and let x0 ∈ X, r := (ri)mi=1 ∈ R

m+

with ri > 0 for each i ∈ {1, 2, . . . , m}. Consider F : B(x0, r) → Pcl(X) a multivalued operator. Oneassumes that

(i) there exist A,B ∈ Mm,m(R+) such that for all x, y ∈ B(x0, r) and u ∈ F(x) there existsv ∈ F(y) with

d(u, v) ≤ Ad(x, y) + Bd(y, u); (2.19)

(ii) there exists x1 ∈ F(x0) such that d(x0, x1)(I −A)−1 ≤ r;

(iii) if u ∈ Rm+ is such that u(I −A)−1 ≤ (I −A)−1r, then u ≤ r.

If A is a matrix convergent towards zero, then Fix(F)/= ∅.

Proof. By (ii) and (iii), there exists x1 ∈ F(x0) such that

d(x0, x1)(I −A)−1 ≤ r ≤ (I −A)−1r =⇒ d(x0, x1) ≤ r =⇒ x1 ∈ B(x0, r). (2.20)

For x1 ∈ F(x0), there exists x2 ∈ F(x1) with

d(x1, x2)(I −A)−1 ≤ Ad(x0, x1)(I −A)−1 + Bd(x1, x1)(I −A)−1 ≤ Ar. (2.21)

Hence

d(x0, x2)(I −A)−1 ≤ d(x0, x1)(I −A)−1 + d(x1, x2)(I −A)−1

≤ Ir +Ar ≤(I +A +A2 + · · ·

)r = (I −A)−1r

=⇒ d(x0, x2) ≤ r =⇒ x2 ∈ B(x0, r).

(2.22)

Next, for x2 ∈ F(x1), there exists x3 ∈ F(x2) with

d(x2, x3)(I −A)−1 ≤ Ad(x1, x2)(I −A)−1 + Bd(x2, x2)(I −A)−1 ≤ A2r, (2.23)

and hence

d(x0, x3)(I −A)−1 ≤ d(x0, x1)(I −A)−1 + d(x1, x2)(I −A)−1 + d(x2, x3)(I −A)−1

≤ Ir +Ar +A2r ≤(I +A +A2 + · · ·

)r = (I −A)−1r

=⇒ d(x0, x3) ≤ r =⇒ x3 ∈ B(x0, r).

(2.24)


By induction, we construct the sequence (xn)n∈N in B(x0, r) such that, for all n ∈ N, wehave

(1) xn+1 ∈ F(xn);(2) d(x0, xn)(I −A)−1 ≤ (I −A)−1r;

(3) d(xn, xn+1)(I −A)−1 ≤ Anr.

By a similar approach as before (see the proof of Theorem 2.1), we get that (xn)n∈N is a Cauchysequence in the complete space (B(x0, r), d). Hence (xn)n∈N is convergent in B(x0, r). Thus,there exists x∗ ∈ B(x0, r) such that x∗ = limn→∞xn.

Next we show that x∗ ∈ F(x∗).Using (i) and the fact that xn ∈ F(xn−1), for all n ∈ N n ≥ 1, we get, for each n ∈ N, the

existence of un ∈ F(x∗) such that

d(xn, un) ≤ Ad(xn−1, x∗) + Bd(x∗, xn). (2.25)

On the other hand

d(x∗, un) ≤ d(x∗, xn) + d(xn, un)≤ d(x∗, xn) +Ad(xn−1, x

∗) + Bd(x∗, xn).(2.26)

Letting n → ∞, we get d(x∗, un) → 0. Hence, we have limn→∞un = x∗ and since un ∈ F(x∗)and F(x∗) is closed set, we get that x∗ ∈ F(x∗).

Remark 2.9. From the proof of the above theorem, we also get the following estimation:

d(xn, x∗) ≤ An(I −A)−1d(x0, x1), for each n ∈ N with n ≥ 1, (2.27)

where x∗ is a fixed point for the multivalued operator F, and the pair (x0, x1) ∈ Graph(F) isarbitrary.

We have also a global variant for the Theorem 2.8 as follows.

Corollary 2.10. Let (X, d) be a complete generalized metric space and F : X → Pcl(X) a multivaluedoperator. One supposes that there exist A,B ∈ Mm,m(R+) such that for each x, y ∈ X and all u ∈F(x), there exists v ∈ F(y) with

d(u, v) ≤ Ad(x, y) + Bd(y, u). (2.28)


Remark 2.11. By a similar approach to that given in Theorem 2.6, one can obtain an existenceresult for a system of operatorial inclusions of the following form:

x1 ∈ T1(x1, x2),

x2 ∈ T1(x1, x2),(2.29)


where T1, T2 : X × X → Pcl(X) are multivalued operators satisfying a contractive typecondition (see also [9]).

The following results are obtained in the case of a set X endowed with two metrics.

Theorem 2.12. Let (X, d) be a complete generalized metric space and ρ another generalized metricon X. Let F : X → P(X) be a multivalued operator. One assumes that

(i) there exists a matrix C ∈Mm,m(R+) such that d(x, y) ≤ ρ(x, y) · C, for all x, y ∈ X;

(ii) F : (X, d) → (P(X),Hd) has closed graph;

(iii) there exist A,B ∈ Mm,m(R+) such that for all x, y ∈ X and u ∈ F(x), there exists v ∈F(y) with

ρ(u, v) ≤ Aρ(x, y) + Bρ(y, u). (2.30)


Proof. Let x0 ∈ X such that x1 ∈ F(x0).For x1 ∈ F(x0), there exists x2 ∈ F(x1) such that

ρ(x1, x2) ≤ Aρ(x0, x1) + Bρ(x1, x1) = Aρ(x0, x1). (2.31)

For x2 ∈ F(x1), there exists x3 ∈ F(x2) such that

ρ(x2, x3) ≤ Aρ(x1, x2) + Bρ(x2, x2) ≤ A2ρ(x0, x1). (2.32)

Consequently, we construct by induction the sequence (xn)n∈N in X which satisfies thefollowing properties:

(1) xn+1 ∈ F(xn), for all n ∈ N;

(2) ρ(xn, xn+1) ≤ Anρ(x0, x1), for all n ∈ N.

We show that (xn)n∈N is a Cauchy sequence in X with respect to ρ. In order to do that,let p ∈ N, p > 0. One has the estimation

ρ(xn, xn+p

) ≤ ρ(xn, xn+1) + ρ(xn+1, xn+2) + · · · + ρ(xn+p−1, xn+p

)

≤ Anρ(x0, x1) +An+1ρ(x0, x1) + · · · +An+p−1ρ(x0, x1)

≤ An(I +A + · · · +Ap−1 + · · ·

)ρ(x0, x1)

= An(I −A)−1ρ(x0, x1).

(2.33)

Since the matrix A converges towards zero, one has An → Θ as n → ∞. Letting n → ∞ oneget ρ(xn, xn+p) → 0 which implies that (xn)n∈N is a Cauchy sequence with respect to ρ.

Using (i), we obtain that d(xn, xn+p) ≤ ρ(xn, xn+p) · C → 0 as n → ∞. Thus, (xn)n∈N isa Cauchy sequence with respect to d too.


Since (X, d) is complete, the sequence (xn)n∈N is convergent in X. Thus there existsx∗ ∈ X such that x∗ = limn→∞xn with respect to d.

Finally, we show that x∗ ∈ F(x∗).Since xn+1 ∈ F(xn), for all n ∈ N and F has closed graph, by using the limit presented

above, we get that x∗ ∈ F(x∗), that is, Fix(F)/= ∅.

Remark 2.13. (1) Theorem 2.12 holds even if the assumption (iii) is replaced by(iii′) there exist A,B ∈ Mm,m(R+) such that for all x, y ∈ X and u ∈ F(x), there exists

v ∈ F(y) such thatρ(u, v) ≤ Aρ(x, y) + Bd(y, u).(2) Letting p → ∞ in the estimation of ρ(xn, xn+p), presented in the proof of

Theorem 2.12, we get

ρ(xn, x∗) ≤ An(I −A)−1ρ(x0, x1). (2.34)

Using the relation between the generalized metrics d and ρ, one has immediately

d(xn, x∗) ≤ CAn(I −A)−1ρ(x0, x1). (2.35)

Theorem 2.14. Let (X, d) be a complete generalized metric space and ρ another generalized metric onX. Let x0 ∈ X, r := (ri)

mi=1 ∈ R

m+ with ri > 0 for each i ∈ {1, 2, . . . , m} and let F : Bρ(x0, r) → P(X)

be a multivalued operator. Suppose that

(i) there exists C ∈Mm,m(R+) such that d(x, y) ≤ Cρ(x, y), for all x, y ∈ X;

(ii) F : (Bρ(x0, r), d) → (Pb(X),Hd) has closed graph;

(iii) there exist A,B ∈ Mm,m(R+) such that A is a matrix that converges to zero and for allx, y ∈ Bρ(x0, r) and u ∈ F(x), there exists v ∈ F(y) such that

ρ(u, v) ≤ Aρ(x, y) + Bρ(y, u); (2.36)

(iv) if u ∈ Rm+ is such that u(I −A)−1 ≤ (I −A)−1r, then u ≤ r;

(v) ρ(x0, x1)(I −A)−1 ≤ r.

Then Fix(F)/= ∅.

Proof. Let x0 ∈ X such that x1 ∈ F(x0). By (v) one has

ρ(x0, x1)(I −A)−1 ≤ r ≤ (I −A)−1r, (2.37)

which implies x1 ∈ Bρ(x0, r).Since x1 ∈ F(x0), there exists x2 ∈ F(x1) such that

ρ(x1, x2)(I −A)−1 ≤ Aρ(x0, x1)(I −A)−1 + Bρ(x1, x1)(I −A)−1 ≤ Ar. (2.38)


Hence,

ρ(x0, x2)(I −A)−1 ≤ ρ(x0, x1)(I −A)−1 + ρ(x1, x2)(I −A)−1

≤ Ir +Ar ≤ (I +A + · · · +An + · · · )r ≤ (I −A)−1r,(2.39)

which implies that ρ(x0, x2) ≤ r, that is, x2 ∈ Bρ(x0, r).For x2 ∈ F(x1), there exists x3 ∈ F(x2) such that

ρ(x2, x3)(I −A)−1 ≤ Aρ(x1, x2)(I −A)−1 + Bρ(x2, x2)(I −A)−1 ≤ A2r. (2.40)

Then the following estimation holds:

ρ(x0, x3)(I −A)−1 ≤ ρ(x0, x1)(I −A)−1 + ρ(x1, x2)(I −A)−1 + ρ(x2, x3)(I −A)−1

≤ Ir +Ar +A2r ≤ (I −A)−1r,(2.41)

and thus ρ(x0, x3) ≤ r, that is, x3 ∈ Bρ(x0, r).Inductively, we can construct the sequence (xn)x∈N which has its elements in the closed

ball Bρ(x0, r) and satisfies the following conditions:

(1) xn+1 ∈ F(xn), for all n ∈ N;

(2) ρ(xn, xn+1)(I −A)−1 ≤ Anr, for all n ∈ N.

By a similar approach as in the proof of Theorem 2.12, the conclusion follows.

A homotopy result for multivalued operators on a set endowed with a vector-valuedmetric is the following.

Theorem 2.15. Let (X, d) be a generalized complete metric space in Perov sense, let U be an opensubset ofX, and let V be a closed subset ofX, withU ⊂ V . LetG : V×[0, 1] → P(X) be a multivaluedoperator with closed (with respect to d) graph, such that the following conditions are satisfied:

(a) x /∈G(x, t), for each x ∈ V \U and each t ∈ [0, 1];

(b) there exist A,B ∈ Mm,m(R+) such that the matrix A is convergent to zero such that foreach t ∈ [0, 1], for each x, y ∈ X and all u ∈ G(x, t)), there exists v ∈ G(y, t) withd(u, v) ≤ Ad(x, y) + Bd(y, u).

(c) there exists a continuous increasing function φ : [0, 1] → Rm such that for all t, s ∈ [0, 1],

each x ∈ V and each u ∈ G(x, t) there exists v ∈ G(x, s) such that d(u, v) ≤ |φ(t)−φ(s)|;(d) if v, r ∈ R

m+ are such that v · (I −A)−1 ≤ (I −A)−1 · r, then v ≤ r;

Then G(·, 0) has a fixed point if and only if G(·, 1) has a fixed point.

Proof. Suppose that G(·, 0) has a fixed point z. From (a) we have that z ∈ U. Define

Q := {(t, x) ∈ [0, 1] ×U | x ∈ G(x, t)}. (2.42)


Clearly Q/= ∅, since (0, z) ∈ Q. Consider on Q a partial order defined as follows:

(t, x) ≤ (s, y

)iff t ≤ s, d

(x, y

) ≤ 2[φ(s) − φ(t)] · (I −A)−1. (2.43)

Let M be a totally ordered subset of Q and consider t∗ := sup{t | (t, x) ∈ M}. Consider asequence (tn, xn)n∈N∗ ⊂ M such that (tn, xn) ≤ (tn+1, xn+1) for each n ∈ N

∗ and tn → t∗, asn → +∞. Then

d(xm, xn) ≤ 2[φ(tm) − φ(tn)

] · (I −A)−1, for each m,n ∈ N∗, m > n. (2.44)

When m,n → +∞, we obtain d(xm, xn) → 0 and, thus, (xn)n∈N∗ is d-Cauchy. Thus (xn)n∈N∗is convergent in (X, d). Denote by x∗ ∈ X its limit. Since xn ∈ G(xn, tn), n ∈ N

∗ and since Gis d-closed, we have that x∗ ∈ G(x∗, t∗). Thus, from (a), we have x∗ ∈ U. Hence (t∗, x∗) ∈ Q.Since M is totally ordered we get that (t, x) ≤ (t∗, x∗), for each (t, x) ∈ M. Thus (t∗, x∗) is anupper bound of M. By Zorn’s Lemma, Q admits a maximal element (t0, x0) ∈ Q. We claimthat t0 = 1. This will finish the proof.

Suppose t0 < 1. Choose r := (ri)mi=1 ∈ R

m+ with ri > 0 for each i ∈ {1, 2, . . . , m} and

t ∈]t0, 1] such that B(x0, r) ⊂ U, where r := 2[φ(t) − φ(t0)] · (I − A)−1. Since x0 ∈ G(x,t0), by(c), there exists x1 ∈ G(x0, t) such that d(x0, x1) ≤ |φ(t) − φ(t0)|. Thus, d(x0, x1)(I − A)−1 ≤|φ(t) − φ(t0)| · (I −A)−1 < r.

Since B(x0, r) ⊂ V , the multivalued operator G(·, t) : B(x0, r) → Pcl(X) satisfies, forall t ∈ [0, 1], the assumptions of Theorem 2.1 Hence, for all t ∈ [0, 1], there exists x ∈ B(x0, r)such that x ∈ G(x, t). Thus (t, x) ∈ Q. Since d(x0, x) ≤ r = 2[φ(t) − φ(t0)](I − A)−1, weimmediately get that (t0, x0) < (t, x). This is a contradiction with the maximality of (t0, x0).

Conversely, if G(·, 1) has a fixed point, then putting t := 1 − t and using first part of theproof we get the conclusion.

Remark 2.16. Usually in the above result, we take Q = U. Notice that in this case, condition(a) becomes

(a′) x /∈G(x, t), for each x ∈ ∂U and each t ∈ [0, 1].

Remark 2.17. If in the above results we consider m = 1, then we obtain, as consequences,several known results in the literature, as those given by M. Berinde and V. Berinde [8],Precup [5], Petrusel and Rus [11], and Feng and Liu [12]. Notice also that the theoremspresented here represent extensions of some results given Bucur et al. [9], O’Regan andPrecup [13], O’Regan et al. [7], Perov [1], and so forth.

Remark 2.18. Notice also that since Rn+ is a particular type of cone in a Banach space, it is a

nice direction of research to obtain extensions of these results for the case of operators on K-metric (or K-normed) spaces (see Zabrejko [14]). For other similar results, open questions,and research directions see [7, 11–13, 15–18].

Acknowledgments

The authors are thankful to anonymous reviewer(s) for remarks and suggestions thatimproved the quality of the paper. The first author wishes to thank for the financial support


provided from programs co-financed by The Sectoral Operational Programme HumanResources Development, Contract POS DRU 6/1.5/S/3-“Doctoral studies: through sciencetowards society”.

References

[1] A. I. Perov, “On the Cauchy problem for a system of ordinary differential equations,” Pviblizhen. Met.Reshen. Differ. Uvavn., vol. 2, pp. 115–134, 1964.

[2] R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, Springer,Berlin, Germany, 2000.

[3] G. Allaire and S. M. Kaber, Numerical Linear Algebra, vol. 55 of Texts in Applied Mathematics, Springer,New York, NY, USA, 2008.

[4] I. A. Rus, Principles and Applications of the Fixed Point Theory, Dacia, Cluj-Napoca, Romania, 1979.[5] R. Precup, “The role of matrices that are convergent to zero in the study of semilinear operator

systems,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 703–708, 2009.[6] M. Turinici, “Finite-dimensional vector contractions and their fixed points,” Studia Universitatis Babes-

Bolyai. Mathematica, vol. 35, no. 1, pp. 30–42, 1990.[7] D. O’Regan, N. Shahzad, and R. P. Agarwal, “Fixed point theory for generalized contractive maps on

spaces with vector-valued metrics,” in Fixed Point Theory and Applications. Vol. 6, pp. 143–149, NovaScience, New York, NY, USA, 2007.

[8] M. Berinde and V. Berinde, “On a general class of multi-valued weakly Picard mappings,” Journal ofMathematical Analysis and Applications, vol. 326, no. 2, pp. 772–782, 2007.

[9] A. Bucur, L. Guran, and A. Petrusel, “Fixed points for multivalued operators on a set endowed withvector-valued metrics and applications,” Fixed Point Theory, vol. 10, no. 1, pp. 19–34, 2009.

[10] V. Berinde and M. Pacurar, “Fixed points and continuity of almost contractions,” Fixed Point Theory,vol. 9, no. 1, pp. 23–34, 2008.

[11] A. Petrusel and I. A. Rus, “Fixed point theory for multivalued operators on a set with two metrics,”Fixed Point Theory, vol. 8, no. 1, pp. 97–104, 2007.

[12] Y. Feng and S Liu, “Fixed point theorems for multi-valued contractive mappings and multi-valuedCaristi type mappings,” Journal of Mathematical Analysis and Applications, vol. 317, no. 1, pp. 103–112,2006.

[13] D. O’Regan and R. Precup, “Continuation theory for contractions on spaces with two vector-valuedmetrics,” Applicable Analysis, vol. 82, no. 2, pp. 131–144, 2003.

[14] P. P. Zabrejko, “K-metric and K-normed linear spaces: survey,” Collectanea Mathematica, vol. 48, no.4–6, pp. 825–859, 1997.

[15] A. Chis-Novac, R. Precup, and I. A. Rus, “Data dependence of fixed points for non-self generalizedcontractions,” Fixed Point Theory, vol. 10, no. 1, pp. 73–87, 2009.

[16] I. A. Rus, A. Petrusel, and G. Petrusel, Fixed Point Theory, Cluj University Press, Cluj-Napoca,Romania, 2008.

[17] C. Chifu and G. Petrusel, “Well-posedness and fractals via fixed point theory,” Fixed Point Theory andApplications, vol. 2008, Article ID 645419, 9 pages, 2008.

[18] F. Voicu, “Fixed-point theorems in vector metric spaces,” Studia Universitatis Babes-Bolyai. Mathematica,vol. 36, no. 4, pp. 53–56, 1991 (French).


Review ArticleNormal Structure and Common Fixed PointProperties for Semigroups of NonexpansiveMappings in Banach Spaces

Anthony To-Ming Lau

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton,AB, Canada T6G 2G1

Correspondence should be addressed to Anthony To-Ming Lau, [email protected]

Received 9 October 2009; Accepted 10 December 2009


Copyright q 2010 Anthony To-Ming Lau. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

In 1965, Kirk proved that if C is a nonempty weakly compact convex subset of a Banach space withnormal structure, then every nonexpansive mapping T : C → C has a fixed point. The purposeof this paper is to outline various generalizations of Kirk’s fixed point theorem to semigroup ofnonexpansive mappings and for Banach spaces associated to a locally compact group.

1. Introduction

A closed convex subset C of a Banach space E has normal structure if for each bounded closedconvex subset D of C which contains more than one point, there is a point x ∈ D which is nota diametral point of D, that is, sup {‖x − y‖ : y ∈ D} < δ(D), where δ(D) = the diameter ofD.

The set C is said to have fixed point property (FPP) if every nonexpansive mappingT : C → C has a fixed point. In [1], Kirk proved the following important celebrated result.

Theorem 1.1 (Kirk [1]). Let E be a Banach space, and C a nonempty closed convex subset of E. If Cis weakly compact and has normal structure, then C has the FPP.

As well known, compact convex subset of a Banach space E always has normalstructure (see [2]). It was an open problem for over 15 years whether every weakly compactconvex subset of E has normal structure. This problem was answered negatively by Alspach[3] when he showed that there is a weakly compact convex subset C of L1[0, 1] which doesnot have the fixed point property. In particular, C cannot have normal structure.


It is the purpose of this paper to outline the relation of normal structure and fixedpoint property for semigroup of nonexpansive mappings. This paper is organized as follows.In Section 3, we will focus on generalizations of Kirk’s fixed point theorem to semigroupsof nonexpansive mappings. In Section 4, we will discuss about fixed point properties andnormal structure on Banach spaces associated to a locally compact group.

2. Some Preliminaries

All topologies in this paper are assumed to be Hausdorff. If E is a Banach space and A ⊆ E,then A and co A will denote the closure of A and the closed convex hull of A in E,respectively.

Let E be a Banach space and let C a subset of E. A mapping T from C into itself is saidto be nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for each x, y ∈ C. A Banach space E is said to beuniformly convex if for each ε > 0, there exists δ > 0 such that ‖(x + y)/2‖ ≤ 1 − δ for eachx, y ∈ E satisfying ‖x‖ ≤ 1, ‖y‖ ≤ 1 and ‖x − y‖ ≥ ε.

Let S be a semigroup, �∞(S) the Banach space of bounded real valued functions on Swith the supremum norm. Then a subspace X of �∞(S) is left (resp., right) translation invariantif �a(X) ⊆ X (resp., ra(X) ⊆ X) for all a ∈ S, where (�af)(s) = f(as) and (raf)(s) = f(sa),s ∈ S.

A semitoplogical semigroup S is a semigroup with Hausdorff topology such that foreach a ∈ S, the mappings s → a · s and s → s · a from S into S are continuous. Examplesof semitopological semigroups include all topological groups, the set M(n,C) of all n × nmatrices with complex entries, matrix multiplication, and the usual topology, the unit ballof �∞ with weak∗-topology and pointwise multiplication, or B(H)(= the space of boundedlinear operators on a Hilbert space H) with the weak∗-topology and composition.

If S is a semitopological semigroup, we denote CB(S) the closed subalgebra of �∞(S)consisting of continuous functions. Let LUC(S) (resp., RUC(S)) be the space of left (resp.,right) uniformly continuous functions on S; that is, all f ∈ CB(S) such that the mapping fromS into CB(S) defined by s → �sf (resp., s → rsf) is continuous when CB(S) has the supnorm topology. Then as is known (see [4]), LUC(S) and RUC(S) are left and right translationinvariant closed subalgebras of CB(S) containing constants. Also let AP(S) (resp., WAP(S))denote the space of almost periodic (resp., weakly almost periodic) functions f in CB(S);that is, all f ∈ CB(S) such that {�af ;a ∈ S} is relatively compact in the norm (resp., weak)topology of CB(S), or equivalently {raf ;a ∈ S} is relatively compact in the norm (resp.,weak) topology of CB(S). Then as is known [4, page 164], AP(S) ⊆ LUC(S) ∩ RUC(S), andAP(S) ⊆WAP(S). When S is a locally compact group, then WAP(S) ⊆ LUC(S)∩RUC(S) (see[4, page 167]).

A semitopological semigroup S is left reversible if any two closed right ideals of S havenonvoid intersection.

The class S of all left reversible semitopological semigroups includes trivially allsemitopological semigroups which are algebraically groups, and all commuting semigroups.

The class S is closed under the following operations.

(a) If S ∈ S and S′ is a continuous homomorphic image of S, then S′ ∈ S.

(b) Let Sα ∈ S, α ∈ I and S be the semitopological semigroup consisting of the set ofall functions f on I such that f(α) ∈ Sα, α ∈ I, the binary operation defined byfg(α) = f(α)g(α) for all α ∈ I and f, g ∈ S, and the product topology. Then S ∈ S.


(c) Let S be a semitopological semigroup and Sα, α ∈ I, semitopological sub-semigroups of S with the property that S = ∪Sα and, if α1, α2 ∈ I, then there existsα3 ∈ I such that Sα3 ⊇ Sα1 ∪ Sα2 . If Sα ∈ S for each α ∈ I, then S ∈ S.

Let S be a nonempty set and X a translation invariant subspace of �∞(S) containingconstants. Then μ ∈ X∗ is called a mean on X if ‖μ‖ = μ(1) = 1. As well known, μ is a mean onX if and only if

infs∈S

f(s) ≤ μ(f) ≤ sups∈S

f(s) (2.1)

for each f ∈ X.Also μ is called a left (resp., right) invariant mean if μ(�af) = μ(f) (resp., μ(raf) = μ(f))

for all a ∈ S, f ∈ X.

Lemma 2.1. Let S be a semitopological semigroup and X a left translation invariant subspace ofCB(S) containing constants and which separates closed subsets of S. If X has a left invariant mean,then S is left reversible.

Proof. Let μ be a left invariant mean of X, I1 and I2 disjoint nonempty closed right ideals of S.By assumption, there exists f ∈ X such that f ≡ 1 on I1 and f ≡ 0 on I2. Now if a1 ∈ I1, then�a1f = 1. So,

μ(f)= μ

(�a1f

)= 1. (2.2)

But if a2 ∈ I2, then �a2f ≡ 0. So μ(f) = μ(�a2f) = 0, which is impossible.

Corollary 2.2. If S is normal and CB(S) has a left invariant mean, then S is left reversible.

See [5] for details.A discrete semigroup S is called left amenable [6] if �∞(S) has a left invariant mean.

In particular every left amenable discrete semigroup is left reversible by Corollary 2.2. Thesemigroup S is amenable if it is both left and right amenable. In this case, there is always aninvariant mean on �∞(S).

Remark 2.3. Lemma 2.1 is not true without normality. Let S be a topological space which isregular and Hausdorff and CB(S) consists of constant functions only [7]. Define on S themultiplication st = s for all s, t ∈ S. Let a ∈ S be fixed. Define μ(f) = f(a) for all a ∈ S. Thenμ is a left invariant mean on C(S), but S is not left reversible.

3. Generalizations of Kirk’s Fixed Point Theorem

By a (nonlinear) submean on X, we will mean a real-valued function μ on X satisfying thefollowing properties:

(1) μ(f + g) ≤ μ(f) + μ(g) for every f, g ∈ X;

(2) μ(αf) = αμ(f) for every f ∈ X and α ≥ 0;


(3) for f, g ∈ X, f ≤ q implies μ(f) ≤ μ(g);

(4) μ(c) = c for every constant function c.

Clearly every mean is a submean. See [8] for details.If S is a semigroup and X is left translation invariant, a submean μ on X is left

subinvariant if μ(�af) ≥ μ(f) for each f ∈ X and a ∈ S.Let S be a semitopological semigroup, C a nonempty subset of a Banach space E, then

a representation S = {Ts : s ∈ S} of S as mappings from C into C is continuous if S × C → Cdefined by (s, x) → Tsx, s ∈ S, x ∈ C is continuous when S×C has the product topology. It iscalled separately continuous if for each x ∈ C and s ∈ S, the maps s → Tsx from S into C andthe map x → Tsx from C into C are continuous.

Theorem 3.1. Let S be a semitopological semigroup, let C a nonempty weakly compact convex subsetof a Banach space E which has normal structure and let S = {Ts; s ∈ S} a continuous representation ofS as nonexpansive self-mappings on C. Suppose that RUC(S) has a left subinvariant submean. ThenS has a common fixed point in C.

Corollary 3.2. Let S be a left reversible semitopological semigroup. Let C be a nonempty weaklycompact convex subset of a Banach space E which has normal structure and let S = {Ts; s ∈ S} acontinuous representation of S as nonexpansive self-mappings on C. Then S has a fixed point in C.

Proof. If S is left reversible, define μ(f) = infssupt∈sSf(t). Then the proof of Lemma 3.6 in [9]shows that μ is a submean on CB(S) such that μ(�af) ≥ μ(f) for all f ∈ CB(S) and a ∈ S, thatis, μ is left subinvariant.

Note that since every compact convex set has normal structure, Corollary 3.2 impliesthe following.

Corollary 3.3 (DeMarr [10]). Let E be a Banach space and C a nonempty compact convex subset ofE. If F is a commuting family of nonexpansive mappings of C into C, then the family F has a commonfixed point in C.

Remark 3.4. Theorem 3.1 is proved by Lau and Takahashi in [11]. Mitchell [12] generalizedthe theorems of DeMarr [10, page 1139] and Takahaski [13, page 384] by showing that if Cis a nonempty compact convex subset of a Banach space and S is a left-reversible discretesemigroup of nonexpansive mappings from C into C, then C contains a common fixed pointfor S. Belluce and Kirk [14] also improved DeMarr’s result in [10] and proved that if C is anonempty weakly compact convex subset of a Banach space and if C has complete normalstructure, then every family of commuting nonexpansive self-maps on C has a common fixedpoint.

This result was extended to the class of left reversible semitopological semigroup byHolmes and Lau in [15]. Corollary 3.2 is due to Lim [16] who showed that normal structureand complete normal structure are equivalent.

The following related theorem was also established in [15].

Theorem 3.5. Let S be a left reversible semitopological semigroup, let C a nonempty, bounded, closedconvex subset of a Banach space E, and let S = {Ts; s ∈ S} a separately continuous representation ofS as nonexpansive self-maps on C. If there is a nonempty compact subsetM ⊆ C and a ∈ S such that


a commutes with all elements of S and for each x ∈ C, the closure of the set {an(x) | n = 1, 2, . . .}contains a point ofM, thenM contains a common fixed point of S.

Let S be a semitopological semigroup and C is a nonempty subset of a Banach spaceE, and S = {Ts; s ∈ S} a separately continuous representation of S as mappings from C intoC. We say that the representation is asymptotically nonexpansive if for each x, y ∈ C, there isa left ideal J ⊆ S such that ‖Tsx − Tsy‖ ≤ ‖x − y‖ for all s ∈ J.

We also say that the representation has property (B) if for each x ∈ C, whenever a net{sα(x); α ∈ I}, sα ∈ S, converges to x, then the net {(sαa)x; α ∈ I} also converges to a(x) foreach a ∈ S.

Clearly condition (B) is automatically satisfied when S is commutative.The semitopological semigroup S is right reversible if

sa ∩ sb /= ∅ for each a, b ∈ S. (3.1)

The following theorem is proved in [17].

Theorem 3.6. Let C be a nonempty compact convex subset of a Banach space E and S a rightreversible semitopological semigroup. If S = {Ts; s ∈ S} is a separately continuous asymptoticallynonexpansive representation of S as mappings from C into C with property (B), then C contains acommon fixed point for S.

The following example from [17] shows a simple situation where our fixed pointtheorem applies, but DeMarr’s fixed point theorem does not.

Let K = {(r, θ) | 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π} be the closed unit disc in R2 with polarcoordinates and the usual Euclidean norm. Define continuous mappings f, g from K into Kby

f(r, θ) =( r

2, θ),

g(r, θ) = (r, 2θ(mod 2π)).(3.2)

Then the semigroup of continuous mappings from K to K generated by f and g under usualcomposition is commutative and asymptotically nonexpansive. However, the action of S (orany ideal of S) on K is not nonexpansive.

Open Problem 1. Can right reversibility of S and property (B) in Theorem 3.6 be replaced byamenability of S?

Let C be a nonempty closed convex subset of a Banach space E. Then C has the fixedpoint property for nonexpansive mappings if every nonexpansive mapping T : C → Chas a fixed point; C has the onlyconditional fixed point property for nonexpansive mappings ifevery nonexpansive mapping T : C → C satisfies either T has no fixed point in C, or Thas a fixed point in every nonempty bounded closed convex T -invariant subset of C. Forcommuting family of nonexpansive mappings, the following is a remarkable common fixedpoint property due to Bruck [18].


Theorem 3.7. Let E be a Banach space and C a nonempty closed convex subset of E. If C has both thefixed point property and the conditional fixed point property for nonexpansive mappings, then for anycommuting family S of nonexpansive mappings of C into C, there is a common fixed point for S.

Theorem 3.7 was proved by Belluce and Kirk [19] when S is finite and C is weaklycompact and has normal structure, by Belluce and Kirk [14] when C is weakly compact andhas complete normal structure, Browder [20] when E is uniformly convex and C is bounded,Lau and Holmes [15] when S is left reversible andC is compact, and finally by Lim [16] whenS is left reversible and C is weakly compact and has normal structure.

Open Problem 2 (Bruck [18]). Can commutativity of S be replaced by left reversibility?

The answer to Problem 2 is not known even when the semigroup is left amenable.Let (Σ, ◦) be a compact right topological semigroup, that is, a semigroup and a compact

Hausdorff topological space such that for each τ ∈ Σ the mapping γ → γ ◦ τ from Σ into Σis continuous. In this case, Σ must contain minimal left ideals. Any minimal left ideal in Σ isclosed and any two minimal left ideals of Σ are homeomorphic and algebraically isomorphic.

Let X be a nonempty weakly compact convex subset of a Banach space E. Let S = {Ts :s ∈ S} be a representation of a semigroup S as nonexpansive and weak-weak continuousmappings from X into X. Let Σ be the closure of S in the product space (X,weak)X. Then Σ isa compact right topological semigroup consisting of nonexpansive mappings from X into X.Further, for any T ∈ Σ, there exists a sequence {Tn} of convex combination of operators fromS such that ‖Tnx − Tx‖ → 0 for every x ∈ X. See [21] for details.

Σ is called the enveloping semigroup of S.

Theorem 3.8. Let X be a nonempty weakly compact convex subset of a Banach space, E and X hasnormal structure. Let S = {Ts : s ∈ S} be a representation of a semigroup as norm nonexpansive andweakly continuous mappings from X into X and let Σ be the enveloping semigroup of S. Let I be aminimal left ideal of Σ and let Y a minimal S-invariant closed convex subset of X. Then there exists anonempty weakly closed subset C of Y such that I is constant on C.

Corollary 3.9. Let Σ and X as in Theorem 3.8. Then there exist T0 ∈ Σ and x ∈ X such that T0Tx =T0x for every T ∈ Σ.

Proof. Pick x ∈ C and T0 ∈ I of the above theorem.

Remark 3.10. If S is commutative, then for any T ∈ Σ and s ∈ S, Ts ◦ T = T ◦ Ts, that is, z = T0xis in fact a common fixed point for Σ (and, hence, for S). Note that if X is norm compact,the weak and norm topology agree on X. Hence every nonexpansive mapping from X intoX must be weakly continuous. Therefore, Corollary 3.9 improves the fixed point theorem ofDeMarr [10] for commuting semigroups of nonexpansive mappings on compact convex sets.

The above theorem proved in [21] provides a new approach using envelopingsemigroups in the study of common fixed point of a semigroup of nonexpansive mappingson a weakly compact convex subset of a Banach space.

Open Problem 3. Can the above technique applied to give a proof of Lim’s fixed point theoremfor left reversible semigroup in [16].

The following generalization of DeMarr’s fixed point theorem was proved in [22].


Theorem 3.11. Let S a be semitopological semigroup.If AP(S) has a left invariant mean, then S has the following fixed point property. Whenever

S = {Ts : s ∈ S} is a separately continuous representation of S as nonexpansive self-mappings on acompact convex subset C of a Banach space, then C contains a common fixed point for S.

Quite recently the Lau and Zhang [23] are able to establish the following related fixedpoint property.

Theorem 3.12. Let S be a separable semitopological semigroup. If WAP(S) has a left invariant mean,then S has the following fixed point property:

Whenever S = {Ts; s ∈ S} is a continuous representation of S as nonexpansive self-mappingson a weakly compact convex subset C of a Banach space E such that the closure of S in CC with theproduct of weak topology consists entirely of continuous functions, then C contains a common fixedpoint of C.

Remark 3.13. (a) The converse of Theorem 3.12 also holds when S has an identity byconsidering S = {rs; r ∈ S}, the semigroup of right translations, on the weakly compactconvex sets Cf = co{rsf ; s ∈ S} for each f ∈WAP(S) (see [24]).

(b) When S is a discrete semigroup, the following implication diagram is known:

��

S ��

S ��

��

The implication “S is left reversible =⇒ AP(S) has a LIM” for any semitopological semigroupwas established in [22]. During the 1984 Richmond, Virginia conference on analysis onsemigroups, T. Mitchell [12] gave two examples to show that for discrete semigroups “AP(S)has LIM” � “S is left reversible” (see [25] or [23]). The implication “S is left reversible=⇒WAP(S) has LIM” for discrete semigroups was proved by Hsu [26]. Recently, it is shownin [23] that if S1 is the bicyclic semigroup generated by {e, a, b, c} such that e is the unit of S1

and ab = e and ac = e, then WAP(S) has a LIM, but S1 is not left reversible. Also if S2 is thebicyclic semigroup generated by {e, a, b, c, d}, where e is the unit element and ac = bd = e,then AP(S2) has a LIM, but WAP(S2) does not have a LIM.

The following is proved in [5] (see also [27]).

Theorem 3.14. Let S be a left reversible discrete semigroup. Then S has the following fixed pointproperty.

Whenever S = {Ts : s ∈ S} is a representation of S as norm nonexpansive weak∗-weak∗

continuous mappings of a norm-separable weak∗-compact convex subset C of a dual Banach space Einto C, then C contains a common fixed point for S.

It can be shown that the following fixed point property on a discrete semigroup Simplies that S is left amenable.


(G) Whenever S = {Ts : s ∈ S} is a representation of S as norm nonexpansive weak∗-weak∗ continuous mappings of a weak∗-compact convex subset C of a dual Banachspace E into C, then C contains a common fixed point for S.

Open Problem 4. Does left amenability of S imply (G)?

Other related results for this section can also be found in [9, 28–38].

4. Normal Structure in Banach Spaces Associated toLocally Compact Groups

A Banach space has weak-normal structure if every nontrivial weakly compact convex subsethas normal structure. If the Banach space is also a dual space then it has weak∗-normalstructure if every nontrivial weak∗ compact convex subset has normal structure. It is clear thata dual Banach space has weak-normal structure whenever it has weak∗-normal structure.

A [dual] Banach space E is said to have the weak-fixed point property (weak-FPP) [(FPP∗)] if for every weakly [weak∗] compact convex subset C of E and for everynonexpansive T : C → C, T has a fixed point in C. Kirk proved that if E has weak-normalstructure then E has property FPP [1]. Subsequently, Lim [39] proved that a dual Banachspace has property FPP∗ whenever it has weak∗-normal structure.

A Banach space E is said to have the Kadec-Klee property (KK) if whenever (xn) is asequence in the unit ball of E that converges weakly to x, and sep((xn)) > 0, where

sep((xn)) ≡ inf{‖xn − xm‖ : n/=m}, (4.1)

then ‖x‖ < 1 (see [40]).For dual Banach spaces, we have the similar properties replacing weak converges by

weak∗ converges.A Banach space E is said to have the uniformly Kadec-Klee property (UKK) if for every

ε > 0 there is a 0 < δ < 1 such that whenever (xn) is a sequence in the unit ball of E convergingweakly to x and sep((xn)) > ε then ‖x‖ ≤ δ. This property was introduced by Huff [40] whoshowed that property UKK is strictly stronger than property KK. van Dulst and Sims showedthat a Banach space with property UKK has property weak FPP [41].

It is natural to define a property similar to UKK by replacing the weak convergenceby weak∗ convergence in UKK and calling it UKK∗. However, van Dulst and Sims found thatthe following definition is more useful.

A dual Banach space E has property UKK∗ if for every ε > 0 there is a 0 < δ < 1such that whenever A is a subset of the closed unit ball of E containing a sequence (xn) withsep((xn)) > ε, then there is an x in weak∗-closure (A) such that ‖x‖ ≤ δ.

They proved that a dual Banach space with property UKK∗ has property FPP∗ [41].Moreover, they observed that if the dual unit ball is weak∗ sequentially compact then propertyUKK∗, as defined above, is equivalent to the condition obtained from UKK by replacing weakconvergence by weak∗ convergence.


We now summarize the various properties defined above by

�� ∗

��∗

��∗��

�� ∗

(where n.s. = normal structure).Let X = (�2 ⊕ �3 ⊕ · · · ⊕ �n ⊕ · · · )2. Then, as noted by Huff [40], X is reflexive and has

property KK but not UKK.Let X be a locally compact Hausdorff space, and C(X) the space of bounded

continuous complex-valued functions defined on X with the supremum norm. Let C0(X) bethe subspace of C(X) consisting of functions “vanishing at infinity,” and M(X) be the spaceof bounded regular Borel measure on X, with the variation norm. Let Md(X) be the subspaceof M(X) consisting of the discrete measures on X. It is well known that the dual of C0(X) canbe identified with M(X), and that Md(X) is isometrically isomorphic to �1(X).

Lennard [42] proved the following theorem.

Theorem 4.1. LetH be a Hilbert space. Then T(H), the trace class operators onH, has the propertyUKK∗ and has FPP∗ when regarded as the dual space of C(H), the C∗-algebra of compact operator onH.

Theorem 4.2. Let G be a locally compact group. Then the following statements are equivalent.

(1) G is discrete.

(2) M(G) is isometrically isomorphic to �1(G).

(3) M(G) has property UKK∗.

(4) M(G) has property KK∗.

(5) Weak∗ convergence and weak convergence of sequences agree on the unit sphere ofM(G).

(6) M(G) has weak∗ normal structure.

(7) M(G) has property FPP∗.

Theorem 4.3. Let G be a locally compact group. Then the group algebra L1(G) has the weak fixedpoint property for left reversible semigroups if and only if G is discrete.

Theorem 4.4. Let G be a locally compact group. Let N be a C∗-subalgebra of WAP(G) containingC0(G) and the constants. Then the following statements are equivalent.

(1) G is finite.

(2) N∗ has property UKK∗.

(3) N∗ has property KK∗.

(4) Weak∗ convergence and weak convergence for sequences agree on the unit sphere ofN∗.

(5) N∗ has weak∗-normal structure.


Theorem 4.5. Let G be a locally compact group. Then

(1) Weak∗ convergence and weak convergence for sequences agree on the unit sphere ofLUC(G)∗ if and only if G is discrete.

(2) LUC(G∗) has weak∗-normal structure if and only if G is finite.

Let G be a locally compact group. We define C∗(G), the group C∗-algebra of G, to bethe completion of L1(G) with respect to the norm

∥∥f∥∥∗ = sup∥∥πf∥∥, (4.2)

where the supremum is taken over all nondegenerate representations π of L1(G) as analgebra of bounded operator on a Hilbert space. Let C(G) be the Banach space of boundedcontinuous complex-valued function on G with the supremum norm. Denote the set ofcontinuous positive definite functions on G by P(G), and the set of continuous functionson G with compact support by C00(G). Define the Fourier-Stieltjes algebra of G, denoted byB(G), to be the linear span of P(G). The Fourier algebra of G, denoted by A(G), is defined tobe the closed linear span of P(G) ∩ C00(G). Finally, let λ be the left regular representationof G, that is, for each f ∈ L1(G), λ(f) is the bounded operator in B(L2(G)) defined onL2(G) by λ(f)(h) = f ∗ h (the convolution of f and h). Then denote by VN(G) to be theclosure of {λ(f) : f ∈ L1(G)} in the weak operator topology in B(L2(G)). It is known thatC∗(G)∗ = B(G) and A(G)∗ = VN(G). Furthermore, ifG is amenable (e.g., when G is compact),then

C∗(G) ∼= norm closure of{λ(f)

: f ∈ L1(G)} ⊆ VN(G). (4.3)

We refer the reader to [43] for more details on these spaces.Notice that when G is an abelian locally compact group, then B(G) ∼= M(G) and

C∗(G) ∼= C0(G), where G is the dual group of G. It follows from Theorem 4.2 that B(G) wasthe weak∗-normal structure if and only if G is discrete, or equivalently, G is compact.

Theorem 4.6. If G is compact, then B(G) has weak∗-normal structure and hence the FPP∗.

For a Banach space (resp., dual Banach space) E, we say that E has the weak-FPP(weak∗-FPP) for left reversible semigroup if whenever S is a left reversible semitopologicalsemigroup and C is a weak (resp., weak∗) compact convex subset of E, and S = {Ts : s ∈ S}is a separately continuous representation of S as nonexpansive mappings from C into C, thenthere is a common fixed point in C for S.

Theorem 4.7. If G is a separable compact group, then B(G) has the weak∗- FPP for left reversiblesemigroups.

Open Problem 5. Can separability be dropped from Theorem 4.7?

A locally compact groupG is called an [IN]-group if there is a compact neighbourhoodof the identity e in G which is invariant under the inner automorphisms. The class of [IN]-group contains all discrete groups, abelian groups and compact groups. Every [IN]-group isunimodular.


We now investigate the weak fixed point property for a semigroup. A group G issaid to an [AU]-group if the von Neumann algebra generated by every continuous unitaryrepresentation of G is atomic (i.e., every nonzero projection in the van Neumann algebramajorizes a nonzero minimal projection). It is an [AR]-group if the von Neumann algebraVN(G) is atomic. Since VN(G) is the von Neumann algebra generated by the regularrepresentation, it is clear that every [AU]-group is an [AR]-group. It was shown in [44,Lemma 3.1] that if the predual M∗ of a von Neumann algebra M has the Radon-Nikodymproperty, then M∗ has the weak fixed point property. In fact, since the property UKK ishereditary, the proof there actually showed that M∗ has property UKK and hence has weaknormal structure. For the two preduals A(G) and B(G), we know from [45, Theorems 4.1 and4.2] that the class of groups for which A(G) and B(G) have the Radon-Nikodym property isprecisely the [AR]-groups and [AU]-groups, respectively. Thus by Lim’s result [16, Theorem3] we have the following proposition

Proposition 4.8. Let G be a locally compact group.

(a) If G is an [AR]-group, then A(G) has the weak fixed point property for left reversiblesemigroups.

(b) If G is an [AU]-group, then B(G) has the weak fixed point property for left reversiblesemigroups.

Proposition 4.9. Let G be an [IN]-group. Then the following are equivalent.

(a) G is compact.

(b) A(G) has property UKK.

(c) A(G) has weak normal structure.

(d) A(G) has the weak fixed point property for left reversible semigroups.

(e) A(G) has the weak fixed point property.

(f) A(G) has the Radon-Nikodym property.

(g) A(G) has the Krein-Milman property.

A Banach space E is said to have the fixed point property (FPP) if every boundedclosed convex subset of E has the fixed point property for nonexpansive mapping. As wellknown, every uniformly convex space has the FPP.

Theorem 4.10. Let G be a locally compact group. Then A(G) has the FPP if and only if G is finite.

Remark 4.11. (a) Theorems 4.1, 4.2, 4.4, 4.5 and 4.6 are proved by Lau and Mah in [46];Theorems 4.3, 4.7, and Propositions 4.8 and 4.9 are proved by Lau and Mah in [47] and byLau and Leinert in [48].

(b) Upon the completion of this paper, the author received a preprint from ProfessorNarcisse Randrianantoanina [49], where he answered an old question in [50] (see also [23])and showed that for any Hilbert space H (separable or not) the trace class opertors onH,T(H) has the weak∗-FPP for left reversible semigroups. He is also able to remove theseparability condition in our Theorem 4.7, and show that for any locally compact group G:


(i) A(G) has the weak FPP if and only if is an [AR]-group;

(ii) B(G) has the weak-FPP if and only if G is an [AU]-group. In this case, B(G) evenhas the weak-FPP for left reversible semigroup.

We are grateful to Professor Randrianantoanina for sending us a copy of his work.(c) An example of an [AU]-group G which is not compact is the Fell group which is

the semidirect product of the additive p-adic number field Qp and the multiplicative compactgroup of p-adic units for a fixed prime p. So G is solvable and hence amenable. We claim thatB(G) cannot have property KK∗. Indeed, the Fell group G is separable. Hence (AG) is normseparable (see [29]). So the proof of [51] shows that there is a bounded approximate identityin A(G) consisting of a sequence {φn}, φn positive definite with norm 1. The sequence φnconverges to 1 in B(G) in the weak∗-topology. Now if B(G) has property KK∗, then ‖φn−1‖ →0, and so 1 ∈ A(G). In particular G is compact. See [52] for a more general result.

(d) Theorem 4.10 is proved by Lau and Leinert in [48]. In a preprint of HernandezLinares and Japon [53] sent to the author just recently, they have shown that if G is compactand separable, then A(G) can be renormed to have the FPP. This generalizes an earlier resultof Lin [54] who proves that �1 can be renormed to have the FPP. Note that if G = T, thecircle group, then A(G) is isometric isomorphic to �1. We are grateful to Professor Japon forproviding us with a preprint of their work.

(e) Other related results for this section can also be found in [55].

Acknowledgment

This research is supported by NSERC Grant A-7679 and is dedicated to Professor William A.Kirk with admiration and respect.

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Research ArticleA Continuation Method for Weakly Kannan Maps

David Ariza-Ruiz and Antonio Jimenez-Melado

Departamento de Analisis Matematico, Facultad de Ciencias, Universidad de Malaga, 29071 Malaga, Spain

Correspondence should be addressed to Antonio Jimenez-Melado, [email protected]

Received 25 September 2009; Revised 4 December 2009; Accepted 6 December 2009


Copyright q 2010 D. Ariza-Ruiz and A. Jimenez-Melado. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

The first continuation method for contractive maps in the setting of a metric space was givenby Granas. Later, Frigon extended Granas theorem to the class of weakly contractive maps,and recently Agarwal and O’Regan have given the corresponding result for a certain type ofquasicontractions which includes maps of Kannan type. In this note we introduce the conceptof weakly Kannan maps and give a fixed point theorem, and then a continuation method, for thisclass of maps.

1. Introduction

Suppose that (X, d) is a metric space and that f : D ⊂ X → X is a map. We say that fis contractive if there exists α ∈ [0, 1) such that d(f(x), f(y)) ≤ αd(x, y) for all x, y ∈ D.The well-known Banach fixed point theorem states that f has a fixed point if D = X and(X, d) is complete. In 1962, Rakotch [1] obtained an extension of Banach theorem replacingthe constant α by a function of d(x, y), α = α(d(x, y)), provided that α is nonincreasingand 0 ≤ α(t) < 1 for all t > 0 (for a recent refinement of this result see [2]). A similargeneralization of the contractive condition was considered by Dugundji and Granas [3], whoextended Banach theorem to the class of weakly contractive mappings (i.e., α = α(x, y), withsup{α(x, y) : a ≤ d(x, y) ≤ b} < 1 for all 0 < a ≤ b).

Another focus of attention in Fixed Point Theory is to establish fixed point theoremsfor non-self mappings. In the setting of a Banach space, Gatica and Kirk [4] proved that iff : U → X is contractive, with U an open neighborhood of the origin, then f has a fixedpoint if it satisfies the well-known Leray-Schauder condition:

f(x)/=λx, for x ∈ ∂U, λ > 1. (L-S)

Recently, Kirk [5] has extended this result to the abstract setting of a certain class ofmetric spaces: the CAT(0) spaces. In the proof, the author uses a homotopy result due to


Granas [6], which is known as continuation method for contractive maps. In fact, the jumpfrom a Banach space setting to the metric space setting was given by Granas himself in [6](for more information on this topic see, for instance, [7–9]). After Granas, Frigon [8] gave asimilar result for weakly contractive maps.

A variant of the Banach contraction principle was given by Kannan [10], who provedthat a map f : X → X, where (X, d) is a complete metric space, has a unique fixed point if fis what we call a Kannan map, that is, there exists α ∈ [0, 1) such that, for all x, y ∈ X,

d(f(x), f

(y)) ≤ α

2[d(x, f(x)

)+ d(y, f(y))]

. (1.1)

In this note, following the pattern of Dugundji and Granas [3], we extend Kannantheorem to the class of weakly Kannan maps (i.e., α = α(x, y), with sup{α(x, y) : a ≤ d(x, y) ≤b} < 1 for all 0 < a ≤ b). This is done in Section 2. In Section 3 we use a local version of theprevious result to obtain a continuation method for weakly Kannan maps.

2. Weakly Kannan Maps

In this section we follow the pattern of Dugundji and Granas [3] to introduce the concept ofweakly Kannan maps.

Definition 2.1. Let (X, d) be a metric space, D ⊂ X, and f : D → X. Therefore f is a weaklyKannan map if there exists α : D × D → [0, 1], with θ(a, b) := sup{α(x, y) : a ≤ d(x, y) ≤b} < 1 for every 0 < a ≤ b such that, for all x, y ∈ D,

d(f(x), f

(y)) ≤ α

(x, y)

2[d(x, f(x)

)+ d(y, f(y))]

. (2.1)

Remark 2.2. Clearly, any weakly Kannan map f has at most one fixed point: if x = f(x) andy = f(y), then

d(x, y)= d(f(x), f

(y)) ≤ 1

2[d(x, f(x)

)+ d(y, f(y))]

= 0. (2.2)

Remark 2.3. Notice that if f : D ⊂ X → X is a weakly Kannan map and we define αf(x, y) onD ×D as

αf(x, y)=

⎧⎪⎨⎪⎩

2d(f(x), f

(y))

d(x, f(x)

)+ d(y, f(y)) if d

(x, f(x)

)+ d(y, f(y))/= 0,

0 otherwise,(2.3)

then αf is well defined, takes values in [0, 1], satisfies sup{αf(x, y) : a ≤ d(x, y) ≤ b} < 1for all 0 < a ≤ b (for αf is smaller than any α associated to f), and also satisfies (2.1), with αreplaced by αf , for all x, y ∈ D. Conversely, if αf is defined as in (2.3) and satisfies the aboveset of conditions, then f is a weakly Kannan map, establishing in this way an equivalentdefinition for Kannan maps.


Remark 2.4. Although Kannan showed that the concept of Kannan map is independent of theconcept of contractive map, Janos [11] observed that any contractive map f : D ⊂ X → Xwhose Lipschitz constant defined by

L(f)= sup

{d(f(x), f

(y))

d(x, y) : x, y ∈ X, x /=y

}(2.4)

is less than 1/3 is a Kannan map. Next, we exhibit an example of a weakly Kannan mapf , with L(f) = 1/3, which is not a Kannan map, thus showing that the constant 1/3 in theaforementioned result by Janos is sharp.

Example 2.5. Consider the metric space X = [0,∞) with the usual metric d(x, y) = |x−y|, andlet f : X → X be the function defined as f(x) = (1/3) log(1 + ex). Then, L(f) = 1/3 and f isa weakly Kannan map, but not a Kannan map.

The equality L(f) = 1/3 follows from the fact that |f ′(x)| < 1/3 for all x ∈ [0,∞)together with

limx→∞

d(f(x), f(0)

)d(x, 0)

=13. (2.5)

We also have that f is not a Kannan map because

limx→∞

2d(f(x), f(0)

)d(x, f(x)

)+ d(0, f(0)

) = 1. (2.6)

To check that f is a weakly Kannan map, consider the function α : X × X → [0,∞)given by (2.3). This function is well defined and also takes values in [0, 1] since L(f) = 1/3.Next, assume that 0 < a ≤ b and let us see that θ(a, b) = sup{α(x, y) : a ≤ d(x, y) ≤ b} < 1. Tosee this, observe that |u − f(u)| → ∞ as u → ∞, so there is M > 0 such that |u − f(u)| > b forall u > M. Observe also that fM, the restriction of f to [0,M], is a Kannan map with constantαM ∈ [0, 1), due to the fact that L(fM) < 1/3, for fM is continuously differentiable on [0,M]and |f ′(u)| < 1/3 for all u ∈ [0,M]. We will see θ(a, b) ≤ max{2/3, αM}. To do it, supposethat x, y ∈ [0,∞) with a ≤ |x − y| ≤ b and 0 ≤ x < y. Then, if y > M, use |y − f(y)| > b andthat L(f) ≤ 1/3 to obtain α(x, y) ≤ 2/3. Otherwise, we would have 0 ≤ x < y ≤M and thenα(x, y) ≤ αM.

Although the way we have introduced the concept of weakly Kannan map has beenby analogy with the work done by Dugundji and Granas in [3], we would like to mentionthat this extension may be done in some different ways. For instance, Pathak et al. [12,Theorem 3.1] have proved the following result.

Theorem A. Let (X, d) be a complete metric space and suppose that f : X → X is a map such that

d(f(x), f

(y)) ≤ α1

(d(x, f(x)

))d(x, f(x)

)+ α2(d(y, f(y)))

d(y, f(y)), (2.7)

for all x, y ∈ X, where αi : R → [0, 1). If, in addition, there exists a sequence {xn} in X withd(xn, f(xn)) → 0, then f has a fixed point in X.


Observe that relation (2.7) can be written in the following more general form:

d(f(x), f

(y)) ≤ A1(x)d

(x, f(x)

)+A2

(y)d(y, f(y)), (2.8)

for all x, y ∈ X, where Ai : X → [0, 1), i = 1, 2, and notice that any map satisfying (2.8) alsosatisfies the relation (2.1) with α(x, y) = 2 max{A1(x), A2(y)}. In fact, the arguments used bythe authors in the proof of Theorem A are also valid for this class of maps. Next, we statethis slightly more general result and include the proof for the sake of completeness. Then, weobtain, as a consequence, a fixed point theorem for weakly Kannan maps.

Theorem 2.6. Let (X, d) be a complete metric space and assume that A : X × X → [0,∞) is abounded function satisfying the following condition: for any sequence {xn} in X and u ∈ X,

xn −→ u =⇒ lim supA(xn, u) < 1. (∗)

Assume also that f : X → X is a map such that

d(f(x), f

(y)) ≤ A(x, y)[d(x, f(x)) + d(y, f(y))], (2.9)

for all x, y ∈ X. If there exists a sequence {xn} in X with d(xn, f(xn)) → 0, then f has a uniquefixed point u in X, and xn → u.

Proof. Since A is bounded, there exists M > 0 such that |A(x, y)| ≤ M for all x, y ∈ X.Suppose that {xn} is a sequence in X with d(xn, f(xn)) → 0 and use (2.9) to obtain that, forall n,m ∈ N,

d(f(xn), f(xm)

) ≤M[d(xn, f(xn)) + d(xm, f(xm))]. (2.10)

This implies that {f(xn)} is a Cauchy sequence. Since (X, d) is complete, the sequence{f(xn)} is convergent, say to u ∈ X. Then xn → u because d(xn, f(xn)) → 0. Thus, by (∗),lim supA(xn, u) < 1.

That u = f(u) is a consequence of the following relation and the fact thatlim supA(xn, u) < 1, then

d(u, f(u)

)= limd

(f(xn), f(u)

) ≤ lim supA(xn, u)[d(xn, f(xn)

)+ d(u, f(u)

)]= d(u, f(u)

)lim supA(xn, u).

(2.11)

Finally, u is the unique fixed point of f because if z = f(z):

d(u, z) = d(f(u), f(z)

) ≤ A(u, z)[d(u, f(u)

)+ d(z, f(z)

)]= 0. (2.12)

Corollary 2.7. Let (X, d) be a complete metric space and suppose that f : X → X is a weaklyKannan map. Then, f has a unique fixed point u ∈ X and, for any x0 ∈ X, the sequence of iterates{fn(x0)} converges to u.


Proof. Since f is a weakly Kannan map, there exists a function α : X × X → [0, 1] withθ(a, b) := sup{α(x, y) : a ≤ d(x, y) ≤ b} < 1 for all 0 < a ≤ b, satisfying (2.1) for all x, y ∈ X.Hence, the function A : X × X → [0, 1/2] given as A(x, y) = (1/2)α(x, y) is bounded andsatisfies the conditions (∗) and (2.9).

Consider any x0 ∈ X and define xn = f(xn−1), n = 1, 2, . . . . We may assume thatd(x0, x1) > 0 because otherwise we have finished. We will prove that d(xn, f(xn)) → 0 andhence, by Theorem 2.6, {xn}will converge to a point u which is the unique fixed point of f .

First of all, observe that the inequality

d(xn+1, xn) ≤ α(xn, xn−1)d(xn, xn−1) (2.13)

holds for all n ≥ 1. In fact, it is a consequence of the following one, which is true by (2.1):

d(xn+1, xn) ≤ α(xn, xn−1)2

[d(xn+1, xn) + d(xn, xn−1)]. (2.14)

From (2.13) we obtain that the sequence {d(xn, xn−1)} is nonincreasing, for 0 ≤α(xn, xn−1) ≤ 1, and then it is convergent to the real number

d = inf{d(xn, xn−1) : n = 1, 2, . . .}. (2.15)

To prove that d = 0, suppose that d > 0 and arrive to a contradiction as follows: use

0 < d ≤ d(xn, xn−1) ≤ d(x1, x0) (2.16)

and the definition of θ = θ(d, d(x1, x0)) to obtain α(xn, xn−1) ≤ θ for all n = 1, 2, . . . . This,together with (2.13), gives that

d ≤ d(xn+1, xn) ≤ θnd(x1, x0), (2.17)

for all n = 1, 2, . . ., which is impossible since d > 0 and 0 ≤ θ < 1.

Remark 2.8. We do not know whether Theorem A is, or not, a particular case of Theorem 2.6,although that is the case if the functions α1, α2 satisfy the additional assumption sup{α1(t) +α2(t) : t ≥ 0} < 2. To see this, suppose that the map f : X → X is in the conditions ofTheorem A, that is, f satisfies relation (2.7) for some given functions αi : R → [0, 1), i = 1, 2,and suppose also that the functions α1, α2 satisfy in addition sup{α1(t) + α2(t) : t ≥ 0} < 2.Define A : X ×X → [0,∞) as A(x, y) = max{a(x), a(y)}, where a : X → [0, 1) is given by

a(z) =12[α1(d(z, f(z)

))+ α2(d(z, f(z)

))]. (2.18)


Let us see that, with this function A, f satisfies the hypotheses of Theorem 2.6. Indeed, A isclearly bounded and also satisfies (∗); if {xn} is a sequence in X and u ∈ X, with xn → u,then

sup{a(xn) : n = 1, 2, . . .} ≤ sup{α1(t) + α2(t)

2: t ≥ 0

}< 1. (2.19)

Since we also have that a(u) < 1, we obtain that sup{A(xn, u) : n = 1, 2, . . .} < 1.Finally, to see that f satisfies relation (2.9), use relation (2.7) with x, y ∈ X, together

with the same relation interchanging the roles of x and y, and the fact that d(f(x), f(y)) =d(f(y), f(x)), to obtain that

d(f(x), f

(y)) ≤ a(x)d(x, f(x)) + a(y)d(y, f(y)), (2.20)

from which the result follows.

To prove the homotopy result of the next section, we will need the following localversion of Corollary 2.7.

Corollary 2.9. Assume that (X, d) is a complete metric space, x0 ∈ X, r > 0, and f : B(x0, r) → Xis a weakly Kannan map with associated function α satisfying (2.1). If θ is defined as usual, and

d(x0, f(x0)

)<

13

min{ r

2, r[1 − θ

( r2, r)]}

, (2.21)

then f has a fixed point.

Proof. In view of Corollary 2.7, it suffices to show that the closed ball B(x0, r) is invariantunder f . To prove it, consider any x ∈ B(x0, r) and obtain the relation

d(x0, f(x)

) ≤ d(x0, f(x0))+ d(f(x0), f(x)

)

≤ d(x0, f(x0))+α(x0, x)

2[d(x0, f(x0)

)+ d(x, f(x)

)]

≤ d(x0, f(x0))+α(x0, x)

2[d(x0, f(x0)

)+ d(x, x0) + d

(x0, f(x)

)],

(2.22)

from which, having in mind that α(x0, x) ≤ 1,

d(x0, f(x)

) ≤ 3d(x0, f(x0)

)+ α(x0, x)d(x0, x). (2.23)

To end the proof, obtain that d(x0, f(x)) ≤ r through the above inequality byconsidering two cases: if d(x0, x) ≤ r/2, then d(x0, f(x)) ≤ r because d(x0, f(x0)) ≤ r/6.Otherwise, we would have r/2 ≤ d(x0, x) ≤ r, and consequently α(x0, x) ≤ θ(r/2, r), from


which

d(x0, f(x)

) ≤ r[1 − θ( r2, r)]

+ rθ( r

2, r)= r. (2.24)

3. A Homotopy Result

In 1974 Ciric [13] introduced the concept of quasicontractions and proved the following fixedpoint theorem: suppose that (X, d) is a complete metric space and that f : X → X is aquasicontraction, that is, there exists q ∈ [0, 1) such that, for all x, y ∈ X,

d(f(x), f

(y)) ≤ qmax

{d(x, y), d(x, f(x)

), d(y, f(y)), d(x, f(y)), d(y, f(x)

)}. (3.1)

Then, f has a fixed point in X.Observe that any contractive map, as well as any Kannan map, is a quasicontraction;

thus, the theorem by Ciric generalizes the well known fixed point theorems by Banach andKannan.

On the other hand, Agarwal and O’Regan [14] considered a certain class of quasi-contractions: those maps f : X → X, where (X, d) is a metric space, for which there existsq ∈ (0, 1) such that, for all x, y ∈ X,

d(f(x), f

(y))

≤ qmax{d(x, y), d(x, f(x)

), d(y, f(y)),

12[d(x, f(y))

+ d(y, f(x)

)]},

(Q)

and gave the following homotopy result.

Theorem B. Let (X, d) be a complete metric space,U an open subset of X, andH : U × [0, 1] → Xsatisfying the following properties:

(i) H(x, λ)/=x for all x ∈ ∂U and all λ ∈ [0, 1],

(ii) there exists q ∈ (0, 1) such that for all x, y ∈ U and λ ∈ [0, 1] we have

d(H(x, λ),H

(y, λ))

≤ qmax{d(x, y), d(x,H(x, λ)), d

(y,H

(y, λ)),

12[d(x,H

(y, λ)), d(y,H(x, λ)

)]},

(3.2)

(iii) H(x, λ) is continuous in λ, uniformly for x ∈ U.

IfH(·, 0) has a fixed point inU, thenH(·, λ) also has a fixed point inU for all λ ∈ [0, 1].

The above homotopy result includes the corresponding one for the class of Kannanmaps, and in the following theorem we show that an analogous result is true for the widerclass of weakly Kannan maps.


Theorem 3.1. Let (X, d) be a complete metric space,U an open subset ofX, andH : U×[0, 1] → Xsatisfying the following properties:

(P1) H(x, λ)/=x for all x ∈ ∂U and all λ ∈ [0, 1],

(P2) there exists α : U ×U → [0, 1] such that for all x, y ∈ U and λ ∈ [0, 1] one has

d(H(x, λ),H

(y, λ)) ≤ α

(x, y)

2[d(x,H(x, λ)) + d

(y,H

(y, λ))]

, (3.3)

and θ(a, b) = sup{α(x, y) : a ≤ d(x, y) ≤ b} < 1 for all 0 < a ≤ b,

(P3) there exists a continuous function φ : [0, 1] → R such that, for every x ∈ U and t, s ∈[0, 1], d(H(x, t),H(x, s)) ≤ |φ(t) − φ(s)|.

IfH(·, 0) has a fixed point inU, thenH(·, λ) also has a fixed point inU for all λ ∈ [0, 1].

Proof. Consider the nonempty set

A = {λ ∈ [0, 1] : H(x, λ) = x for some x ∈ U}. (3.4)

We will prove that A = [0, 1], and for this it suffices to show that A is both closed and openin [0, 1].

We start showing that A is closed in [0, 1]: suppose that {λn} is a sequence in Aconverging to λ ∈ [0, 1] and let us show that λ ∈ A. By definition of A, there exists a sequence{xn} in U with xn = H(xn, λn). We will prove that {xn} converges to a point x0 ∈ U withH(x0, λ) = x0, thus showing that λ ∈ A.

That {xn} is a Cauchy sequence is a consequence of the following relation, where wehave used (P2), (P3), and the fact that xm = H(xm, λm):

d(xn, xm) = d(H(xn, λn),H(xm, λm))

≤ d(H(xn, λn),H(xn, λm)) + d(H(xn, λm),H(xm, λm))

≤ ∣∣φ(λn) − φ(λm)∣∣ + α(xn, xm)2

[d(xn,H(xn, λm)) + d(xm,H(xm, λm))]

=∣∣φ(λn) − φ(λm)∣∣ + α(xn, xm)

2d(H(xn, λn),H(xn, λm))

≤ 32∣∣φ(λn) − φ(λm)∣∣.

(3.5)


Write x0 = limxn and let us see that x0 ∈ U and also that x0 = H(x0, λ). That x0 = H(x0, λ) isa consequence of the following relation:

d(x0,H(x0, λ)) ≤ d(x0, xn) + d(xn,H(x0, λ))

≤ d(x0, xn) + d(H(xn, λn),H(xn, λ)) + d(H(xn, λ),H(x0, λ))

≤ d(x0, xn) +∣∣φ(λn) − φ(λ)∣∣ + 1

2[d(xn,H(xn, λ)) + d(x0,H(x0, λ))]

≤ d(x0, xn) +32∣∣φ(λn) − φ(λ)∣∣ + 1

2d(x0,H(x0, λ)),

(3.6)

and that x0 ∈ U is straightforward from (P1).Next we prove that A is open in [0, 1]: suppose that λ0 ∈ A and let us show that

(λ0−δ, λ0+δ)∩[0, 1] ⊂ A, for some δ > 0. Since λ0 ∈ A, there exists x0 ∈ U with x0 = H(x0, λ0).Consider r > 0 with B(x0, r) ⊂ U and use the continuity of φ to obtain δ > 0 such that

∣∣φ(λ) − φ(λ0)∣∣ < min

{ r2, r[1 − θ

( r2, r)]}

, (3.7)

for all λ ∈ (λ0 − δ, λ0 + δ) ∩ [0, 1].To show now that any λ ∈ (λ0 − δ, λ0 + δ) ∩ [0, 1] is also in A, it suffices to prove that

the map H(·, λ) : B(x0, r) → X has a fixed point. And this is true by Corollary 2.9, since

d(x0,H(x0, λ)) = d(H(x0, λ0),H(x0, λ))

≤ ∣∣φ(λ0) − φ(λ)∣∣

< min{ r

2, r[1 − θ

( r2, r)]}

.

(3.8)

Remark 3.2. A careful reading of the proof shows that hypothesis (P3) in Theorem 3.1 can beeasily replaced by the weaker hypothesis (iii) in Theorem B.

Remark 3.3. The counterpart to Theorem 3.1 for weakly contractive maps was proved byFrigon [8]. In that result, it was assumed, in place of our (3.3), an equivalent formulationof the following condition (H’):

d(H(x, λ),H

(y, λ)) ≤ α(x, y)d(x, y). (H’)

Observe that condition (H’) means that all the maps H(·, λ) : U → X, λ ∈ [0, 1] areweakly contractive, and with the same function α. Our condition (3.3) is no surprise then. Italso means that all the maps H(·, λ) are of weakly Kannan type, and with the same functionα.

We end the section with an example of a homotopy H satisfying (P1), (P2), and (P3)but not the hypotheses of Theorem B. In fact, the function f = H(·, 1) will be of weaklyKannan type, but will not satisfy the quasicontractivity condition (Q) (hence, it will not be


of Kannan type since any Kannan map satisfies (Q)). Moreover, f will not be of weaklycontractive type.

Example 3.4. Consider the metric space (X, d), where X = [−1, 1] and d(x, y) = |x − y|, and letf : [−1, 1] → [−1, 1] be the map given as

f(x) =

⎧⎨⎩− sin(x), −1 ≤ x < 1,

0, x = 1.(3.9)

First of all, we will see that the map f does not satisfy condition (Q). Define, for x, y ∈[−1, 1],

β(x, y)= max

{d(x, y), d(x, f(x)

), d(y, f(y)),

12[d(x, f(y))

+ d(y, f(x)

)]}. (3.10)

Then, for x ∈ (0, 1), we have that β(x,−x) = 2|x|, since | sin(x)| ≤ |x|. Hence,

limx→ 0

d(f(x), f(−x))β(x,−x) = lim

x→ 0

|sin(x)||x| = 1, (3.11)

showing that no q ∈ (0, 1) can be found to satisfy (Q).Secondly, observe that f is not weakly contractive, since any weakly contractive map

is continuous.Next, let us check that f is a weakly Kannan map. Since f has 0 as unique fixed point

then, the function α : [−1, 1]×[−1, 1] → [0,∞) given by α(x, y) = 2d(f(x), f(y))/(d(x, f(x))+d(y, f(y))) if (x, y)/= (0, 0), α(0, 0) = 0, is well defined. We have to check that α only takesvalues in [0, 1] and that θ(a, b) = sup{α(x, y) : a ≤ |x − y| ≤ b, x, y ∈ [−1, 1]} < 1 for all0 < a ≤ b. In fact, all this will follow if we just show that, for 0 < a ≤ 2,

θ(a, 2) ≤ max{

23, 1 − a

8

(1 − cos

(a4

))}. (3.12)

Thus, take 0 < a ≤ 2 and assume that x, y ∈ [−1, 1], with a ≤ |x − y|. If any of the points x, yequals 1, for example y = 1, then use |x + sin(x)| = |x| + | sin(x)| and | sin(x)| ≤ |x| to obtainthat

α(x, y)=

2|sin(x)||x + sin(x)| + 1

=2|sin(x)|

|x| + |sin(x)| + 1≤ 2

3. (3.13)

Otherwise, we would have that x, y ∈ [−1, 1). In this case, since |x − y| ≥ a, then we mayassume additionally that |x| ≥ a/2, and we claim that

α(x, y) ≤ 1 − a

8

(1 − cos

(a4

)). (3.14)


To be convinced of this, check the following chain of inequalities having in mind that|z + sin(z)| = |z| + | sin(z)| for all z ∈ [−1, 1], that | sin(z)| ≤ |z|, and also that | cos(x/2)| ≤cos(a/4):

α(x, y)=

2∣∣sin(x) − sin

(y)∣∣

|x + sin(x)| + ∣∣y + sin(y)∣∣

≤ 2|sin(x)| + 2∣∣sin(y)∣∣

|x| + |sin(x)| + ∣∣y∣∣ + ∣∣sin(y)∣∣

= 1 − |x| − |sin(x)| + ∣∣y∣∣ − ∣∣sin(y)∣∣

|x| + |sin(x)| + ∣∣y∣∣ + ∣∣sin(y)∣∣

≤ 1 − |x| − |sin(x)|4

≤ 1 − 14

(|x| − 2

∣∣∣sin(x

2

)cos(x

2

)∣∣∣)

≤ 1 − |x|4

(1 −∣∣∣cos(x

2

)∣∣∣)

≤ 1 − a8

(1 −∣∣∣cos(a

4

)∣∣∣).

(3.15)

Next, define H : [−1, 1] × [0, 1] → [−1, 1] by H(x, λ) = λf(x) and let us see that Hsatisfies (P1), (P2), and (P3).

It is obvious that H satisfies (P1). To check (P2), observe that

∣∣x − λf(x)∣∣ = |x| + ∣∣λf(x)∣∣, (3.16)

for all λ ∈ [0, 1] and all x ∈ [−1, 1], and hence, if α(x, y) is the function previously defined,we have that, for all λ ∈ [0, 1] and all x, y ∈ [−1, 1],

d(H(x, λ),H

(y, λ))

= λ∣∣f(x) − f(y)∣∣

≤ λα(x, y)

2[∣∣x − f(x)∣∣ + ∣∣y − f(y)∣∣]

= λα(x, y)

2[|x| + ∣∣f(x)∣∣ + ∣∣y∣∣ + ∣∣f(y)∣∣]

≤ α(x, y)

2[|x| + ∣∣λf(x)∣∣ + ∣∣y∣∣ + ∣∣λf(y)∣∣]

=α(x, y)

2[∣∣x − λf(x)∣∣ + ∣∣y − λf(y)∣∣]

=α(x, y)

2[d(x,H(x, λ)) + d

(y,H

(y, λ))]

.

(3.17)

Finally, (P3) is trivially satisfied with φ(t) = t.


Acknowledgments

This research was partially supported by the Spanish (Grant no. MTM2007-60854) andregional Andalusian (Grants no. FQM210 and no. FQM1504) Governments.

References

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[5] W. A. Kirk, “Fixed point theorems in CAT(0) spaces and R-trees,” Fixed Point Theory and Applications,vol. 2004, no. 4, pp. 309–316, 2004.

[6] A. Granas, “Continuation method for contractive maps,” Topological Methods in Nonlinear Analysis,vol. 3, no. 2, pp. 375–379, 1994.

[7] R. P. Agarwal, M. Meehan, and D. O’Regan, Fixed Point Theory and Applications, vol. 141 of CambridgeTracts in Mathematics, Cambridge University Press, Cambridge, UK, 2001.

[8] M. Frigon, “On continuation methods for contractive and nonexpansive mappings,” in RecentAdvances on Metric Fixed Point Theory, T. Dominguez Benavides, Ed., vol. 48 of Ciencias, pp. 19–30,University of Seville, Seville, Spain, 1996.

[9] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, vol. 3 of Series inMathematical Analysis and Applications, Gordon and Breach Science, Amsterdam, The Netherlands,2001.

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Research ArticleFixed Point Theorems for NonlinearOperators with and without Monotonicity inPartially Ordered Banach Spaces

Hui-Sheng Ding,1 Jin Liang,2 and Ti-Jun Xiao3

1 College of Mathematics and Information Science, Jiangxi Normal University, Nanchang,Jiangxi 330022, China

2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China3 Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences,Fudan University, Shanghai 200433, China

Correspondence should be addressed to Ti-Jun Xiao, [email protected]

Received 30 September 2009; Revised 5 December 2009; Accepted 6 December 2009


Copyright q 2010 Hui-Sheng Ding et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We establish two fixed point theorems for nonlinear operators on Banach spaces partially orderedby a cone. The first fixed point theorem is concerned with a class of mixed monotone operators.In the second fixed point theorem, the nonlinear operators are neither monotone nor mixedmonotone. We also provide an illustrative example for our second result.

1. Introduction

Fixed point theorems for nonlinear operators on partially ordered Banach spaces have manyapplications in nonlinear equations and many other subjects (cf., e.g., [1–7] and referencestherein); in particular, various kinds of fixed point theorems for mixed monotone operatorsare proved and applied (see, e.g., [1, 3, 5, 7] and references therein).

Stimulated by [7, 8], we investigate further, in this paper, the existence of fixed pointsof nonlinear operators with and without monotonicity in partially ordered Banach spaces.

In Section 2, a fixed point theorem for a class of mixed monotone operators isestablished. In Section 3, without any monotonicity assumption for a class of nonlinearoperators, we obtain a fixed point theorem by using Hilbert’s projection metric.

Let us recall some basic notations about cone (for more details, we refer the reader to[2]). Let X be a real Banach space. A closed convex set P in X is called a convex cone if the


following conditions are satisfied:

(i) if x ∈ P , then λx ∈ P for any λ ≥ 0,(ii) if x ∈ P and −x ∈ P , then x = 0.

A cone P induces a partial ordering ≤ in X by

x ≤ y iff y − x ∈ P. (1.1)

For any given u, v ∈ P ,

[u, v] := {x ∈ X | u ≤ x ≤ v}. (1.2)

A cone P is called normal if there exists a constant k > 0 such that

0 ≤ x ≤ y implies that ‖x‖ ≤ k∥∥y∥∥, (1.3)

where ‖ · ‖ is the norm on X.Throughout this paper, we denote by N the set of nonnegative integers, R the set of

real numbers, X a real Banach space, P a convex cone in X, e an element in P \ {θ} (θ is thezero element of X), and Pe the following set:

Pe ={x ∈ P : ∃α, β > 0 such that αe ≤ x ≤ βe}. (1.4)

2. Monotonic Operators

Theorem 2.1. Suppose that the operator A : Pe × Pe × Pe → Pe satisfies the following.

(S1) A(·, y, z) is increasing, A(x, ·, z) is decreasing, and A(x, y, ·) is decreasing.(S2) There exist a constant t0 ∈ [0, 1) and a function φ : (0, 1) × Pe × Pe → (0,+∞) such that

for each x, y, z ∈ Pe and t ∈ (t0, 1), φ(t, x, y) > t and

A(tx, t−1y, z

)≥ φ(t, x, y)A(x, y, z). (2.1)

(S3) There exist x0, y0 ∈ Pe such that x0 ≤ y0, x0 ≤ A(x0, y0, x0), A(y0, x0, y0) ≤ y0 and

infx,y∈[x0,y0]

φ(t, x, y

)> t, ∀t ∈ (t0, 1). (2.2)

(S4) There exists a constant L > 0 such that, for all x, y, z1, z2 ∈ Pe with z1 ≥ z2,

A(x, y, z1

) −A(x, y, z2) ≥ −L · (z1 − z2). (2.3)

Then A has a unique fixed point x∗ in [x0, y0], that is, A(x∗, x∗, x∗) = x∗.


Proof. The proof is divided into 4 steps.

Step 1. Let t1 ∈ (t0, 1) and

ψ(t, x, y

)=φ(t1, x, y

)t1

t, t ∈ (0, 1), x, y ∈ Pe. (2.4)

For each t ∈ (0, 1), there exists a nonnegative integer k such that tk+11 ≤ t < tk1 , that is, t1 ≤

t/tk1 < 1. Now, by (S2), we deduce, for all x, y, z ∈ Pe,

A(tx, t−1y, z

)= A

(t

tk1· tk1x,

tk1t· t−k1 y, z

)

≥ φ(

t

tk1, tk1x, t

−k1 y

)A(tk1x, t

−k1 y, z

)

≥ t

tk1A(tk1x, t

−k1 y, z

)

≥ t

t1A(t1x, t

−11 y, z

)

≥ t

t1φ(t1, x, y

)A(x, y, z

)

= ψ(t, x, y

)A(x, y, z

).

(2.5)

Moreover, by (S3), we get

infx,y∈[x0,y0]

ψ(t, x, y

)=

infx,y∈[x0,y0] φ(t1, x, y

)t1

· t > t, ∀t ∈ (0, 1). (2.6)

Hence, in the following proof, one can assume that t0 = 0 in (S2) and (S3) without loss.

Step 2. Fix x, y ∈ Pe. Then, there exists α ∈ (0, 1] such that x, y ∈ [αx0, α−1y0]. Let

Ψxy(z) =A(x, y, z

)+ Lz

1 + L, z ∈ Pe. (2.7)

Then Ψxy is an operator from Pe to Pe, and by (S4), Ψxy is increasing in Pe. Combining (S1)–(S3), we have

A(x, y, αx0

) ≥ A(αx0, α−1y0, x0

)≥ φ(α, x0, y0

)A(x0, y0, x0

) ≥ αx0, (2.8)


provided that α ∈ (0, 1). Moreover, it is easy to see that (2.8) holds when α = 1. Similarly, onecan show that

A(x, y, α−1y0

)≤ α−1y0. (2.9)

Then, it follows that

Ψxy(αx0) ≥ αx0, Ψxy

(α−1y0

)≤ α−1y0. (2.10)

Let

xnxy = Ψxy

(xn−1xy

), ynxy = Ψxy

(yn−1xy

), n = 1, 2, . . . ,

x0xy = αx0, y

0xy = α−1y0.

(2.11)

Then, using arguments similar to those in the proof of [7, Theorem 2.1], one can show thatΨxy has a unique fixed point x∗xy in [αx0, α

−1y0], and

xnxy −→ x∗xy, ynxy −→ x∗xy (n −→ ∞). (2.12)

We claim that x∗xy is the unique fixed point of Ψxy in Pe. In fact, let y∗xy be a fixed point of Ψxy

in Pe, and β ∈ (0, α) such that y∗xy ∈ [βx0, β−1y0]. By the above proof, Ψxy has a unique fixed

point in [βx0, β−1y0], which means that x∗xy = y∗xy. In addition, it follows from

x∗xy = Ψxy

(x∗xy)=A(x, y, x∗xy

)+ Lx∗xy

1 + L(2.13)

that x∗xy = A(x, y, x∗xy).

Step 3. By Step 2, we can define an operator Φ : Pe × Pe → Pe by

Φ(x, y)= x∗xy = Ψxy

(x∗xy)= A(x, y, x∗xy

). (2.14)

Let x, x′ ∈ [x0, y0] with x ≤ x′ and α ∈ (0, 1] with x, x′, y ∈ [αx0, α−1y0]. Denote by

{xnxy}, {xnx′y} the corresponding sequences in the proof of Step 2. Then

x1xy = Ψxy

(x0xy

)= Ψxy(αx0) =

A(x, y, αx0

)+ Lαx0

1 + L

≤ A(x′, y, αx0

)+ Lαx0

1 + L= Ψx′y(αx0) = x1

x′y.

(2.15)


Next, by induction and Ψxy being increasing, one can show that xnxy ≤ xnx′y for all n ∈ N. So

x∗xy = limn→∞

xnxy ≤ limn→∞

xnx′y = x∗x′y, (2.16)

that is, Φ(x, y) ≤ Φ(x′, y). Thus, Φ(·, y) is increasing. By a similar method, one can prove thatΦ(x, ·) is decreasing. On the other hand, by (S3), for x, y ∈ Pe and t ∈ (0, 1),

Φ(tx, t−1y

)= A(tx, t−1y,Φ

(tx, t−1y

))

≥ A(tx, t−1y,Φ

(x, y))

≥ φ(t, x, y)A(x, y,Φ(x, y))= φ(t, x, y

)Φ(x, y).

(2.17)

Let u0 = x0, v0 = y0, and

un = Φ(un−1, vn−1), vn = Φ(vn−1, un−1), for n = 1, 2, . . . . (2.18)

By choosing α = 1 in Step 1, we get x∗x0y0∈ [x0, y0]. Then

u1 = Φ(x0, y0

)= x∗x0y0

≥ x0 = u0, v1 = Φ(y0, x0

)= x∗y0x0

≤ y0 = v0. (2.19)

As Φ(·, y) is increasing and Φ(x, ·) is decreasing, it follows immediately that

u0 ≤ u1 ≤ · · · ≤ un ≤ · · · ≤ vn ≤ · · · ≤ v0. (2.20)

Next, by making some needed modifications in the proof of [3, Theorem 2.11], one can showthat Φ has a fixed point x∗ ∈ [x0, y0]. Suppose that y∗ ∈ [x0, y0] is a fixed point of Φ. It followsfrom the definition of un and vn that un ≤ y∗ ≤ vn for all n ∈ N. Then, by the normality of Φ,we get y∗ = x∗. So x∗ is the unique fixed point of Φ in [x0, y0].

Step 4. By Steps 2 and 3, we get

x∗ = Φ(x∗, x∗) = A(x∗, x∗,Φ(x∗, x∗)) = A(x∗, x∗, x∗). (2.21)

Let x ∈ [x0, y0] such that x = A(x, x, x). Then it follows from Step 2 that Φ(x, x) = x, that is,x is a fixed point of Φ in [x0, y0]. Thus, by Step 3, x = x∗, which means that x∗ is the uniquefixed point of A in [x0, y0].

Remark 2.2. Compared with [7, Remark 2.4], the nonlinear operator A in Theorem 2.1 is moregeneral, and so Theorem 2.1 may have a wider range of applications.


3. Nonmonotonic Case

First, let us recall some definitions and basic results about Hilbert’s projection metric (formore details, see [6]).

Definition 3.1. Elements x and y belonging to P (not both zero) are said to be linked if thereexist λ, μ > 0 such that

λx ≤ y ≤ μx. (3.1)

This defines an equivalence relation on P and divides P into disjoint subsets which we callconstituents of P .

Let x and y be linked. Define

M(x, y)= inf

{μ > 0 : y ≤ μx},

d(x, y)= ln[max

{M(x, y),M(y, x)}]

.(3.2)

Then, the following holds.

Theorem 3.2. d(·, ·) defines a complete metric on each constituent of P .

Proof. See [6].

We will also need the following result.

Theorem 3.3. [9] LetM be a complete metric space and suppose that f : M → M satisfies

d(f(x), f

(y)) ≤ Ψ

(d(x, y)), ∀x, y ∈M, (3.3)

where Ψ : [0,+∞) → [0,+∞) is upper semicontinuous from the right and satisfies Ψ(t) < t for allt > 0. Then f has a unique fixed point inM.

Theorem 3.3 is a generalization of the classical Banach’s contraction mappingprinciple. There are many generalizations of the classical Banach’s contraction mappingprinciple (see, e.g., [10, 11] and references therein), and these generalizations play animportant role in research work about fixed points of nonlinear operators in partially orderedBanach spaces; see, for example, [1] and the proof of the following theorem.

Now, we are ready to present our fixed point theorem, in which no monotonecondition is assumed on the nonlinear operator.

Theorem 3.4. Let T be an operator from Pe to Pe. Assume that there exist a constant ε ∈ (0, 1) and afunction φ : [ε, 1) → (0,+∞) such that φ(λ) > λ for all λ ∈ [ε, 1), and

Ty ≥ φ(λ)Tx, (3.4)

for all x, y ∈ Pe and λ ∈ [ε, 1) satisfying λx ≤ y ≤ λ−1x. Then T has a unique fixed point in Pe.


Proof. We divided the proof into 2 steps.

Step 1. Let λ ∈ (0, 1), x, y ∈ Pe, and λx ≤ y ≤ λ−1x. Then, there exists k ∈ N such that

ε ≤ λ

εk< 1. (3.5)

In view of

λ

εk·(εkx)= λx ≤ y ≤ λ−1x ≤ ε2kλ−1x =

(λ

εk

)−1

·(εkx), (3.6)

by the assumptions, we have

Ty ≥ φ(λ

εk

)· T(εkx)≥ λ

εk· T(εkx). (3.7)

Similar to the above proof, since ε · εk−1x = εkx ≤ ε−1 · εk−1x, one can deduce

Ty ≥ λ

εk· T(εkx)≥ λ

εk· φ(ε) · T

(εk−1x

)≥ λ

εk−1· T(εk−1x

). (3.8)

Continuing by this way, one can get

Ty ≥ λε· T(εx) ≥ φ(ε)

ελ · Tx. (3.9)

Let

ψ(λ) =φ(ε)ε

λ, λ ∈ (0, 1). (3.10)

Then ψ is continuous, ψ(λ) > λ for all λ ∈ (0, 1), and

Ty ≥ ψ(λ)Tx, (3.11)

for all x, y ∈ Pe and λ ∈ (0, 1) satisfying λx ≤ y ≤ λ−1x.

Step 2. Next, let x, y ∈ Pe with x /=y and

λ =1

max{M(x, y),M(y, x)} . (3.12)

Then λ ∈ (0, 1), λx ≤ y ≤ λ−1x, and d(x, y) = ln(λ−1). Moreover, by Step 1, we have

Ty ≥ ψ(λ)Tx. (3.13)


On the other hand, since λy ≤ x ≤ λ−1y, we also have

Tx ≥ ψ(λ)Ty. (3.14)

Thus, we get

ψ(λ)Tx ≤ Ty ≤ Tx

ψ(λ). (3.15)

Now, by the definition of d(·, ·), we have

d(Tx, Ty

) ≤ ln(

1ψ(λ)

). (3.16)

Let

Ψ(t) =

⎧⎨⎩− ln[ψ(e−t)], t ∈ (0,+∞),

0, t = 0.(3.17)

Then, Ψ is a continuous function from [0,+∞) to [0,+∞), and

d(Tx, Ty

) ≤ Ψ(d(x, y)). (3.18)

Moreover, since ψ(λ) > λ for all λ ∈ (0, 1), we get

Ψ(t) = ln1

ψ(e−t)< ln

1e−t

= t, t > 0. (3.19)

On the other hand, Pe is obviously a constituent of P , and thus (Pe, d) is complete byTheorem 3.2. Now, Theorem 3.3 yields that T has a unique fixed point in Pe.

Corollary 3.5. Assume that A : Pe × Pe → Pe is a mixed monotone operator, that is, A(·, y) isincreasing and A(x, ·) is decreasing. Moreover, there exist a constant ε ∈ (0, 1) and a function φ :[ε, 1) → (0,+∞) such that φ(λ) > λ for all λ ∈ [ε, 1), and

A(λx, λ−1y

)≥ φ(λ)A(x, y), (3.20)

for all x, y ∈ Pe and λ ∈ [ε, 1). Then A has a unique fixed point in Pe.

Proof. Let Tz = A(z, z), z ∈ Pe. Then, since A is a mixed monotone operator, we have

Ty = A(y, y) ≥ A(λx, λ−1x

)≥ φ(λ)A(x, x) = φ(λ)Tx, (3.21)


for all x, y ∈ Pe and λ ∈ [ε, 1) satisfying λx ≤ y ≤ λ−1x. Then, Theorem 3.4 yields theconclusion.

Remark 3.6. Corollary 3.5 is an improvement of [1, Corollary 3.2] in the sense that there φ islower semicontinuous on (0, 1), and the corresponding conditions need to hold on the wholeinterval (0, 1).

4. An Example

In this section, we give an example to illustrate Theorem 3.4. Let us consider the followingnonlinear delay integral equation:

x(t) =∫ tt−τf(s, x(s))ds, (4.1)

which is a classical model for the spread of some infectious disease (cf. [12]). In fact, (4.1) hasbeen of great interest for many authors (see, e.g., [3, 8] and references therein).

In the rest of this paper, let τ = 1 and

f(t, x) =

⎧⎪⎪⎨⎪⎪⎩

(1 + sin2t + sin2πt

)√x, t ∈ (−∞,+∞), 0 ≤ x ≤ 1,

1 + sin2t + sin2πt√x

, t ∈ (−∞,+∞), x ≥ 1.(4.2)

Next, let us investigate the existence of positive almost periodic solution to (4.1). For thereader’s convenience, we recall some definitions and basic results about almost periodicfunctions (for more details, see [13]).

Definition 4.1. A continuous function f : R → R is called almost periodic if for each ε > 0there exists l(ε) > 0 such that every interval I of length l(ε) contains a number τ with theproperty that

∥∥f(t + τ) − f(t)∥∥ < ε ∀ t ∈ R. (4.3)

Denote by AP(R) the set of all such functions.

Lemma 4.2. Assume that f , g ∈ AP(R). Then the following hold.

(a) The range Rf = {f(t) : t ∈ R} is precompact in R, and so f is bounded.

(b) F(f) ∈ AP(R) provided that F is continuous on Rf .(c) f + g, f · g ∈ AP(R). Moreover, f/g ∈ AP(R) provided that inft∈R|g(t)| > 0.

(d) Equipped with the sup norm

∥∥f∥∥ = supt∈R

∣∣f(t)∣∣, (4.4)

AP(R) turns out to be a Banach space.


Now, let P = {x ∈ AP(R) : x(t) ≥ 0, ∀t ∈ R}, and e ∈ P is defined by e(t) ≡ 1. It is notdifficult to verify that P is a normal cone in AP(R), and

Pe = {x ∈ AP(R) : ∃ ε > 0 such that x(t) > ε, ∀t ∈ R}. (4.5)

Define a nonlinear operator T on Pe by

(Tx)(t) =∫ tt−1f(s, x(s))ds, x ∈ Pe, t ∈ R. (4.6)

By Lemma 4.2 and [3, Corollary 3.3], it is not difficult to verify that T is an operator from Peto Pe. In addition, in view of (4.2), one can verify that

(Ty)(t) =

∫ tt−1f(s, y(s)

)ds ≥

√λ

∫ tt−1f(s, x(s))ds =

√λ(Tx)(t), t ∈ R, (4.7)

that is, Ty ≥√λTx for all x, y ∈ Pe and λ ∈ (0, 1) with λx ≤ y ≤ λ−1x. Then, by Theorem 3.4,

T has a unique fixed point in Pe, that is, (4.1) has a unique almost periodic solution withpositive infimum.

Acknowledgments

The authors are very grateful to the referees for valuable suggestions and comments. Inaddition, Hui-Sheng Ding acknowledges support from the NSF of China (10826066), the NSFof Jiangxi Province of China (2008GQS0057), and the Youth Foundation of Jiangxi ProvincialEducation Department(GJJ09456); Jin Liang and Ti-Jun Xiao acknowledge support from theNSF of China (10771202), the Research Fund for Shanghai Key Laboratory for ContemporaryApplied Mathematics (08DZ2271900), and the Specialized Research Fund for the DoctoralProgram of Higher Education of China (2007035805).

References

[1] Y. Z. Chen, “Thompson’s metric and mixed monotone operators,” Journal of Mathematical Analysis andApplications, vol. 177, no. 1, pp. 31–37, 1993.

[2] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.[3] H.-S. Ding, T.-J. Xiao, and J. Liang, “Existence of positive almost automorphic solutions to nonlinear

delay integral equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2216–2231, 2009.

[4] J. A. Gatica and W. A. Kirk, “A fixed point theorem for k-set-contractions defined in a cone,” PacificJournal of Mathematics, vol. 53, pp. 131–136, 1974.

[5] K. Li, J. Liang, and T.-J. Xiao, “New existence and uniqueness theorems of positive fixed points formixed monotone operators with perturbation,” Journal of Mathematical Analysis and Applications, vol.328, no. 2, pp. 753–766, 2007.

[6] A. C. Thompson, “On certain contraction mappings in a partially ordered vector space,” Proceedingsof the American Mathematical Society, vol. 14, pp. 438–443, 1963.

[7] Z. Zhang and K. Wang, “On fixed point theorems of mixed monotone operators and applications,”Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3279–3284, 2009.


[8] E. Ait Dads, P. Cieutat, and L. Lhachimi, “Positive pseudo almost periodic solutions for somenonlinear infinite delay integral equations,” Mathematical and Computer Modelling, vol. 49, no. 3-4,pp. 721–739, 2009.

[9] D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American MathematicalSociety, vol. 20, pp. 458–464, 1969.



[12] K. L. Cooke and J. L. Kaplan, “A periodicity threshold theorem for epidemics and populationgrowth,” Mathematical Biosciences, vol. 31, no. 1-2, pp. 87–104, 1976.

[13] A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics, Springer,Berlin, Germany, 1974.


Research ArticleWeak and Strong Convergence of an ImplicitIteration Process for an AsymptoticallyQuasi-I-Nonexpansive Mapping in Banach Space

Farrukh Mukhamedov and Mansoor Saburov

Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic UniversityMalaysia, P.O. Box 141, 25710 Kuantan, Malaysia

Correspondence should be addressed to Farrukh Mukhamedov, [email protected]

Received 31 August 2009; Accepted 6 December 2009


Copyright q 2010 F. Mukhamedov and M. Saburov. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We prove the weak and strong convergence of the implicit iterative process to a commonfixed point of an asymptotically quasi-I-nonexpansive mapping T and an asymptotically quasi-nonexpansive mapping I, defined on a nonempty closed convex subset of a Banach space.

1. Introduction

LetK be a nonempty subset of a real normed linear spaceX and let T : K → K be a mapping.Denote by F(T) the set of fixed points of T , that is, F(T) = {x ∈ K : Tx = x}. Throughout thispaper, we always assume that F(T)/= ∅. Now let us recall some known definitions.

Definition 1.1. A mapping T : K → K is said to be

(i) nonexpansive, if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ K;

(ii) asymptotically nonexpansive, if there exists a sequence {λn} ⊂ [1,∞) withlimn→∞λn = 1 such that ‖Tnx − Tny‖ ≤ λn‖x − y‖ for all x, y ∈ K and n ∈ N;

(iii) quasi-nonexpansive, if ‖Tx − p‖ ≤ ‖x − p‖ for all x ∈ K, p ∈ F(T);

(iv) asymptotically quasi-nonexpansive, if there exists a sequence {μn} ⊂ [1,∞) withlimn→∞μn = 1 such that ‖Tnx − p‖ ≤ μn‖x − p‖ for all x ∈ K, p ∈ F(T) and n ∈ N.


Note that from the above definitions, it follows that a nonexpansive mapping mustbe asymptotically nonexpansive, and an asymptotically nonexpansive mapping must beasymptotically quasi-nonexpansive, but the converse does not hold (see [1]).

If K is a closed nonempty subset of a Banach space and T : K → K is nonexpansive,then it is known that T may not have a fixed point (unlike the case if T is a strict contraction),and even when it has, the sequence {xn} defined by xn+1 = Txn (the so-called Picard sequence)may fail to converge to such a fixed point.

In [2, 3] Browder studied the iterative construction for fixed points of nonexpansivemappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years orso, the studies of the iterative processes for the approximation of fixed points of nonexpansivemappings and fixed points of some of their generalizations have been flourishing areas ofresearch for many mathematicians (see for more details [1, 4]).

In [5] Diaz and Metcalf studied quasi-nonexpansive mappings in Banach spaces.Ghosh and Debnath [6] established a necessary and sufficient condition for convergence ofthe Ishikawa iterates of a quasi-nonexpansive mapping on a closed convex subset of a Banachspace. The iterative approximation problems for nonexpansive mapping, asymptoticallynonexpansive mapping and asymptotically quasi-nonexpansive mapping were studiedextensively by Goebel and Kirk [7], Liu [8], Wittmann [9], Reich [10], Gornicki [11], Schu[12] Shioji and Takahashi [13], and Tan and Xu [14] in the settings of Hilbert spaces anduniformly convex Banach spaces.

There are many methods for approximating fixed points of a nonexpansive mapping.Xu and Ori [15] introduced implicit iteration process to approximate a common fixed point ofa finite family of nonexpansive mappings in a Hilbert space. Recently, Sun [16] has extendedan implicit iteration process for a finite family of nonexpansive mappings, due to Xu and Ori,to the case of asymptotically quasi-nonexpansive mappings in a setting of Banach spaces.In [17] it has been studied the weak and strong convergence of implicit iteration processwith errors to a common fixed point for a finite family of nonexpansive mappings in Banachspaces, which extends and improves the mentioned papers (see also [18, 19] for applicationsand other methods of implicit iteration processes).

There are many concepts which generalize a notion of nonexpansive mapping. One ofsuch concepts is I-nonexpansivity of a mapping T ([20]). Let us recall some notions.

Definition 1.2. Let T : K → K, I : K → K be two mappings of a nonempty subset K of a realnormed linear space X. Then T is said to be

(i) I-nonexpansive, if ‖Tx − Ty‖ ≤ ‖Ix − Iy‖ for all x, y ∈ K;

(ii) asymptotically I-nonexpansive, if there exists a sequence {λn} ⊂ [1,∞) withlimn→∞λn = 1 such that ‖Tnx − Tny‖ ≤ λn‖Inx − Iny‖ for all x, y ∈ K and n ≥ 1;

(iii) asymptotically quasi I-nonexpansive mapping, if there exists a sequence {μn} ⊂[1,∞) with limn→∞μn = 1 such that ‖Tnx − p‖ ≤ μn‖Inx − p‖ for all x ∈ K, p ∈F(T) ∩ F(I) and n ≥ 1.

Remark 1.3. If F(T) ∩ F(I)/= ∅ then an asymptotically I-nonexpansive mapping is asymptot-ically quasi-I-nonexpansive. But, there exists a nonlinear continuous asymptotically quasiI-nonexpansive mappings which is asymptotically I-nonexpansive.

In [21] a weakly convergence theorem for I-asymptotically quasi-nonexpansivemapping defined in Hilbert space was proved. In [22] strong convergence of Manniterations of I-nonexpansive mapping has been proved. Best approximation properties of


I-nonexpansive mappings were investigated in [20]. In [23] the weak convergence of three-step Noor iterative scheme for an I-nonexpansive mapping in a Banach space has beenestablished. Recently, in [24] the weak and strong convergence of implicit iteration processto a common fixed point of a finite family of I-asymptotically nonexpansive mappings werestudied. Assume that the family consists of one I-asymptotically nonexpansive mapping T .Now let us consider an iteration method used in [24], for T , which is defined by

x1 ∈ K,xn+1 = (1 − αn)xn + αnInyn,yn =

(1 − βn

)xn + βnTnxn.

n ≥ 1, (1.1)

where {αn} and {βn} are two sequences in [0, 1]. From this formula one can easily see that theemployed method, indeed, is not implicit iterative processes. The used process is some kindof modified Ishikawa iteration.

Therefore, in this paper we will extend of the implicit iterative process, defined in [16],to I-asymptotically quasi-nonexpansive mapping defined on a uniformly convex Banachspace. Namely, let K be a nonempty convex subset of a real Banach space X and T : K → Kbe an asymptotically quasi I-nonexpansive mapping, and let I : K → K be an asymptoticallyquasi-nonexpansive mapping. Then for given two sequences {αn} and {βn} in [0, 1] we willconsider the following iteration scheme:

x0 ∈ K,xn = (1 − αn)xn−1 + αnTnyn,

yn =(1 − βn

)xn + βnInxn.

n ≥ 1, (1.2)

In this paper we will prove the weak and strong convergences of the implicit iterativeprocess (1.2) to a common fixed point of T and I. All results presented here generalize andextend the corresponding main results of [15–17] in a case of one mapping.

2. Preliminaries

Throughout this paper, we always assume that X is a real Banach space. We denote by F(T)and D(T) the set of fixed points and the domain of a mapping T, respectively. Recall thata Banach space X is said to satisfy Opial condition [25], if for each sequence {xn} in X, xnconverging weakly to x implies that

lim infn→∞

‖xn − x‖ < lim infn→∞

∥∥xn − y∥∥. (2.1)

for all y ∈ X with y /=x. It is well known that (see [26]) inequality (2.1) is equivalent to

lim supn→∞

‖xn − x‖ < lim supn→∞

∥∥xn − y∥∥. (2.2)


Definition 2.1. Let K be a closed subset of a real Banach space X and let T : K → K be amapping.

(i) A mapping T is said to be semiclosed (demiclosed) at zero, if for each boundedsequence {xn} in K, the conditions xn converges weakly to x ∈ K and Txnconverges strongly to 0 imply Tx = 0.

(ii) A mapping T is said to be semicompact, if for any bounded sequence {xn} in Ksuch that ‖xn − Txn‖ → 0, n → ∞, then there exists a subsequence {xnk} ⊂ {xn}such that xnk → x∗ ∈ K strongly.

(iii) T is called a uniformly L-Lipschitzian mapping, if there exists a constant L > 0 suchthat ‖Tnx − Tny‖ ≤ L‖x − y‖ for all x, y ∈ K and n ≥ 1.

The following lemmas play an important role in proving our main results.

Lemma 2.2 (see [12]). Let X be a uniformly convex Banach space and let b, c be two constants with0 < b < c < 1. Suppose that {tn} is a sequence in [b, c] and {xn} and {yn} are two sequences in Xsuch that

limn→∞

∥∥tnxn + (1 − tn)yn∥∥ = d, lim sup

n→∞‖xn‖ ≤ d, lim sup

n→∞

∥∥yn∥∥ ≤ d, (2.3)

holds some d ≤ 0. Then limn→∞‖xn − yn‖ = 0.

Lemma 2.3 (see [14]). Let {an} and {bn} be two sequences of nonnegative real numbers with∑∞n=1 bn <∞. If one of the following conditions is satisfied:

(i) an+1 ≤ an + bn, n ≥ 1,

(ii) an+1 ≤ (1 + bn)an, n ≥ 1,

then the limit limn→∞an exists.

3. Main Results

In this section we will prove our main results. To formulate one, we need some auxiliaryresults.

Lemma 3.1. Let X be a real Banach space and let K be a nonempty closed convex subset of X. LetT : K → K be an asymptotically quasi I-nonexpansive mapping with a sequence {λn} ⊂ [1,∞) andI : K → K be an asymptotically quasi-nonexpansive mapping with a sequence {μn} ⊂ [1,∞) suchthat F = F(T) ∩ F(I)/= ∅. Suppose A∗ = supnαn, Λ = supnλn ≥ 1, M = supnμn ≥ 1 and {αn} and{βn} are two sequences in [0, 1] which satisfy the following conditions:

(i)∑∞

n=1(λnμn − 1)αn <∞,(ii) A∗ < 1/Λ2M2.

If {xn} is the implicit iterative sequence defined by (1.2), then for each p ∈ F = F(T) ∩ F(I) the limitlimn→∞‖xn − p‖ exists.


Proof. Since F = F(T) ∩ F(I)/= ∅, for any given p ∈ F, it follows from (1.2) that

∥∥xn − p∥∥ =∥∥(1 − αn)(xn−1 − p

)+ αn

(Tnyn − p

)∥∥≤ (1 − αn)

∥∥xn−1 − p∥∥ + αn

∥∥Tnyn − p∥∥≤ (1 − αn)

∥∥xn−1 − p∥∥ + αnλn

∥∥Inyn − p∥∥≤ (1 − αn)

∥∥xn−1 − p∥∥ + αnλnμn

∥∥yn − p∥∥.

(3.1)

Again from (1.2) we derive that

∥∥yn − p∥∥ =∥∥(1 − βn

)(xn − p

)+ βn

(Inxn − p

)∥∥≤ (

1 − βn)∥∥xn − p∥∥ + βnμn

∥∥xn − p∥∥≤ (

1 − βn)μn

∥∥xn − p∥∥ + βnμn∥∥Inxn − p∥∥

≤ μn∥∥xn − p∥∥,

(3.2)

which means

∥∥yn − p∥∥ ≤ μn∥∥xn − p∥∥ ≤ λnμn∥∥xn − p∥∥. (3.3)

Then from (3.3) one finds

∥∥xn − p∥∥ ≤ (1 − αn)∥∥xn−1 − p

∥∥ + αnλ2nμ

2n

∥∥xn − p∥∥, (3.4)

and so

(1 − αnλ2

nμ2n

)∥∥xn − p∥∥ ≤ (1 − αn)∥∥xn−1 − p

∥∥. (3.5)

By condition (ii) we have αnλ2nμ

2n ≤ A∗Λ2M2 < 1, and therefore

1 − αnλ2nμ

2n ≥ 1 −A∗Λ2M2 > 0. (3.6)

Hence from (3.5) we obtain

∥∥xn − p∥∥ ≤ 1 − αn1 − αnλ2

nμ2n

∥∥xn−1 − p∥∥

=

(1 +

(λ2nμ

2n − 1

)αn

1 − αnλ2nμ

2n

)∥∥xn−1 − p∥∥

≤(

1 +

(λ2nμ

2n − 1

)αn

1 −A∗Λ2M2

)∥∥xn−1 − p∥∥.

(3.7)


By putting bn = (λ2nμ

2n − 1)αn/(1 −A∗Λ2M2) the last inequality can be rewritten as follows:

∥∥xn − p∥∥ ≤ (1 + bn)∥∥xn−1 − p

∥∥. (3.8)

From condition (i) we find

∞∑n=1

bn =1

1 −A∗Λ2M2

∞∑n=1

(λ2nμ

2n − 1

)αn

=1

1 −A∗Λ2M2

∞∑n=1

(λnμn − 1

)(λnμn + 1

)αn

≤ ΛM + 11 −A∗Λ2M2

∞∑n=1

(λnμn − 1

)αn <∞.

(3.9)

Denoting an = ‖xn−1 − p‖ in (3.8) one gets

an+1 ≤ (1 + bn)an, (3.10)

and Lemma 2.3 implies the existence of the limit limn→∞an. This means the limit

limn→∞

∥∥xn − p∥∥ = d (3.11)

exists, where d ≥ 0 is a constant. This completes the proof.

Now we prove the following result.

Theorem 3.2. Let X be a real Banach space and let K be a nonempty closed convex subset of X. LetT : K → K be a uniformly L1-Lipschitzian asymptotically quasi-I-nonexpansive mapping with asequence {λn} ⊂ [1,∞) and let I : K → K be a uniformly L2-Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence {μn} ⊂ [1,∞) such that F = F(T) ∩ F(I)/= ∅. SupposeA∗ = supnαn, Λ = supnλn ≥ 1, M = supnμn ≥ 1, and {αn} and {βn} are two sequences in [0, 1]which satisfy the following conditions:

(i)∑∞

n=1(λnμn − 1)αn <∞,(ii) A∗ < 1/Λ2M2.

Then the implicitly iterative sequence {xn} defined by (1.2) converges strongly to a common fixedpoint in F = F(T) ∩ F(I)/= ∅ if and only if

lim infn→∞

d(xn, F) = 0. (3.12)

Proof. The necessity of condition (3.12) is obvious. Let us proof the sufficiency part oftheorem.

Since T, I : K → K are uniformly L-Lipschitzian mappings, so T and I are continuousmappings. Therefore the sets F(T) and F(I) are closed. Hence F = F(T) ∩ F(I) is a nonemptyclosed set.


For any given p ∈ F, we have (see (3.8))

∥∥xn − p∥∥ ≤ (1 + bn)∥∥xn−1 − p

∥∥, (3.13)

here as before bn = (λ2nμ

2n − 1)αn/(1 −A∗Λ2M2) with

∑∞n=1 bn <∞. Hence, one finds

d(xn, F) ≤ (1 + bn)d(xn−1, F). (3.14)

From (3.14) due to Lemma 2.3 we obtain the existence of the limit limn→∞d(xn, F). Bycondition (3.12), one gets

limn→∞

d(xn, F) = lim infn→∞

d(xn, F) = 0. (3.15)

Let us prove that the sequence {xn} converges to a common fixed point of T and I. Infact, due to 1 + t ≤ exp(t) for all t > 0, and from (3.13), we obtain

∥∥xn − p∥∥ ≤ exp(bn)∥∥xn−1 − p

∥∥. (3.16)

Hence, for any positive integers m,n, from (3.16) with∑∞

n=1 bn <∞ we find

∥∥xn+m − p∥∥ ≤ exp(bn+m)∥∥xn+m−1 − p

∥∥≤ exp(bn+m + bn+m−1)

∥∥xn+m−2 − p∥∥

≤ · · ·

≤ exp

(n+m∑i=n+1

bi

)∥∥xn − p∥∥

≤ exp

( ∞∑i=1

bi

)∥∥xn − p∥∥,

(3.17)

which means that

∥∥xn+m − p∥∥ ≤W∥∥xn − p∥∥ (3.18)

for all p ∈ F, where W = exp(∑∞

i=1 bi) <∞.Since limn→∞d(xn, F) = 0, then for any given ε > 0, there exists a positive integer

number n0 such that

d(xn0 , F) <ε

W. (3.19)

Therefore there exists p1 ∈ F such that

∥∥xn0 − p1∥∥ < ε

W. (3.20)


Consequently, for all n ≥ n0 from (3.18) we derive

∥∥xn − p1∥∥ ≤W∥∥xn0 − p1

∥∥< W · ε

W

= ε,

(3.21)

which means that the strong convergence of the sequence {xn} is a common fixed point p1 ofT and I. This proves the required assertion.

We need one more auxiliary result.

Proposition 3.3. Let X be a real uniformly convex Banach space and let K be a nonempty closedconvex subset of X. Let T : K → K be a uniformly L1-Lipschitzian asymptotically quasi-I-nonexpansive mapping with a sequence {λn} ⊂ [1,∞) and let I : K → K be a uniformly L2-Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence {μn} ⊂ [1,∞) such thatF = F(T) ∩ F(I)/= ∅. Suppose A∗ = infnαn, A∗ = supnαn, Λ = supnλn ≥ 1, M = supnμn ≥ 1 and{αn} and {βn} are two sequences in [0, 1] which satisfy the following conditions:

(i)∑∞

n=1(λnμn − 1)αn <∞,(ii) 0 < A∗ ≤ A∗ < 1/Λ2M2,

(iii) 0 < B∗ = infnβn ≤ supnβn = B∗ < 1.

Then the implicitly iterative sequence {xn} defined by (1.2) satisfies the following:

limn→∞

‖xn − Txn‖ = 0, limn→∞

‖xn − Ixn‖ = 0. (3.22)

Proof. First, we will prove that

limn→∞

‖xn − Tnxn‖ = 0, limn→∞

‖xn − Inxn‖ = 0. (3.23)

According to Lemma 3.1 for any p ∈ F = F(T) ∩ F(I) we have limn→∞‖xn − p‖ = d. Itfollows from (1.2) that

∥∥xn − p∥∥ =∥∥(1 − αn)(xn−1 − p

)+ αn

(Tnyn − p

)∥∥ −→ d, n −→ ∞. (3.24)

By means of asymptotically quasi-I-nonexpansivity of T and asymptotically quasi-nonexpansivity of I from (3.3) we get

lim supn→∞

∥∥Tnyn − p∥∥ ≤ lim supn→∞

λnμn∥∥yn − p∥∥ ≤ lim sup

n→∞λ2nμ

2n

∥∥xn − p∥∥ = d. (3.25)

Now using

lim supn→∞

∥∥xn−1 − p∥∥ = d (3.26)


with (3.25) and applying Lemma 2.2 to (3.24) one finds

limn→∞

∥∥xn−1 − Tnyn∥∥ = 0. (3.27)

Now from (1.2) and (3.27) we infer that

limn→∞

‖xn − xn−1‖ = limn→∞

∥∥αn(Tnyn − xn−1)∥∥ = 0. (3.28)

On the other hand, we have

∥∥xn−1 − p∥∥ ≤ ∥∥xn−1 − Tnyn

∥∥ +∥∥Tnyn − p∥∥

≤ ∥∥xn−1 − Tnyn∥∥ + λnμn

∥∥yn − p∥∥,(3.29)

which implies

∥∥xn−1 − p∥∥ − ∥∥xn−1 − Tnyn

∥∥ ≤ λnμn∥∥yn − p∥∥. (3.30)

The last inequality with (3.3) yields that

∥∥xn−1 − p∥∥ − ∥∥xn−1 − Tnyn

∥∥ ≤ λnμn∥∥yn − p∥∥ ≤ λ2nμ

2∥∥xn − p∥∥. (3.31)

Then (3.27) and (3.24) with the Squeeze theorem imply that

limn→∞

∥∥yn − p∥∥ = d. (3.32)

Again from (1.2) we can see that

∥∥yn − p∥∥ =∥∥(1 − βn

)(xn − p

)+ βn

(Inxn − p

)∥∥ −→ d, n −→ ∞. (3.33)

From (3.11) one finds

lim supn→∞

∥∥Inxn − p∥∥ ≤ lim supn→∞

μn∥∥xn − p∥∥ = d. (3.34)

Now applying Lemma 2.2 to (3.33) we obtain

limn→∞

‖xn − Inxn‖ = 0. (3.35)


Consider

‖xn − Tnxn‖ ≤ ‖xn − xn−1‖ +∥∥xn−1 − Tnyn

∥∥ +∥∥Tnyn − Tnxn∥∥

≤ ‖xn − xn−1‖ +∥∥xn−1 − Tnyn

∥∥ + L1∥∥yn − xn∥∥

= ‖xn − xn−1‖ +∥∥xn−1 − Tnyn

∥∥ + L1∥∥βn(Inxn − xn)∥∥

= ‖xn − xn−1‖ +∥∥xn−1 − Tnyn

∥∥ + L1βn‖Inxn − xn‖.

(3.36)

Then from (3.27), (3.28), and (3.35) we get

limn→∞

‖xn − Tnxn‖ = 0. (3.37)

Finally, from

‖xn − Txn‖ ≤ ‖xn − Tnxn‖ + ‖Tnxn − Txn‖

≤ ‖xn − Tnxn‖ + L1

∥∥∥Tn−1xn − xn∥∥∥

≤ ‖xn − Tnxn‖ + L1

(∥∥∥Tn−1xn − Tn−1xn−1

∥∥∥+∥∥∥Tn−1xn−1 − xn−1

∥∥∥ + ‖xn−1 − xn‖)

≤ ‖xn − Tnxn‖ + L1

(L1‖xn − xn−1‖

+∥∥∥Tn−1xn−1 − xn−1

∥∥∥ + ‖xn−1 − xn‖)

≤ ‖xn − Tnxn‖ + L1(L1 + 1)‖xn − xn−1‖ + L1

∥∥∥Tn−1xn−1 − xn−1

∥∥∥

(3.38)

with (3.28) and (3.37) we obtain

limn→∞

‖xn − Txn‖ = 0. (3.39)

Analogously, one has

‖xn − Ixn‖ ≤ ‖xn − Inxn‖ + L2(L2 + 1)‖xn − xn−1‖ + L2

∥∥∥In−1xn−1 − xn−1

∥∥∥, (3.40)

which with (3.28) and (3.35) implies

limn→∞

‖xn − Ixn‖ = 0. (3.41)

Now we are ready to formulate one of main results concerning weak convergence ofthe sequence {xn}.


Theorem 3.4. Let X be a real uniformly convex Banach space satisfying Opial condition and letK be a nonempty closed convex subset of X. Let E : X → X be an identity mapping, letT : K → K be a uniformly L1-Lipschitzian asymptotically quasi-I-nonexpansive mapping with asequence {λn} ⊂ [1,∞), and, I : K → K be a uniformly L2-Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence {μn} ⊂ [1,∞) such that F = F(T) ∩ F(I)/= ∅. SupposeA∗ = infnαn, A∗ = supnαn, Λ = supnλn ≥ 1, M = supnμn ≥ 1, and {αn} and {βn} are twosequences in [0, 1] satisfying the following conditions:

(i)∑∞

n=1(λnμn − 1)αn <∞,(ii) 0 < A∗ ≤ A∗ < 1/Λ2M2.

(iii) 0 < B∗ = infnβn ≤ supnβn = B∗ < 1.

If the mappings E − T and E − I are semiclosed at zero, then the implicitly iterative sequence {xn}defined by (1.2) converges weakly to a common fixed point of T and I.

Proof. Let p ∈ F, then according to Lemma 3.1 the sequence {‖xn − p‖} converges. Thisprovides that {xn} is a bounded sequence. Since X is uniformly convex, then every boundedsubset of X is weakly compact. Since {xn} is a bounded sequence in K, then there exists asubsequence {xnk} ⊂ {xn} such that {xnk} converges weakly to q ∈ K. Hence from (3.39) and(3.41) it follows that

limnk→∞

‖xnk − Txnk‖ = 0, limnk→∞

‖xnk − Ixnk‖ = 0. (3.42)

Since the mappings E − T and E − I are semiclosed at zero, therefore, we find Tq = q andIq = q, which means q ∈ F = F(T) ∩ F(I).

Finally, let us prove that {xn} converges weakly to q. In fact, suppose the contrary, thatis, there exists some subsequence {xnj} ⊂ {xn} such that {xnj} converges weakly to q1 ∈ K andq1 /= q. Then by the same method as given above, we can also prove that q1 ∈ F = F(T) ∩F(I).

Taking p = q and p = q1 and using the same argument given in the proof of (3.11), wecan prove that the limits limn→∞‖xn − q‖ and limn→∞‖xn − q1‖ exist, and we have

limn→∞

∥∥xn − q∥∥ = d, limn→∞

∥∥xn − q1∥∥ = d1, (3.43)

where d and d1 are two nonnegative numbers. By virtue of the Opial condition of X, onefinds

d = lim supnk→∞

∥∥xnk − q∥∥ < lim supnk→∞

∥∥xnk − q1∥∥ = d1

= lim supnj →∞

∥∥∥xnj − q1

∥∥∥ < lim supnj →∞

∥∥∥xnj − q∥∥∥ = d.

(3.44)

This is a contradiction. Hence q1 = q. This implies that {xn} converges weakly to q. Thiscompletes the proof of Theorem 3.4.

Now we formulate next result concerning strong convergence of the sequence {xn}.


Theorem 3.5. Let X be a real uniformly convex Banach space and let K be a nonempty closedconvex subset of X. Let T : K → K be a uniformly L1-Lipschitzian asymptotically quasi-I-nonexpansive mapping with a sequence {λn} ⊂ [1,∞) and I : K → K be a uniformly L2-Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence {μn} ⊂ [1,∞) such thatF = F(T) ∩ F(I)/= ∅. Suppose A∗ = infnαn, A∗ = supnαn, Λ = supnλn ≥ 1, M = supnμn ≥ 1 and{αn} and {βn} are two sequences in [0, 1] satisfying the following conditions:

(i)∑∞

n=1(λnμn − 1)αn <∞,(ii) 0 < A∗ ≤ A∗ < 1/Λ2M2.

(iii) 0 < B∗ = infnβn ≤ supnβn = B∗ < 1

If at least one mapping of the mappings T and I is semicompact, then the implicitly iterative sequence{xn} defined by (1.2) converges strongly to a common fixed point of T and I.

Proof. Without any loss of generality, we may assume that T is semicompact. This with (3.39)means that there exists a subsequence {xnk} ⊂ {xn} such that xnk → x∗ strongly and x∗ ∈ K.Since T, I are continuous, then from (3.39) and (3.41) we find

‖x∗ − Tx∗‖ = limnk→∞

‖xnk − Txnk‖ = 0, ‖x∗ − Ix∗‖ = limnk→∞

‖xnk − Ixnk‖ = 0. (3.45)

This shows that x∗ ∈ F = F(T) ∩ F(I). According to Lemma 3.1 the limit limn→∞‖xn − x∗‖exists. Then

limn→∞

‖xn − x∗‖ = limnk→∞

‖xnk − x∗‖ = 0, (3.46)

which means that {xn} converges to x∗ ∈ F. This completes the proof.

Note that all results presented here generalize and extend the corresponding mainresults of [15–17] in a case of one mapping.

Acknowledgment

The authors acknowledge the MOSTI Grant 01-01-08-SF0079.

References



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Research ArticleA Kirk Type Characterization of Completeness forPartial Metric Spaces

Salvador Romaguera

Insitituto Universitario de Matematica Pura y Aplicada, Universidad Politecnica de Valencia,46071 Valencia, Spain

Correspondence should be addressed to Salvador Romaguera, [email protected]

Received 1 October 2009; Accepted 25 November 2009


Copyright q 2010 Salvador Romaguera. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We extend the celebrated result of W. A. Kirk that a metric space X is complete if and only if everyCaristi self-mapping for X has a fixed point, to partial metric spaces.


Caristi proved in [1] that if f is a selfmapping of a complete metric space (X, d) such thatthere is a lower semicontinuous function φ : X → [0,∞) satisfying

d(x, fx

) ≤ φ(x) − φ(fx) (1.1)

for all x ∈ X, then f has a fixed point.This classical result suggests the following notion. A selfmapping f of a metric space

(X, d) for which there is a function φ : X → [0,∞) satisfying the conditions of Caristi’stheorem is called a Caristi mapping for (X, d).

There exists an extensive and well-known literature on Caristi’s fixed point theoremand related results (see, e.g., [2–10], etc.).

In particular, Kirk proved in [7] that a metric space (X, d) is complete if and onlyif every Caristi mapping for (X, d) has a fixed point. (For other characterizations of metriccompleteness in terms of fixed point theory see [11–14], etc., and also [15, 16] for recentcontributions in this direction.)

In this paper we extend Kirk’s characterization to a kind of complete partial metricspaces.


Let us recall that partial metric spaces were introduced by Matthews in [17] as a partof the study of denotational semantics of dataflow networks. In fact, it is widely recognizedthat partial metric spaces play an important role in constructing models in the theory ofcomputation (see [18–25], etc.).

A partial metric [17] on a set X is a function p : X × X → [0,∞) such that for allx, y, z ∈ X: (i) x = y ⇔ p(x, x) = p(x, y) = p(y, y); (ii) p(x, x) ≤ p(x, y); (iii) p(x, y) = p(y, x);(iv) p(x, z) ≤ p(x, y) + p(y, z) − p(y, y).

A partial metric space is a pair (X, p) where p is a partial metric on X.Each partial metric p onX induces a T0 topology τp onX which has as a base the family

of open balls {Bp(x, ε) : x ∈ X, ε > 0}, where Bp(x, ε) = {y ∈ X : p(x, y) < p(x, x) + ε} for allx ∈ X and ε > 0.

Next we give some pertinent concepts and facts on completeness for partial metricspaces.

If p is a partial metric on X, then the function ps : X ×X → [0,∞) given by ps(x, y) =2p(x, y) − p(x, x) − p(y, y) is a metric on X.

A sequence (xn)n∈N in a partial metric space (X, p) is called a Cauchy sequence if thereexists (and is finite) limn,mp(xn, xm) ([17, Definition 5.2]).

Note that (xn)n∈N is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequencein the metric space (X, ps) (see, e.g., [17, page 194]).

A partial metric space (X, p) is said to be complete if every Cauchy sequence (xn)n∈Nin X converges, with respect to τp, to a point x ∈ X such that p(x, x) = limn,mp(xn, xm) ([17,Definition 5.3]).

It is well known and easy to see that a partial metric space (X, p) is complete if andonly if the metric space (X, ps) is complete.

In order to give an appropriate notion of a Caristi mapping in the framework of partialmetric spaces, we naturally propose the following two alternatives.

(i) A selfmapping f of a partial metric space (X, p) is called a p-Caristi mapping on Xif there is a function φ : X → [0,∞) which is lower semicontinuous for (X, p) andsatisfies p(x, fx) ≤ φ(x) − φ(fx), for all x ∈ X.

(ii) A selfmapping f of a partial metric space (X, p) is called a ps-Caristi mapping on Xif there is a function φ : X → [0,∞) which is lower semicontinuous for (X, ps) andsatisfies p(x, fx) ≤ φ(x) − φ(fx), for all x ∈ X.

It is clear that every p-Caristi mapping is ps-Caristi but the converse is not true, ingeneral.

In a first attempt to generalize Kirk’s characterization of metric completeness to thepartial metric framework, one can conjecture that a partial metric space (X, p) is complete ifand only if every p-Caristi mapping on X has a fixed point.

The following easy example shows that this conjecture is false.

Example 1.1. On the set N of natural numbers construct the partial metric p given by

p(n,m) = max{

1n,

1m

}. (1.2)

Note that (N, p) is not complete, because the metric ps induces the discrete topologyon N, and (n)n∈N is a Cauchy sequence in (N, ps). However, there is no p-Caristi mappings onN as we show in the next.


Indeed, let f : N → N and suppose that there is a lower semicontinuous function φfrom (N, τp) into [0,∞) such that p(n, fn) ≤ φ(n) − φ(fn) for all n ∈ N. If 1 < f1, we havep(1, f1) = 1 = p(1, 1), which means that f1 ∈ Bp(1, ε) for any ε > 0, so φ(1) ≤ φ(f1) bylower semicontinuity of φ, which contradicts condition p(1, f1) ≤ φ(1) − φ(f1). Therefore1 = f1, which again contradicts condition p(1, f1) ≤ φ(1) − φ(f1). We conclude that f is nota p-Caristi mapping on N.

Unfortunately, the existence of fixed point for each ps-Caristi mapping on a partialmetric space (X, p) neither characterizes completeness of (X, p) as follows from ourdiscussion in the next section.

2. The Main Result

In this section we characterize those partial metric spaces for which every ps-Caristi mappinghas a fixed point in the style of Kirk’s characterization of metric completeness. This will bedone by means of the notion of a 0-complete partial metric space which is introduced asfollows.

Definition 2.1. A sequence (xn)n∈N in a partial metric space (X, p) is called 0-Cauchy iflimn,mp(xn, xm) = 0. We say that (X, p) is 0-complete if every 0-Cauchy sequence in Xconverges, with respect to τp, to a point x ∈ X such that p(x, x) = 0.

Note that every 0-Cauchy sequence in (X, p) is Cauchy in (X, ps), and that everycomplete partial metric space is 0-complete.

On the other hand, the partial metric space (Q ∩ [0,∞), p), where Q denotes the setof rational numbers and the partial metric p is given by p(x, y) = max{x, y}, provides aparadigmatic example of a 0-complete partial metric space which is not complete.

In the proof of the “only if” part of our main result we will use ideas from [11, 26],whereas the following auxiliary result will be used in the proof of the “if” part.

Lemma 2.2. Let (X, p) be a partial metric space. Then, for each x ∈ X, the function px : X → [0,∞)given by px(y) = p(x, y) is lower semicontinuous for (X, ps).

Proof. Assume that limnps(y, yn) = 0, then

px(y) ≤ px(yn) + p(yn, y) − p(yn, yn) = px

(yn

)+ ps

(yn, y

) − p(yn, y) + p(y, y). (2.1)

This yields lim infnpx(yn) ≥ px(y) because p(y, y) ≤ p(y, yn).

Theorem 2.3. A partial metric space (X, p) is 0-complete if and only if every ps-Caristi mapping fon X has a fixed point.

Proof. Suppose that (X, p) is 0-complete and let f be a ps-Caristi mapping on X, then, there isa φ : X → [0,∞) which is lower semicontinuous function for (X, ps) and satisfies

p(x, fx

) ≤ φ(x) − φ(fx), (2.2)

for all x ∈ X.


Now, for each x ∈ X define

Ax :={y ∈ X : p

(x, y

) ≤ φ(x) − φ(y)}. (2.3)

Observe that Ax /=φ because fx ∈ Ax. Moreover Ax is closed in the metric space (X, ps) sincey → p(x, y) + φ(y) is lower semicontinuous for (X, ps).

Fix x0 ∈ X. Take x1 ∈ Ax0 such that φ(x1) < infy∈Ax0φ(y) + 2−1. Clearly Ax1 ⊆ Ax0 .

Hence, for each x ∈ Ax1 we have

p(x1, x) ≤ φ(x1) − φ(x) < infy∈Ax0

φ(y)+ 2−1 − φ(x)

≤ φ(x) + 2−1 − φ(x) = 2−1.

(2.4)

Following this process we construct a sequence (xn)n∈ω in X such that its associated sequence(Axn)n∈ω of closed subsets in (X, ps) satisfies

(i) Axn+1 ⊆ Axn, xn+1 ∈ Axn for all n ∈ ω,(ii) p(xn, x) < 2−n for all x ∈ Axn, n ∈ N.

Since p(xn, xn) ≤ p(xn, xn+1), and, by (i) and (ii), p(xn, xm) < 2−n for allm > n, it followsthat limn,mp(xn, xm) = 0, so (xn)n∈ω is a 0-Cauchy sequence in (X, p), and by our hypothesis,there exists z ∈ X such that limnp(z, xn) = p(z, z) = 0, and thus limnp

s(z, xn) = 0. Thereforez ∈ ⋂

n∈ωAxn.Finally, we show that z = fz. To this end, we first note that

p(xn, fz

) ≤ p(xn, z) + p(z, fz)≤ φ(xn) − φ(z) + φ(z) − φ

(fz

),

(2.5)

for all n ∈ ω. Consequently fz ∈ ⋂n∈ωAxn, so by (ii), p(xn, fz) < 2−n for all n ∈ N. Since

p(z, fz) ≤ p(z, xn) + p(xn, fz), and limnp(z, xn) = 0, it follows that p(z, fz) = 0. Henceps(z, fz) = 0 since ps(z, fz) ≤ 2p(z, fz), so z = fz.

Conversely, suppose that there is a 0-Cauchy sequence (xn)n∈ω of distinct points in(X, p) which is not convergent in (X, ps). Construct a subsequence (yn)n∈ω of (xn)n∈ω suchthat p(yn, yn+1) < 2−(n+1) for all n ∈ ω.

Put A = {yn : n ∈ ω}, and define f : X → X by fx = y0 if x ∈ X \A, and fyn = yn+1

for all n ∈ ω.Observe that A is closed in (X, ps).Now define φ : X → [0,∞) by φ(x) = p(x, y0) + 1 if x ∈ X \A, and φ(yn) = 2−n for all

n ∈ ω.Note that φ(yn+1) < φ(yn) for all n ∈ ω and that φ(y0) ≤ φ(x) for all x ∈ X \A.From this fact and the preceding lemma we deduce that φ is lower semicontinuous for

(X, ps).Moreover, for each x ∈ X \A we have

p(x, fx

)= p

(x, y0

)= φ(x) − φ(y0

)= φ(x) − φ(fx), (2.6)


and for each yn ∈ A we have

p(yn, fyn

)= p

(yn, yn+1

)< 2−(n+1) = φ

(yn

) − φ(yn+1)

= φ(yn

) − φ(fyn).(2.7)

Therefore f is a Caristi ps-mapping on X without fixed point, a contradiction. Thisconcludes the proof.

Acknowledgments

The author is very grateful to the referee for his/her useful suggestions. This work waspartially supported by the Spanish Ministry of Science and Innovation, and FEDER, GrantMTM2009-12872-C02-01.

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Research ArticleGeneralized Asymptotic Pointwise Contractionsand Nonexpansive Mappings Involving Orbits

Adriana Nicolae

Department of Applied Mathematics, Babes-Bolyai University, Kogalniceanu 1,400084 Cluj-Napoca, Romania

Correspondence should be addressed to Adriana Nicolae, [email protected]

Received 30 September 2009; Accepted 25 November 2009


Copyright q 2010 Adriana Nicolae. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We give fixed point results for classes of mappings that generalize pointwise contractions,asymptotic contractions, asymptotic pointwise contractions, and nonexpansive and asymptoticnonexpansive mappings. We consider the case of metric spaces and, in particular, CAT(0) spaces.We also study the well-posedness of these fixed point problems.

1. Introduction

Four recent papers [1–4] present simple and elegant proofs of fixed point results for pointwisecontractions, asymptotic pointwise contractions, and asymptotic nonexpansive mappings.Kirk and Xu [1] study these mappings in the context of weakly compact convex subsetsof Banach spaces, respectively, in uniformly convex Banach spaces. Hussain and Khamsi[2] consider these problems in the framework of metric spaces and CAT(0) spaces. In [3],the authors prove coincidence results for asymptotic pointwise nonexpansive mappings.Espınola et al. [4] examine the existence of fixed points and convergence of iterates forasymptotic pointwise contractions in uniformly convex metric spaces.

In this paper we do not consider more general spaces, but instead we formulate lessrestrictive conditions for the mappings and show that the conclusions of the theorems stillstand even in such weaker settings.

2. Preliminaries

Let (X, d) be a metric space. For z ∈ X and r > 0 we denote the closed ball centered at z withradius r by B(z, r) := {x ∈ X : d(x, z) ≤ r}.


Let K ⊆ X and let T : K → K. Throughout this paper we will denote the fixed pointset of T by Fix(T). The mapping T is called a Picard operator if it has a unique fixed point zand (Tn(x))n∈N converges to z for each x ∈ K.

A sequence (xn)n∈N ⊆ K is said to be an approximate fixed point sequence for themapping T if limn→∞d(xn, T(xn)) = 0.

The fixed point problem for T is well-posed (see [5, 6]) if T has a unique fixed pointand every approximate fixed point sequence converges to the unique fixed point of T .

A mapping T : X → X is called a pointwise contraction if there exists a functionα : X → [0, 1) such that

d(T(x), T

(y)) ≤ α(x)d(x, y) for every x, y ∈ X. (2.1)

Let T : X → X and for n ∈ N let αn : X → R+ such that

d(Tn(x), Tn

(y)) ≤ αn(x)d(x, y) for every x, y ∈ X. (2.2)

If the sequence (αn)n∈N converges pointwise to the function α : X → [0, 1), then T is calledan asymptotic pointwise contraction.If for every x ∈ X, lim supn→∞αn(x) ≤ 1, then T is called an asymptotic pointwisenonexpansive mapping.If there exists 0 < k < 1 such that for every x ∈ X, lim supn→∞αn(x) ≤ k, then T is called astrongly asymptotic pointwise contraction.

For a mapping T : X → X and x ∈ X we define the orbit starting at x by

OT (x) ={x, T(x), T2(x), . . . , Tn(x), . . .

}, (2.3)

where Tn+1(x) = T(Tn(x)) for n ≥ 0 and T0(x) = x. Denote also OT (x, y) = OT (x) ∪OT (y).Given D ⊆ X and x ∈ X, the number rx(D) = supy∈Dd(x, y) is called the radius of D

relative to x. The diameter of D is diam(D) = supx,y∈Dd(x, y) and the cover of D is definedas cov(D) =

⋂{B : B is a closed ball and D ⊆ B}.As in [2], we say that a family F of subsets of X defines a convexity structure on X if

it contains the closed balls and is stable by intersection. A subset of X is admissible if it is anonempty intersection of closed balls. The class of admissible subsets of X denoted byA(X)defines a convexity structure on X. A convexity structure F is called compact if any family(Aα)α∈Γ of elements of F has nonempty intersection provided

⋂α∈FAα /= ∅ for any finite subset

F ⊆ Γ.According to [2], for a convexity structure F, a function ϕ : X → R+ is called F-

convex if {x : ϕ(x) ≤ r} ∈ F for any r ≥ 0. A type is defined as ϕ : X → R+, ϕ(u) =lim supn→∞d(u, xn) where (xn)n∈N is a bounded sequence in X. A convexity structure F isT -stable if all types are F-convex.

The following lemma is mentioned in [2].

Lemma 2.1. Let X be a metric space and F a compact convexity structure on X which is T -stable.Then for any type ϕ there is x0 ∈ X such that

ϕ(x0) = infx∈X

ϕ(x). (2.4)


A metric space (X, d) is a geodesic space if every two points x, y ∈ X can be joinedby a geodesic. A geodesic from x to y is a mapping c : [0, l] → X, where [0, l] ⊆ R, suchthat c(0) = x, c(l) = y, and d(c(t), c(t′)) = |t − t′| for every t, t′ ∈ [0, l]. The image c([0, l])of c forms a geodesic segment which joins x and y. A geodesic triangle Δ(x1, x2, x3) consistsof three points x1, x2, and x3 in X (the vertices of the triangle) and three geodesic segmentscorresponding to each pair of points (the edges of the triangle). For the geodesic traingleΔ = Δ(x1, x2, x3), a comparison triangle is the triangle Δ = Δ(x1, x2, x3) in the Euclideanspace E

2 such that d(xi, xj) = dE2(xi, xj) for i, j ∈ {1, 2, 3}. A geodesic triangle Δ satisfies theCAT(0) inequality if for every comparison triangle Δ of Δ and for every x, y ∈ Δ we have

d(x, y) ≤ dE2

(x, y), (2.5)

where x, y ∈ Δ are the comparison points of x and y. A geodesic metric space is a CAT(0)space if every geodesic traingle satisfies the CAT(0) inequality. In a similar way we can defineCAT(k) spaces for k > 0 or k < 0 using the model spaces M2

k.

A geodesic space is a CAT(0) space if and only if it satisfies the following inequalityknown as the (CN) inequality of Bruhat and Tits [7]. Let x, y1, y2 be points of a CAT(0) spaceand let m be the midpoint of [y1, y2]. Then

d(x,m)2 ≤ 12d(x, y1

)2 +12d(x, y2

)2 − 14d(y1, y2

)2. (2.6)

It is also known (see [8]) that in a complete CAT(0) space, respectively, in a closed convexsubset of a complete CAT(0) space every type attains its infimum at a single point. For moredetails about CAT(k) spaces one can consult, for instance, the papers [9, 10].

In [2], the authors prove the following fixed point theorems.

Theorem 2.2. Let X be a bounded metric space. Assume that the convexity structure A(X) iscompact. Let T : X → X be a pointwise contraction. Then T is a Picard operator.

Theorem 2.3. Let X be a bounded metric space. Assume that the convexity structure A(X) iscompact. Let T : X → X be a strongly asymptotic pointwise contraction. Then T is a Picard operator.

Theorem 2.4. LetX be a bounded metric space. Assume that there exists a convexity structureF thatis compact and T -stable. Let T : X → X be an asymptotic pointwise contraction. Then T is a Picardoperator.

Theorem 2.5. Let X be a complete CAT(0) space and let K be a nonempty, bounded, closed andconvex subset of X. Then any mapping T : K → K that is asymptotic pointwise nonexpansive has afixed point. Moreover, Fix(T) is closed and convex.

The purpose of this paper is to present fixed point theorems for mappings that satisfymore general conditions than the ones which appear in the classical definitions of pointwisecontractions, asymptotic contractions, asymptotic pointwise contractions and asymptoticnonexpansive mappings. Besides this, we show that the fixed point problems are well-posed.Some generalizations of nonexpansive mappings are also considered. We work in the contextof metric spaces and CAT(0) spaces.


3. Generalizations Using the Radius of the Orbit

In the sequel we extend the results obtained by Hussain and Khamsi [2] using the radiusof the orbit. We also study the well-posedness of the fixed point problem. We start byintroducing a property for a mapping T : X → X, where X is a metric space. Namely, wewill say that T satisfies property (S) if

(S) for every approximate fixed point sequence (xn)n∈N and for every m ∈ N, thesequence (d(xn, Tm(xn)))n∈N converges to 0 uniformly with respect to m.

For instance, if for every x ∈ X, d(x, T2(x)) ≤ d(x, T(x)) then property (S) is fulfilled.

Proposition 3.1. Let X be a metric space and let T : X → X be a mapping which satisfies (S). If(xn)n∈N is an approximate fixed point sequence, then for everym ∈ N and every x ∈ X,

lim supn→∞

d(x, Tm(xn)) = lim supn→∞

d(x, xn), (3.1)

lim supn→∞

rx(OT (xn)) = lim supn→∞

d(x, xn), (3.2)

limn→∞

diam OT(xn) = 0. (3.3)

Proof. Since T satisfies (S) and (xn)n∈N is an approximate fixed point sequence, it easilyfollows that (3.1) holds. To prove (3.2), let ε > 0. Then there exists m ∈ N such that

rx(OT (xn)) ≤ d(x, Tm(xn)) + ε ≤ d(x, xn) + d(xn, Tm(xn)) + ε. (3.4)

Taking the superior limit,

lim supn→∞

rx(OT (xn)) ≤ lim supn→∞

d(x, xn) + ε. (3.5)

Hence, (3.2) holds. Now let again ε > 0. Then there exist m1, m2 ∈ N such that

diamOT (xn) ≤ d(Tm1(xn), Tm2(xn)) + ε ≤ d(xn, Tm1(xn)) + d(xn, Tm2(xn)) + ε. (3.6)

We only need to let n → ∞ in the above relation to prove (3.3).

Theorem 3.2. Let X be a bounded metric space such thatA(X) is compact. Also let T : X → X forwhich there exists α : X → [0, 1) such that

d(T(x), T

(y)) ≤ α(x)rx(OT

(y))

for every x, y ∈ X. (3.7)

Then T is a Picard operator. Moreover, if additionally T satisfies (S), then the fixed point problem iswell-posed.


Proof. BecauseA(X) is compact, there exists a nonempty minimal T -invariant K ∈ A(X) forwhich cov(T(K)) = K. If x, y ∈ K then rx(OT (y)) ≤ rx(K). In a similar way as in the proof ofTheorem 3.1 of [2] we show now that T has a fixed point. Let x ∈ K. Then,

d(T(x), T

(y)) ≤ α(x)rx(OT

(y)) ≤ α(x)rx(K) for every y ∈ X. (3.8)

This means that T(K) ⊆ B(T(x), α(x)rx(K)), so K = cov(T(K)) ⊆ B(T(x), α(x)rx(K)).Therefore,

rT(x)(K) ≤ α(x)rx(K). (3.9)

Denote

Kx ={y ∈ K : ry(K) ≤ rx(K)

}. (3.10)

Kx ∈ A(X) since it is nonempty and Kx =⋂y∈KB(y, rx(K)) ∩K.

Let y ∈ Kx. As above we have K ⊆ B(T(y), α(y)ry(K)) ⊆ B(T(y), α(y)rx(K)) andhence T(y) ∈ Kx. Because K is minimal T -invariant it follows that Kx = K. This yieldsry(K) = rx(K) for every x, y ∈ K. In particular, rT(x)(K) = rx(K) and using (3.9) we obtainrx(K) = 0 which implies that K consists of exactly one point which will be fixed under T .

Now suppose x, y ∈ X, x /=y are fixed points of T . Then

d(x, y) ≤ α(x)rx(OT

(y))

= α(x)d(x, y). (3.11)

This means that α(x) ≥ 1 which is impossible.Let z denote the unique fixed point of T , let x ∈ X and lx = lim supn→∞d(z, T

n(x)).Observe that the sequence (rz(OT (Tn(x))))n∈N is decreasing and bounded below by 0 so itslimit exists and is precisely lx. Then

lx ≤ α(z) limn→∞

rz(OT

(Tn−1(x)

))= α(z)lx. (3.12)

This implies that lx = 0 and hence limn→∞Tn(x) = z.Next we prove that the problem is well-posed. Let (xn)n∈N be an approximate fixed

point sequence. We know that

d(z, xn) ≤ d(xn, T(xn)) + d(T(xn), T(z)) ≤ d(xn, T(xn)) + α(z)rz(OT (xn)). (3.13)

Taking the superior limit and applying (3.2) of Proposition 3.1 for z,

lim supn→∞

d(z, xn) ≤ α(z)lim supn→∞

d(z, xn), (3.14)

which implies limn→∞d(z, xn) = 0.


We remark that if in the above result T is, in particular, a pointwise contraction thenthe fixed point problem is well-posed without additional assumptions for T .

Next we give an example of a mapping which is not a pointwise contraction, but fulfills(3.7).

Example 3.3. Let T : [0, 1] → [0, 1],

T(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1 − x2

, if x ≥ 12,

34x, if x <

12,

(3.15)

and let α : [0, 1] → [0, 1),

α(x) =

⎧⎪⎪⎨⎪⎪⎩

12, if x ≥ 1

2,

34+ x2, if x <

12.

(3.16)

Then T is not a pointwise contraction, but (3.7) is verified.

Proof. T is not continuous, so it is not nonexpansive and hence it cannot be a pointwisecontraction. If x, y ≥ 1/2 or x, y < 1/2 the conclusion is immediate. Suppose x ≥ 1/2 andy < 1/2. Then

rx(OT

(y))

= x, ry(OT (x)) = max{x − y, y}. (3.17)

(i) If T(x) − T(y) ≥ 0, then

1 − x2− 3

4y ≤ x

2= α(x)rx

(OT

(y)),

1 − x2− 3

4y ≤(

34+ y2

)(x − y) ≤ α(y)ry(OT (x)).

(3.18)

The above is true because 1/2 − 5/4x < 0 ≤ y2(x − y).(ii) If T(x) − T(y) < 0, then

34y − 1 − x

2≤ −1

8+x

2<x

2= α(x)rx

(OT

(y)),

34y − 1 − x

2≤(

34+ y2

)y ≤ α(y)ry(OT (x)).

(3.19)


Theorem 3.4. Let X be a bounded metric space, T : X → X, and suppose there exists a convexitystructure F which is compact and T -stable. Assume

d(Tn(x), Tn

(y)) ≤ αn(x)rx(OT

(y))

for every x, y ∈ X, (3.20)

where for each n ∈ N, αn : X → R+, and the sequence (αn)n∈N converges pointwise to a functionα : X → [0, 1). Then T is a Picard operator. Moreover, if additionally T satisfies (S), then the fixedpoint problem is well-posed.

Proof. Assume T has two fixed points x, y ∈ X, x /=y. Then for each n ∈ N,

d(x, y) ≤ αn(x)d(x, y). (3.21)

When n → ∞we obtain α(x) ≥ 1 which is false. Hence, T has at most one fixed point.Let x ∈ X. We consider ϕ : X → R+,

ϕ(u) = lim supn→∞

d(u, Tn(x)) for u ∈ X. (3.22)

Because F is compact and T -stable there exists z ∈ X such that

ϕ(z) = infu∈X

ϕ(u). (3.23)

For p ∈ N,

ϕ(z) ≤ ϕ(Tp(z)) ≤ αp(z) limn→∞

rz(OT (Tn(x))) = αp(z)ϕ(z). (3.24)

Letting p → ∞ in the above relation yields ϕ(z) = 0 so (Tn(x))n∈N converges to z whichwill be the unique fixed point of T because d(T(z), Tn+1(x)) ≤ α1(z)rz(OT (Tn(x))) andlimn→∞rz(OT (Tn(x))) = 0. Thus, all the Picard iterates will converge to z.

Let (xn)n∈N be an approximate fixed point sequence and let m ∈ N. Then

d(z, xn) ≤ d(xn, Tm(xn)) + d(Tm(xn), Tm(z)) ≤ d(xn, Tm(xn)) + αm(z)rz(OT (xn)). (3.25)

Taking the superior limit and applying (3.2) of Proposition 3.1,

lim supn→∞

d(z, xn) ≤ αm(z)lim supn→∞

d(z, xn). (3.26)

Letting m → ∞we have limn→∞d(z, xn) = 0.


Theorem 3.5. Let X be a complete CAT(0) space and let K ⊆ X be nonempty, bounded, closed, andconvex. Let T : K → K and for n ∈ N, let αn : K → R+ be such that lim supn→∞αn(x) ≤ 1 for allx ∈ K. If for all n ∈ N,

d(Tn(x), Tn


(y))

for every x, y ∈ K, (3.27)

then T has a fixed point. Moreover, Fix(T ) is closed and convex.

Proof. The idea of the proof follows to a certain extend the proof of Theorem 5.1 in [2]. Letx ∈ K. Denote ϕ : K → R+,


d(u, Tn(x)) for u ∈ K. (3.28)

Since K is a nonempty, closed, and convex subset of a complete CAT(0) space there exists aunique z ∈ K such that

ϕ(z) = infu∈K

ϕ(u). (3.29)

For p ∈ N,

ϕ(Tp(z)) ≤ αp(z) limn→∞

rz(OT (Tn(x))) = αp(z)ϕ(z). (3.30)

Let p, q ∈ N and let m denote the midpoint of the segment [Tp(z), Tq(z)]. Using the (CN)inequality, we have

d(m, Tn(x))2 ≤ 12d(Tp(z), Tn(x))2 +

12d(Tq(z), Tn(x))2 − 1

4d(Tp(z), Tq(z))2. (3.31)

Letting n → ∞ and considering ϕ(z) ≤ ϕ(m), we have

ϕ(z)2 ≤ 12ϕ(Tp(z))2 +

12ϕ(Tq(z))2 − 1

4d(Tp(z), Tq(z))2

≤ 12αp(z)2ϕ(z)2 +

12αq(z)2ϕ(z)2 − 1

4d(Tp(z), Tq(z))2.

(3.32)

Letting p, q → ∞we obtain that (Tn(z))n∈N is a Cauchy sequence which converges to ω ∈ K.As in the proof of Theorem 3.4 we can show that ω is a fixed point for T . To prove that Fix(T)is closed take (xn)n∈N a sequence of fixed points which converges to x∗ ∈ K. Then

d(T(x∗), T(xn)) ≤ α1(x∗)d(x∗, xn), (3.33)

which shows that x∗ is a fixed point of T .


The fact that Fix(T) is convex follows from the (CN) inequality. Let x, y ∈ Fix(T) andlet m be the midpoint of [x, y]. For n ∈ N we have

d(m, Tn(m))2 ≤ 12d(x, Tn(m))2 +

12d(y, Tn(m)

)2 − 14d(x, y)2

≤ 12αn(m)2rm(OT (x))2 +

12αn(m)2rm

(OT

(y))2 − 1

4d(x, y)2

=12αn(m)2

(d(m,x)2 + d

(m,y

)2)− 1

4d(x, y)2

=14

(αn(m)2 − 1

)d(x, y)2.

(3.34)

Letting n → ∞ we obtain limn→∞Tn(m) = m. This yields m which is a fixed point since

lim supn→∞

d(T(m), Tn+1(m)

)≤ α1(m)lim sup

n→∞d(m, Tn(m)). (3.35)

Hence, Fix(T) is convex.

We conclude this section by proving a demi-closed principle similarly to [2,Proposition 1]. To this end, for K ⊆ X, K closed and convex and ϕ : K → R+, ϕ(x) =lim supn→∞d(x, xn), as in [2], we introduce the following notation:

xnϕ⇀ ω iff ϕ(ω) = inf

x∈Kϕ(x), (3.36)

where the bounded sequence (xn)n∈N is contained in K.

Theorem 3.6. LetX be a CAT(0) space and letK ⊆ X,K bounded, closed, and convex. Let T : K →K satisfy (S) and for n ∈ N, let αn : K → R+ be such that lim supn→∞αn(x) ≤ 1 for all x ∈ K.Suppose that for n ∈ N,

d(Tn(x), Tn


(y))

for every x, y ∈ K. (3.37)

Let also (xn)n∈N ⊆ K be an approximate fixed point sequence such that xnϕ⇀ ω. Then ω ∈ Fix(T ).

Proof. Using (3.1) of Proposition 3.1 we obtain that for every x ∈ K and every p ∈ N,

ϕ(x) = lim supn→∞

d(x, Tp(xn)). (3.38)

Applying (3.2) of Proposition 3.1 for ω, we have

ϕ(Tp(ω)) = lim supn→∞

d(Tp(ω), Tp(xn)) ≤ αp(ω)lim supn→∞

rω(OT (xn)) = αp(ω)ϕ(ω). (3.39)


Let p ∈ N and let m be the midpoint of [ω, Tp(ω)]. As in the above proof, using the (CN)inequality we have

ϕ(m)2 ≤ 12ϕ(ω)2 +

12ϕ(Tp(ω))2 − 1

4d(ω, Tp(ω))2. (3.40)

Since ϕ(ω) ≤ ϕ(m),

ϕ(ω)2 ≤ 12ϕ(ω)2 +

12αp(ω)2ϕ(ω)2 − 1

4d(ω, Tp(ω))2. (3.41)

Letting p → ∞, we have limp→∞Tp(ω) = ω. This means ω ∈ Fix(T) because

lim supp→∞

d(T(ω), Tp+1(ω)

)≤ α1(ω)lim sup

p→∞d(ω, Tp(ω)). (3.42)

4. Generalized Strongly Asymptotic Pointwise Contractions

In this section we generalize the strongly asymptotic pointwise contraction condition, byusing the diameter of the orbit. We begin with a fixed point result that holds in a completemetric space.

Theorem 4.1. Let X be a complete metric space and let T : X → X be a mapping with boundedorbits that is orbitally continuous. Also, for n ∈ N, let αn : X → R+ for which there exists 0 < k < 1such that for every x ∈ X, lim supn→∞αn(x) ≤ k. If for each n ∈ N,

d(Tn(x), Tn

(y)) ≤ αn(x)diam OT

(x,y)

for every x,y ∈ X, (4.1)

then T is a Picard operator. Moreover, if additionally T satisfies (S), then the fixed point problem iswell-posed.

Proof. First, suppose that T has two fixed points x, y ∈ X, x /=y. Then for each n ∈ N,

d(x, y) ≤ αn(x)d(x, y). (4.2)

Letting n → ∞ we obtain that k ≥ 1 which is impossible. Hence, T has at most one fixedpoint. Let x ∈ X. Notice that the sequence (diam OT(Tn(x)))n∈N is decreasing and boundedbelow by 0 so it converges to lx ≥ 0. For n, p1, p2 ∈ N, p1 ≤ p2 we have

d(Tn+p1(x), Tn+p2(x)) ≤ αn+p1(x)diamOT (x). (4.3)

Taking the supremum with respect to p1 and p2 and then letting n → ∞we obtain

lx ≤ k diamOT (x). (4.4)


For p ∈ N,

lx = limn→∞

diamOT (Tn(Tp(x))) ≤ k diamOT (Tp(x)). (4.5)

Letting p → ∞ in the above relation we have lx ≤ klx which implies that lx = 0. This meansthat the sequence (Tn(x))n∈N is Cauchy so it converges to a point z ∈ X. Because T is orbitallycontinuous it follows that z is a fixed point, which is unique. Therefore, all Picard iteratesconverge to z.

Next we prove that the problem is well-posed. Let (xn)n∈N be an approximate fixedpoint sequence. Taking into account (3.2) applied for z and (3.3) of Proposition 3.1,

lim supn→∞

diamOT (z, xn) = lim supn→∞

diam({z} ∪OT (xn)) = lim supn→∞

d(z, xn). (4.6)

Knowing that

d(z, xn) ≤ d(xn, Tm(xn)) + d(Tm(xn), Tm(z)) ≤ d(xn, Tm(xn)) + αm(z)diamOT (z, xn),(4.7)

and taking the superior limit we obtain

lim supn→∞

d(z, xn) ≤ αm(z)lim supn→∞

d(z, xn). (4.8)

If we let here m → ∞ it is clear that (xn)n∈N converges to z.

A similar result can be given in a bounded metric space where the convexity structuredefined by the class of admissible subsets is compact.

Theorem 4.2. Let X be a bounded metric space such that A(X) is compact and let T : X → X bean orbitally continuous mapping. Also, for n ∈ N, let αn : X → R+ for which there exists 0 < k < 1such that for every x ∈ X, lim supn→∞αn(x) ≤ k. If for each n ∈ N,

d(Tn(x), Tn

(y)) ≤ αn(x)diam OT

(x,y)

for every x,y ∈ X, (4.9)

then T is a Picard operator. Moreover, if additionally T satisfies (S), then the fixed point problem iswell-posed.

Proof. Let x ∈ X. Denote ϕ : X → R+,


d(u, Tn(x)) for u ∈ X. (4.10)

As in the proof of Theorem 4.1 one can show that T has at most one fixed point and for eachx ∈ X, the sequence (Tn(x))n∈N is Cauchy. This means that limn→∞ϕ(Tn(x)) = 0 for eachx ∈ X. BecauseA(X) is compact we can choose

ω ∈⋂n≥1

cov({Tk(x) : k ≥ n

}). (4.11)


Following the argument of [2, Theorem 4.1] we can show that ϕ(ω) = 0. For the sake ofcompleteness we also include this part of the proof. The definition of ϕ yields that for u ∈ Xand every ε > 0 there exists n0 ∈ N such that for any n ≥ n0,

d(u, Tn(x)) ≤ ϕ(u) + ε. (4.12)

Hence, Tn(x) ∈ B(u, ϕ(u) + ε) for every n ≥ n0 and so

cov({Tn(x) : n ≥ n0}) ⊆ B(u, ϕ(u) + ε

). (4.13)

Therefore, ω ∈ B(u, ϕ(u)+ε) for each ε > 0. This implies d(ω, u) ≤ ϕ(u) which holds for everyu ∈ X. Thus,

ϕ(ω) = lim supn→∞

d(ω, Tn(x)) ≤ lim supn→∞

ϕ(Tn(x)) = 0. (4.14)

Now it is clear that (Tn(x))n∈N converges to ω. Because T is orbitally continuous, ω will bethe unique fixed point and all the Picard iterates will converge to this unique fixed point.

The fact that every approximate fixed point sequence (xn)n∈N converges to ω can beproved identically as in Theorem 4.1.

In connection with the use of the diameter of the orbit, Walter [11] obtained a fixedpoint theorem that may be stated as follows.

Theorem 4.3 (Walter [11]). Let (X, d) be a complete metric space and let T : X → X be a mappingwith bounded orbits. If there exists a continuous, increasing function ϕ : R+ → R+ for which ϕ(r) < rfor every r > 0 and

d(T(x), T

(y)) ≤ ϕ(diam

(OT(x,y)))


then T is a Picard operator.

We conclude this section by proving an asymptotic version of this result. In this waywe extend the notion of asymptotic contraction introduced by Kirk in [12].

Theorem 4.4. Let (X, d) be a complete metric space and let T : X → X be an orbitally continuousmapping with bounded orbits. Suppose there exist a continuous function ϕ : R+ → R+ satisfyingϕ(t) < t for all t > 0 and the functions ϕn : R+ → R+ such that the sequence (ϕn)n∈N convergespointwise to ϕ and for each n ∈ N,

d(Tn(x), Tn

(y)) ≤ ϕn(diamOT

(x,y))

for all x, y ∈ X, (4.16)

then T is a Picard operator. Moreover, if additionally T satisfies (S) and ϕn is continuous for eachn ∈ N, then the fixed point problem is well-posed.

Proof. The proof follows closely the ideas presented in the proof of Theorem 4.1.


We begin by supposing that T has two fixed points x, y ∈ X, x /=y. Then for each n ∈ N,

d(x, y) ≤ ϕn(d(x, y)). (4.17)

Letting n → ∞we obtain that d(x, y) ≤ ϕ(d(x, y)) which is impossible. Hence, T has at mostone fixed point.

Notice that for x ∈ X the sequence (diamOT (Tn(x)))n∈N is decreasing and boundedbelow by 0 so it converges to lx ≥ 0. For n, p1, p2 ∈ N, p1 ≤ p2 we have

d(Tn+p1(x), Tn+p2(x)) ≤ ϕn+p1(diamOT (x)). (4.18)

Thus, lx ≤ ϕ(diamOT (x)).For p ∈ N,

lx = limn→∞

diamOT (Tn(Tp(x))) ≤ ϕ(diamOT (Tp(x))). (4.19)

Hence, lx ≤ ϕ(lx) which implies that lx = 0 and the proof may be continued as in Theorem 4.1in order to conclude that T is a Picard operator.

Let z ∈ X be the unique fixed point of T and let (xn)n∈N be an approximate fixed pointsequence. To show that the problem is well-posed, take (xnp)p∈N a subsequence of (xn)n∈Nsuch that

lim supn→∞

d(z, xn) = limp→∞

d(z, xnp

). (4.20)

Because every subsequence of (xn)n∈N is also an approximate fixed point sequence, theconclusions of Proposition 3.1 still stand for (xnp)p∈N. This yields

lim supp→∞

diamOT

(z, xnp

)= lim sup

p→∞diam

({z} ∪OT

(xnp

))= lim

p→∞d(z, xnp

). (4.21)

But since

d(z, xnp

)≤ diamOT

(z, xnp

), (4.22)

by passing to the inferior limit follows limp→∞diamOT (z, xnp) = limp→∞d(z, xnp).For m ∈ N,

d(z, xnp

)≤ d(xnp , T

m(xnp

))+ d(Tm(xnp

), Tm(z)

)

≤ d(xnp , T

m(xnp

))+ ϕm

(diamOT

(z, xnp

)).

(4.23)

If we let here p → ∞, we have limp→∞d(z, xnp) ≤ ϕm(limp→∞d(z, xnp)). Passing here tothe limit with respect to m implies limp→∞d(z, xnp) ≤ ϕ(limp→∞d(z, xnp)) and this meanslimp→∞d(z, xnp) = 0. Because of (4.20) it follows that (xn)n∈N converges to z.


5. Some Generalized Nonexpansive Mappings in CAT(0) Spaces

In this section we give fixed point results in CAT(0) spaces for two classes of mappings whichare more general than the nonexpansive ones.

Theorem 5.1. Let X be a bounded complete CAT(0) space and let T : X → X be such that for everyx, y ∈ X,

d(T(x), T

(y)) ≤ rx(OT

(y)). (5.1)

Then T has a fixed point. Moreover, Fix(T ) is closed and convex.



d(u, Tn(x)) for u ∈ X. (5.2)

Since X is a complete CAT(0) space there exists a unique z ∈ X such that

ϕ(z) = infu∈X

ϕ(u). (5.3)

Supposing z is not a fixed point of T , we have

ϕ(z) < ϕ(T(z)) ≤ limn→∞

rz(OT

(Tn−1(x)

))= ϕ(z). (5.4)

This is a contradiction and thus z ∈ Fix(T).Let (xn)n∈N be a sequence of fixed points which converges to x∗ ∈ X. Then,

d(T(x∗), T(xn)) ≤ d(x∗, xn) (5.5)

which proves that x∗ is a fixed point of T so Fix(T) is closed.Now take x, y ∈ Fix(T). We show that the midpoint of [x, y] denoted by m is a fixed

point of T using the (CN) inequality. More precisely we have

d(m, T(m))2 ≤ 12d(T(m), T(x))2 +

12d(T(m), T

(y))2 − 1

4d(x, y)2

≤ 12d(m,x)2 +

12d(m,y

)2 − 14d(x, y)2 = 0.

(5.6)

Hence, Fix(T) is convex.


A simple example of a mapping which is not nonexpansive, but satisfies (5.1), is thefollowing.

Example 5.2. Let T : [0, 1] −→ [0, 1],

T(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

x

2, if x ≥ 1

2,

x

4, if x <

12.

(5.7)

Then T is not nonexpansive but (5.1) is verified.

Proof. T is not continuous, so it cannot be nonexpansive. To show that (5.1) holds, we onlyconsider the situation when x ≥ 1/2 and y < 1/2 because in all other the condition is clearlysatisfied. Then |T(x) − T(y)| = x/2 − y/4. We can easily observe that

rx(OT

(y))

= x ≥ x2− y

4,

ry(OT (x)) = max{x − y, y}.

(5.8)

If (3/4)y ≤ x/2 then x/2 − y/4 ≤ x − y. Otherwise, x/2 − y/4 ≤ (3/4)y − y/4 = y/2 ≤ y. Inthis way we have shown that (5.1) is accomplished.

Theorem 5.3. Let X be a bounded complete CAT(0) space and let T : X → X be such that for everyx, y ∈ X,

d(T(x), T

(y)) ≤ diam

({x} ∪OT

(y)), (5.9)

d(T(x), T

(y)) ≤ rx(OT

(y))

+ supk,p∈N

(diam

({Tk(x)

}∪OT

(Tk+p

(y))) − diamOT

(Tk+p

(y)))

.

(5.10)

Then T has a fixed point. Moreover, Fix(T ) is closed and convex.



d(u, Tn(x)) for u ∈ X. (5.11)

Since X is a complete CAT(0) space there exists a unique z ∈ X such that

ϕ(z) = infu∈X

ϕ(u). (5.12)

Let lx = limn→∞diamOT (Tn(x)). This limit exists since the sequence is decreasing andbounded below by 0.


Suppose z is not a fixed point of T . Then

lim supn→∞

d(z, Tn(x)) = ϕ(z) < ϕ(T(z)) ≤ limn→∞

diam({z} ∪OT (Tn(x))). (5.13)

This means that

limn→∞

diam({z} ∪OT (Tn(x))) = lx, (5.14)

lim supn→∞

d(T(z), Tn(x)) ≤ lx,

limn→∞

diam({T(z)} ∪OT (Tn(x))) = lx.(5.15)

Inductively, it follows that for k ≥ 0,

lim supn→∞

d(Tk(z), Tn(x)

)≤ lx. (5.16)

Let k, p, n ∈ N and let dk,n = diam({Tk(z)} ∪OT (Tn+k(x))). Obviously,

diam({Tk(z)

}∪OT

(Tn+p+k(x)

))≤ dk,n, (5.17)

since OT (Tn+p+k(x)) ⊆ OT (Tn+k(x)).Because of (5.9) we have

rTk(z)(OT

(Tn+k(x)

))≤ diam

({Tk−1(z)

}∪OT

(Tn+k−1(x)

)). (5.18)

Since diam (OT (Tn+k(x))) ≤ diam(OT (Tn+k−1(x))), it is clear that dk,n ≤ dk−1,n.Hence,

supk∈N

dk,n = diam({z} ∪OT (Tn(x))). (5.19)

Let sn = supk,p∈N(diam({Tk(z)} ∪OT (Tn+p+k(x))) − diamOT (Tn+p+k(x))).Then,

sn ≤ supk∈N

dk,n − infk,p∈N

diamOT

(Tn+p+k(x)

)≤ diam({z} ∪OT (Tn(x))) − lx. (5.20)


Taking into account (5.14), limn→∞sn = 0. Now,

ϕ(T(z)) = lim supn→∞

d(T(z), Tn(x)) ≤ limn→∞

rz(OT

(Tn−1(x)

))+ limn→∞

sn−1

= lim supn→∞

d(z, Tn−1(x)

)= ϕ(z),

(5.21)

which is a contradiction. Hence, T(z) = z.The fact that Fix(T) is closed and convex follows as in the previous proof.

Remark 5.4. It is clear that nonexpansive mappings and mappings for which (5.1) holdssatisfy (5.9) and (5.10). However, there are mappings which satisfy these two conditionswithout verifying (5.1) as the following example shows.

Example 5.5. The set [0, 1] with the usual metric is a CAT(0) space. Let us take T : [0, 1] →[0, 1],

T(x) =

⎧⎪⎪⎨⎪⎪⎩

23x, if x ≥ 1

2,

x

4, if x <

12.

(5.22)

Then T does not satisfy (5.1) but conditions (5.9), (5.10) hold.

Proof. To prove that T does not verify (5.1) we take x = 1/2 and y = 1/4. Then |T(x)−T(y)| =1/3 − 1/16 = 13/48. However,

r1/4

(OT

(12

))=

14<

1348. (5.23)

Next we show that (5.9) and (5.10) hold. We only need to consider the case when x ≥ 1/2 andy < 1/2 because in all the other situations this is evident. Then |T(x) − T(y)| = (2/3)x − y/4.Since

diam({x} ∪OT

(y))

= diam({y} ∪OT (x)

)= x ≥ 2

3x − y

4, (5.24)

relation (5.9) is satisfied.Also,

rx(OT

(y)) ≥ x − y

4≥ 2

3x − y

4,

ry(OT (x)) ≥ x − y.(5.25)

Since supp∈N(diam({y} ∪OT (Tp(x)))−diamOT (Tp(x))) ≥ (3/4)y, we obtain x −y + (3/4)y ≥2/3x − y/4. Hence, relation (5.10) is also accomplished.


Remark 5.6. If we replace condition (5.9) of Theorem 5.3 with

d(T(x), T

(y)) ≤ α(x)diam

({x} ∪OT

(y))


where α : X → [0, 1), then we may conclude that T has s unique fixed point.It is also clear that a pointwise contraction satisfies these conditions so we can apply

this result to prove that it has a unique fixed point.

We next prove a demi-closed principle. We will use the notations introduced at the endof Section 3.

Theorem 5.7. Let X be a CAT(0) space, K ⊆ X, K bounded, closed, and convex. Let T : K → K bea mapping that safisfies (S) and (5.9) for each x, y ∈ K and let (xn)n∈N ⊆ K be an approximate fixed

point sequence such that xnϕ⇀ ω. Then ω ∈ Fix(T ).

Proof. Using (3.1) of Proposition 3.1 we have ϕ(x) = lim supn→∞d(x, T(xn)). Applying (3.2)and (3.3) of Proposition 3.1 for ω,

lim supn→∞

diam({ω} ∪OT (xn)) = lim supn→∞

d(ω, xn). (5.27)

Then,

ϕ(T(ω)) = lim supn→∞

d(T(ω), T(xn)) ≤ lim supn→∞

diam({ω} ∪OT (xn)) = ϕ(ω). (5.28)

Let m denote the midpoint of [ω, T(ω)]. The (CN) inequality yields

d(m,xn)2 ≤ 12d(ω, xn)2 +

12d(T(ω), xn)2 − 1

4d(ω, T(ω))2. (5.29)

Taking the superior limit, we have

ϕ(m)2 ≤ 12ϕ(ω)2 +

12ϕ(T(ω))2 − 1

4d(ω, T(ω))2. (5.30)

But since xnϕ⇀ ω,

14d(ω, T(ω))2 ≤ 1

2ϕ(ω)2 +

12ϕ(ω)2 − ϕ(ω)2 = 0. (5.31)

Hence, ω ∈ Fix(T).

We conclude this paper with the following remarks.

Remark 5.8. All the above results obtained in the context of CAT(0) spaces also hold in themore general setting used in [4] of uniformly convex metric spaces with monotone modulusof convexity.


Remark 5.9. In a similar way as for nonexpansive mappings, one can develop a theory forthe classes of mappings introduced in this section. An interesting idea would be to studythe approximate fixed point property of such mappings. A nice synthesis in the case ofnonexpansive mappings can be found in the recent paper of Kirk [13].

Acknowledgment

The author wishes to thank the financial support provided from programs cofinanced byThe Sectoral Operational Programme Human Resources Development, Contract POS DRU6/1.5/S/3—“Doctoral studies: through science towards society.”

References


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[3] R. Espınola and N. Hussain, “Common fixed points for multimaps in metric spaces,” Fixed PointTheory and Applications, vol. 2010, Article ID 204981, 14 pages, 2010.

[4] R. Espınola, A. Fernandez-Leon, and B. Piatek, “Fixed points of single- and set-valued mappings inuniformly convex metric spaces with no metric convexity,” Fixed Point Theory and Applications, vol.2010, Article ID 169837, 16 pages, 2010.

[5] S. Reich and A. J. Zaslavski, “Well-posedness of fixed point problems,” Far East Journal of MathematicalSciences, Special Volume Part III, pp. 393–401, 2001.

[6] I. A. Rus, “Picard operators and well-posedness of fixed point problems,” Studia Universitatis Babes-Bolyai Mathematica, vol. 52, no. 3, pp. 147–156, 2007.

[7] F. Bruhat and J. Tits, “Groupes reductifs sur un corps locall: I. Donnees radicielles valuees,” Institutdes Hautes Etudes Scientifiques. Publications Mathematiques, no. 41, pp. 5–251, 1972.

[8] S. Dhompongsa, W. A. Kirk, and B. Sims, “Fixed points of uniformly lipschitzian mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 4, pp. 762–772, 2006.

[9] W. A. Kirk, “Geodesic geometry and fixed point theory,” in Seminar of Mathematical Analysis(Malaga/Seville, 2002/2003), D. Girela, G. Lopez, and R. Villa, Eds., vol. 64, pp. 195–225, Universities ofMalaga and Seville, Sevilla, Spain, 2003.

[10] W. A. Kirk, “Geodesic geometry and fixed point theory II,” in Fixed Point Theory and Applications, J.Garcıa-Falset, E. Llorens-Fuster, and B. Sims, Eds., pp. 113–142, Yokohama, Yokohama, Japan, 2004.

[11] W. Walter, “Remarks on a paper by F. Browder about contraction,” Nonlinear Analysis: Theory, Methods& Applications, vol. 5, no. 1, pp. 21–25, 1981.


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Research ArticleHalpern’s Iteration in CAT(0) Spaces

Satit Saejung1, 2

1 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand2 Centre of Excellence in Mathematics, CHE, Sriayudthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Satit Saejung, [email protected]



Copyright q 2010 Satit Saejung. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Motivated by Halpern’s result, we prove strong convergence theorem of an iterative sequence inCAT(0) spaces. We apply our result to find a common fixed point of a family of nonexpansivemappings. A convergence theorem for nonself mappings is also discussed.

1. Introduction

Let (X, d) be a metric space and x, y ∈ X with l = d(x, y). A geodesic path from x to y isan isometry c : [0, l] → X such that c(0) = x and c(l) = y. The image of a geodesic path iscalled a geodesic segment. A metric spaceX is a (uniquely) geodesic space if every two points ofXare joined by only one geodesic segment. A geodesic triangle �(x1, x2, x3) in a geodesic spaceX consists of three points x1, x2, x3 of X and three geodesic segments joining each pair ofvertices. A comparison triangle of a geodesic triangle �(x1, x2, x3) is the triangle �(x1, x2, x3) :=�(x1, x2, x3) in the Euclidean space R

2 such that d(xi, xj) = dR2(xi, xj) for all i, j = 1, 2, 3.A geodesic space X is a CAT(0) space if for each geodesic triangle � := �(x1, x2, x3) in

X and its comparison triangle � := �(x1, x2, x3) in R2, the CAT(0) inequality

d(x, y

) ≤ dR2(x, y

)(1.1)

is satisfied by all x, y ∈ � and x, y ∈ �. The meaning of the CAT(0) inequality is thata geodesic triangle in X is at least thin as its comparison triangle in the Euclidean plane.A thorough discussion of these spaces and their important role in various branches ofmathematics are given in [1, 2]. The complex Hilbert ball with the hyperbolic metric is anexample of a CAT(0) space (see [3]).


The concept of Δ-convergence introduced by Lim in 1976 was shown by Kirk andPanyanak [4] in CAT(0) spaces to be very similar to the weak convergence in Banach spacesetting. Several convergence theorems for finding a fixed point of a nonexpansive mappinghave been established with respect to this type of convergence (e.g., see [5–7]). The purposeof this paper is to prove strong convergence of iterative schemes introduced by Halpern [8]in CAT(0) spaces. Our results are proved under weaker assumptions as were the case inprevious papers and we do not use Δ-convergence. We apply our result to find a commonfixed point of a countable family of nonexpansive mappings. A convergence theorem fornonself mappings is also discussed.

In this paper, we write (1 − t)x ⊕ ty for the the unique point z in the geodesic segmentjoining from x to y such that

d(z, x) = td(x, y

), d

(z, y

)= (1 − t)d(x, y). (1.2)

We also denote by [x, y] the geodesic segment joining from x to y, that is, [x, y] = {(1 − t)x ⊕ty : t ∈ [0, 1]}. A subset C of a CAT(0) space is convex if [x, y] ⊂ C for all x, y ∈ C. Forelementary facts about CAT(0) spaces, we refer the readers to [1] (or, briefly in [5]).

The following lemma plays an important role in our paper.

Lemma 1.1. A geodesic space X is a CAT(0) space if and only if the following inequality

d2((1 − t)x ⊕ ty, z) ≤ (1 − t)d2(x, z) + td2(y, z) − t(1 − t)d2(x, y) (1.3)

is satisfied by all x, y, z ∈ X and all t ∈ [0, 1]. In particular, if x, y, z are points in a CAT(0) spaceand t ∈ [0, 1], then

d((1 − t)x ⊕ ty, z) ≤ (1 − t)d(x, z) + td(y, z). (1.4)

Recall that a continuous linear functional μ on �∞, the Banach space of bounded realsequences, is called a Banach limit if ‖μ‖ = μ(1, 1, . . .) = 1 and μn(an) = μn(an+1) for all {an} ∈�∞.

Lemma 1.2 (see [9, Proposition 2]). Let (a1, a2, . . .) ∈ l∞ be such that μn(an) ≤ 0 for all Banachlimits μ and lim supn(an+1 − an) ≤ 0. Then lim supnan ≤ 0.

Lemma 1.3 (see [10, Lemma 2.3]). Let {sn} be a sequence of nonnegative real numbers, {αn} asequence of real numbers in [0, 1] with

∑∞n=1 αn = ∞, {un} a sequence of nonnegative real numbers

with∑∞

n=1 un <∞, and {tn} a sequence of real numbers with lim supn→∞tn ≤ 0. Suppose that

sn+1 ≤ (1 − αn)sn + αntn + un ∀n ∈ N. (1.5)

Then limn→∞sn = 0.


2. Halpern’s Iteration for a Single Mapping

Lemma 2.1. Let C be a closed convex subset of a complete CAT(0) space X and let T : C → C be anonexpansive mapping. Let u ∈ C be fixed. For each t ∈ (0, 1), the mapping St : C → C defined by

Stx = tu ⊕ (1 − t)Tx for x ∈ C (2.1)

has a unique fixed point xt ∈ C, that is,

xt = Stxt = tu ⊕ (1 − t)Txt. (2.2)

Proof. For x, y ∈ C, we consider the triangle �(u, Tx, Ty) and its comparison triangle and wehave the following:

d(tu ⊕ (1 − t)Tx, tu ⊕ (1 − t)Ty) ≤ dR2

(tu ⊕ (1 − t)Tx, tu ⊕ (1 − t)Ty

)

= (1 − t)dR2

(Tx, Ty

)

= (1 − t)d(Tx, Ty)≤ (1 − t)d(x, y).

(2.3)

This implies that St is a contraction mapping and hence the conclusion follows.

The following result is proved by Kirk in [11, Theorem 26] under the boundednessassumption on C. We present here a new proof which is modified from Kirk’s proof.

Lemma 2.2. Let C, T be as the preceding lemma. Then F(T)/=∅ if and only if {xt} given by theformula (2.2) remains bounded as t → 0. In this case, the following statements hold:

(1) {xt} converges to the unique fixed point z0 of T which is nearest u;

(2) d2(u, z0) ≤ μnd2(u, xn) for all Banach limits μ and all bounded sequences {xn} with xn −Txn → 0.

Proof. If F(T)/=∅, then it is clear that {xt} is bounded. Conversely, suppose that {xt} isbounded. Let {tn} be any sequence in (0, 1) such that limn→∞tn = 0 and define g : C → R by

g(z) = lim supn→∞

d2(xtn , z) (2.4)

for all z ∈ C. By the boundedness of {xtn}, we have δ := inf{g(z) : z ∈ C} < ∞. We choose asequence {zm} in C such that limm→∞g(zm) = δ. It follows from Lemma 1.1 that

d2(xtn ,

12zm ⊕ 1

2zk

)≤ 1

2d2(xtn , zm) +

12d2(xtn , zk) −

14d2(zm, zk). (2.5)


Then, by the convexity of C,

δ ≤ lim supn→∞

d2(xtn ,

12zm ⊕ 1

2zk

)≤ 1

2g(zm) +

12g(zk) − 1

4d2(zm, zk). (2.6)

This implies that {zm} is a Cauchy sequence in C and hence it converges to a point z0 ∈ C.Suppose that z is a point in C satisfying g(z) = δ. It follows then that

δ ≤ lim supn→∞

d2(xtn ,

12z0 ⊕ 1

2z

)≤ 1

2g(z0) +

12g(z) − 1

4d2(z0, z), (2.7)

and hence z = z0. Moreover, z0 is a fixed point of T . To see this, we consider

d(xtn , Txtn) =tn

1 − tn d(u, xtn) −→ 0, (2.8)

and

lim supn→∞

d2(xtn , Tz0) ≤ lim supn→∞

(d(xtn , Txtn) + d(Txtn , Tz0))2

≤ lim supn→∞

(d(xtn , Txtn) + d(xtn , z0))2

= lim supn→∞

d2(xtn , z0) = δ.

(2.9)

This implies that z0 = Tz0 and hence F(T)/=∅.(1) is proved in [12, Theorem 26]. In fact, it is shown that z0 is the nearest point of F(T)

to u. Finally, we prove (2). Suppose that {ztm} is a sequence given by the formula (2.2), where{tm} is a sequence in (0, 1) such that limm→∞tm = 0. We also assume that z0 = limm→∞ztm isthe nearest point of F(T) to u. By the first inequality in Lemma 1.1, we have

d2(xn, ztm) = d2(xn, tmu ⊕ (1 − tm)Tztm)

≤ tmd2(xn, u) + (1 − tm)d2(xn, Tztm) − tm(1 − tm)d2(u, Tztm)

≤ tmd2(xn, u) + (1 − tm)(d(xn, Txn) + d(Txn, Tztm))2 − tm(1 − tm)d2(u, Tztm)

≤ tmd2(xn, u) + (1 − tm)(d(xn, Txn) + d(xn, ztm))2 − tm(1 − tm)d2(u, Tztm).(2.10)

Let μ be a Banach limit. Then

μnd2(xn, ztm) ≤ tmμnd2(xn, u) + (1 − tm)μnd2(xn, ztm) − tm(1 − tm)d2(u, Tztm). (2.11)

This implies that

μnd2(xn, ztm) ≤ μnd2(xn, u) − (1 − tm)d2(u, Tztm). (2.12)


Letting m → ∞ gives

μnd2(xn, z) ≤ μnd2(xn, u) − d2(u, z). (2.13)

In particular,

d2(u, z) ≤ μnd2(xn, u) for all Banach limits μ. (2.14)

Inspired by the results of Wittmann [13] and of Shioji and Takahashi [9], we usethe iterative scheme introduced by Halpern to obtain a strong convergence theorem for anonexpansive mapping in CAT(0) space setting. A part of the following theorem is proved in[14].

Theorem 2.3. Let C be a closed convex subset of a complete CAT(0) space X and let T : C → C be anonexpansive mapping with a nonempty fixed point set F(T). Suppose that u, x1 ∈ C are arbitrarilychosen and {xn} is iteratively generated by

xn+1 = αnu ⊕ (1 − αn)Txn ∀n ≥ 1, (2.15)

where {αn} is a sequence in (0, 1) satisfying

(C1) limn→∞αn = 0;

(C2)∑∞

n=1 αn =∞;

(C3)∑∞

n=1 |αn − αn+1| <∞ or limn→∞(αn/αn+1) = 1.

Then {xn} converges to z ∈ F(T) which is the nearest point of F(T) to u.

Proof. We first show that the sequence {xn} is bounded. Let p ∈ F(T). Then

d(xn+1, p

)= d

(αnu ⊕ (1 − αn)Txn, p

)≤ αnd

(u, p

)+ (1 − αn)d

(Txn, p

)≤ αnd

(u, p

)+ (1 − αn)d

(xn, p

)≤ max

{d(u, p

), d

(xn, p

)}.

(2.16)

By induction, we have

d(xn+1, p

) ≤ max{d(u, p

), d

(x1, p

)}(2.17)

for all n ∈ N. This implies that {xn} is bounded and so is the sequence {Txn}.


Next, we show that d(xn+1, xn) → 0. To see this, we consider the following:

d(xn+1, xn) = d(αnu ⊕ (1 − αn)Txn, αn−1u ⊕ (1 − αn−1)Txn−1)

≤ d(αnu ⊕ (1 − αn)Txn, αnu ⊕ (1 − αn)Txn−1)

+ d(αnu ⊕ (1 − αn)Txn−1, αn−1u ⊕ (1 − αn−1)Txn−1)

≤ (1 − αn)d(Txn, Txn−1) + |αn − αn−1|d(u, Txn−1)

≤ (1 − αn)d(xn, xn−1) + |αn − αn−1|d(u, Txn−1).

(2.18)

By the conditions (C2) and (C3), we have

d(xn+1, xn) −→ 0. (2.19)

Consequently, by the condition (C1),

d(xn, Txn) ≤ d(xn, xn+1) + d(xn+1, Txn)

= d(xn, xn+1) + d(αnu ⊕ (1 − αn)Txn, Txn)= d(xn, xn+1) + αnd(u, Txn) −→ 0.

(2.20)

From Lemma 2.2, let z = limt→ 0xt where xt is given by the formula (2.2). Then z is the nearestpoint of F(T) to u. We next consider the following:

d2(xn+1, z) = d2(αnu ⊕ (1 − αn)Txn, z)

≤ αnd2(u, z) + (1 − αn)d2(Txn, z) − αn(1 − αn)d2(u, Txn)

≤ (1 − αn)d2(xn, z) + αn(d2(u, z) − (1 − αn)d2(u, Txn)

).

(2.21)

By Lemma 2.2, we have μn(d2(u, z) − d2(u, xn)) ≤ 0 for all Banach limits μ. Moreover, sincexn+1 − xn → 0,

lim supn→∞

(d2(u, z) − d2(u, xn)

)−(d2(u, z) − d2(u, xn+1)

)= 0. (2.22)

It follows from xn − Txn → 0 and Lemma 1.2 that

lim supn→∞

(d2(u, z) − (1 − αn)d2(u, Txn)

)= lim sup

n→∞

(d2(u, z) − d2(u, xn)

)≤ 0. (2.23)

Hence the conclusion follows by Lemma 1.3.


3. Halpern’s Iteration for a Family of Mappings

3.1. Finitely Many Mappings

We use the “cyclic method” [15] and Bauschke’s condition [16] to obtain the following strongconvergence theorem for a finite family of nonexpansive mappings.

Theorem 3.1. Let X be a complete CAT(0) space and C a closed convex subset of X. LetT1, T2, . . . , TN : C → C be nonexpansive mappings with

⋂Ni=1 F(Ti)/=∅ and let u, x1 ∈ C be

arbitrarily chosen. Define an iterative sequence {xn} by

xn+1 = αnu ⊕ (1 − αn)Tn mod Nxn ∀n ≥ 1, (3.1)


(C1) limn→∞αn = 0;

(C2)∑∞

n=1 αn =∞;

(C3)∑∞

n=1 |αn − αn+N | <∞ or limn→∞(αn/αn+N) = 1.

Suppose, in addition, that

N⋂i=1

F(Ti) = F(TN ◦ TN−1 ◦ · · · ◦ T1). (3.2)

Then {xn} converges to z ∈⋂Ni=1 F(Ti) which is nearest u.

Here the modN function takes values in {1, 2, . . . ,N}.

Proof. By [16, Theorem 2], we have

N⋂i=1

F(Ti) = F(T1 ◦ TN ◦ TN−1 ◦ · · · ◦ T2) = · · · = F(TN−1 ◦ TN ◦ T1 ◦ · · · ◦ TN−2). (3.3)

The proof line now follows from the proofs of Theorem 2.3 and [15, Theorem 3.1].

3.2. Countable Mappings

The following concept is introduced by Aoyama et al. [10]. Let X be a complete CAT(0) spaceand C a subset of X. Let {Tn}∞n=1 be a countable family of mappings from C into itself. We saythat a family {Tn} satisfies AKTT-condition if

∞∑n=1

sup{d(Tn+1z, Tnz) : z ∈ B} <∞ (3.4)

for each bounded subset of B of C.


IfC is a closed subset and {Tn} satisfies AKTT-condition, then we can define T : C → Csuch that

Tx = limn→∞

Tnx (x ∈ C). (3.5)

In this case, we also say that ({Tn}, T) satisfies AKTT-condition.

Theorem 3.2. Let X be a complete CAT(0) space and C a closed convex subset of X. Let {Tn} : C →C be a countable family of nonexpansive mappings with

⋂∞n=1 F(Tn)/=∅. Suppose that u, x1 ∈ C are

arbitrarily chosen and {xn} is defined by

xn+1 = αnu ⊕ (1 − αn)Tnxn ∀n ≥ 1, (3.6)


(C1) limn→∞αn = 0;

(C2)∑∞

n=1 αn =∞;

(C3)∑∞


Suppose, in addition, that

(M1) ({Tn}, T) satisfies AKTT-condition;(M2) F(T) =

⋂∞n=1 F(Tn).

Then {xn} converges to z ∈⋂∞n=1 F(Tn) which is nearest u.

Proof. Since the proof of this theorem is very similar to that of Theorem 2.3, we present hereonly the sketch proof. First, we notice that both sequences {xn} and {Tnxn} are bounded and

d(xn+1, xn) = d(αnu ⊕ (1 − αn)Tnxn, αn−1u ⊕ (1 − αn−1)Tn−1xn−1)

≤ d(αnu ⊕ (1 − αn)Tnxn, αnu ⊕ (1 − αn)Tnxn−1)

+ d(αnu ⊕ (1 − αn)Tnxn−1, αnu ⊕ (1 − αn)Tn−1xn−1)

+ d(αnu ⊕ (1 − αn)Tn−1xn−1, αn−1u ⊕ (1 − αn−1)Tn−1xn−1)

≤ (1 − αn)d(Tnxn, Tnxn−1) + (1 − αn)d(Tnxn−1, Tn−1xn−1)

+ |αn − αn−1|d(u, Tn−1xn−1)

≤ (1 − αn)d(xn, xn−1) + d(Tnxn−1, Tn−1xn−1)

+ |αn − αn−1|d(u, Tn−1xn−1)

≤ (1 − αn)d(xn, xn−1) + |αn − αn−1|d(u, Tn−1xn−1)

+ sup{d(Tny, Tn−1y

): y ∈ {xn}

}.

(3.7)

By conditions (C2), (C3), AKTT-condition, and Lemma 1.3, we have

d(xn+1, xn) −→ 0. (3.8)


Consequently, d(xn, Tnxn) → 0 and hence

d(xn, Txn) ≤ d(xn, Tnxn) + d(Tnxn, Txn)≤ d(xn, Tnxn) + sup{d(Tnz, Tz) : z ∈ {xn}}

≤ d(xn, Tnxn) +∞∑k=n

sup{d(Tkz, Tk+1z) : z ∈ {xn}} −→ 0.

(3.9)

Let z ∈ F(T) = ⋂∞n=1 F(Tn) be the nearest point of F(T) to u. As in the proof of Theorem 2.3,

we have d2(u, z) ≤ μnd2(u, xn) for all Banach limits μ and lim supn→∞(d2(u, z) − d2(u, xn)) −

(d2(u, z) − d2(u, xn+1)) = 0. We observe that

d2(xn+1, z) = d2(αnu ⊕ (1 − αn)Tnxn, z)

≤ αnd2(u, z) + (1 − αn)d2(Tnxn, z) − αn(1 − αn)d2(u, Tnxn)

≤ (1 − αn)d2(xn, z) + αn(d2(u, z) − (1 − αn)d2(u, Tnxn)

),

(3.10)

and this implies that

lim supn→∞

(d2(u, z) − (1 − αn)d2(u, Tnxn)

)= lim sup

n→∞

(d2(u, z) − d2(u, xn)

)≤ 0. (3.11)

Therefore, limn→∞d2(xn, z) = 0 and hence {xn} converges to z.

We next show how to generate a family of mappings from a given family of mappingsto satisfy conditions (M1) and (M2) of the preceding theorem. The following is an analogueof Bruck’s result [17] in CAT(0) space setting. The idea using here is from [10].

Theorem 3.3. Let X be a complete CAT(0) space and C a closed convex subset of X. Suppose that{Tn} : C → X is a countable family of nonexpansive mappings with

⋂∞n=1 F(Tn)/=∅. Then there

exist a family of nonexpansive mappings {Sn} : C → X and a nonexpansive mapping S : C → Xsuch that

(M1) ({Sn}, S) satisfies AKTT-condition;

(M2) F(S) =⋂∞n=1 F(Tn).

Lemma 3.4. Let X and C be as above. Suppose that S, T : C → X are nonexpansive mappings andF(S) ∩ F(T)/=∅. Then, for any 0 < t < 1, the mappingU := (1 − t)S ⊕ tT : C → X is nonexpansiveand F(U) = F(S) ∩ F(T).

Proof. To see that U is nonexpansive, we only apply the triangle inequality and twoapplications of the second inequality in Lemma 1.1. We next prove the latter. It is clear that


F(S) ∩ F(T) ⊂ F(U). To see the reverse inclusion, let p ∈ F(U) and q ∈ F(S) ∩ F(T). Then, bythe first inequality of Lemma 1.1,

d2(q, p) = d2(q,Up)

= d2(q, (1 − t)Sp ⊕ tTp)

≤ (1 − t)d2(q, Sp) + td2(q, Tp) − t(1 − t)d2(Sp, Tp)

≤ d2(q, p) − t(1 − t)d2(Sp, Tp).

(3.12)

This implies Sp = Tp. As p = Up, we have p ∈ F(S) ∩ F(T), as desired.

Proof of Theorem 3.3. We first define a family of mappings {Sn} : C → X by

S1x =12x ⊕ 1

2T1x

S2x =22 − 1

22S1x ⊕ 1

22T2x

...

Snx =2n − 1

2nSn−1x ⊕ 1

2nTnx

...

(3.13)

By Lemma 3.4, each Sn is a nonexpansive mapping satisfying F(Sn) =⋂nk=1 F(Tk). Notice that,

for fixed p ∈ ⋂∞n=1 F(Tn),

d2(Sn+1x, Snx) = d2

(2n+1 − 1

2n+1Snx ⊕ 1

2n+1Tn+1x, Snx

)

=1

2n+1d2(Tn+1x, Snx)

=1

2n+1

(d(Tn+1x, p) + d(p, Snx)

)2

≤ 12n−1

d2(x, p).

(3.14)

From the estimation above, we have

∞∑n=1

sup{d(Sn+1x, Snx) : x ∈ B} <∞ (3.15)


for each bounded subset B of C. In particular, {Snx} is a Cauchy sequence for each x ∈ C. Wenow define the nonexpansive mapping S : C → X by

Sx = limn→∞

Snx. (3.16)

Finally, we prove that

F(S) =∞⋂n=1

F(Sn) =∞⋂n=1

F(Tn). (3.17)

The latter equality is clearly verified and⋂∞n=1 F(Sn) ⊂ F(S) holds. On the other hand, let

p ∈ F(S) and q ∈ ⋂∞n=1 F(Tn). We consider the following:

d2(q, Snp) = d2(q,

2n − 12n

Sn−1p ⊕ 12nTnp

)

≤ 2n − 12n

d2(q, Sn−1p)+

12nd2(q, Tnp)

≤ 2n − 12n

d2(q, Sn−1p)+

12nd2(q, p).

(3.18)

Then

d2(q, Snp) ≤(

n∏k=2

2k − 12k

)d2(q, S1p

)+

(1 −

n∏k=2

2k − 12k

)d2(q, p)

≤(

n∏k=2

2k − 12k

)(12d2(q, p) + 1

2d2(q, T1p

) − 14d2(p, T1p

))

+

(1 −

n∏k=2

2k − 12k

)d2(q, p)

≤(

n∏k=2

2k − 12k

)(d2(q, p) − 1

4d2(p, T1p

))+

(1 −

n∏k=2

2k − 12k

)d2(q, p).

(3.19)

Letting n → ∞ yields

d2(q, p) ≤( ∞∏

k=2

2k − 12k

)(d2(q, p) − 1

4d2(p, T1p

))+

(1 −

∞∏k=2

2k − 12k

)d2(q, p). (3.20)


Because∏∞

k=2((2k − 1)/2k) > 0, we have p = T1p. Continuing this procedure we obtain that

p ∈ ⋂∞n=1 F(Tn) and hence F(S) ⊂ ⋂∞

n=1 F(Tn). This completes the proof.

4. Nonself Mappings

From Bridson and Haefliger’s book (page 176), the following result is proved.

Theorem 4.1. Let X be a complete CAT(0) space and C a closed convex subset of X. Then thefollowings hold true.

(i) For each x ∈ X, there exists an element π(x) ∈ C such that

d(x, π(x)) = dist(x,C). (4.1)

(ii) π(x) = π(x′) for all x′ ∈ [x, π(x)].

(iii) The mapping x �→ π(x) is nonexpansive.

The mapping π in the preceding theorem is called the metric projection from X onto C.From this, we have the following result.

Theorem 4.2. Let X be a complete CAT(0) space and C a closed convex subset of X. Let T : C → Xbe a nonself nonexpansive mapping with F(T)/=∅ and π : X → C the metric projection from X ontoC. Then the mapping π ◦ T is nonexpansive and F(π ◦ T) = F(T).

Proof. It follows from Theorem 4.1 that π ◦ T is nonexpansive. To see the latter, it suffices toshow that F(π ◦ T) ⊂ F(T). Let p ∈ F(π ◦ T) and q ∈ F(T). Since

d2(q, p) = d2(π(q), π

(12Tp ⊕ 1

2p

))

≤ d2(q,

12Tp ⊕ 1

2p

)

≤ 12d2(q, Tp) + 1

2d2(q, p) − 1

4d2(Tp, p)

≤ d2(q, p) − 14d2(Tp, p),

(4.2)

we have p = Tp and this finishes the proof.

By the preceding theorem and Theorem 2.3, we obtain the following result.

Theorem 4.3. Let X, C, T : C → X, and π : X → C be as the same as Theorem 4.2. Suppose thatu, x1 ∈ C are arbitrarily chosen and the sequence {xn} is defined by

xn+1 = αnu ⊕ (1 − αn)(π ◦ Txn) ∀n ≥ 1, (4.3)



(C1) limn→∞αn = 0;

(C2)∑∞

n=1 αn =∞;

(C3)∑∞


Then {xn} converges to z ∈ F(T) which is nearest u.

Acknowledgments

The author would like to thank the referee for the information that a part of Theorem 2.3 wasproved in [14]. This work was supported by the Centre of Excellence in Mathematics, theCommission on Higher Education, Thailand.

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Research ArticleEquivalent Extensions to Caristi-Kirk’s FixedPoint Theorem, Ekeland’s Variational Principle,and Takahashi’s Minimization Theorem

Zili Wu

Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren Ai Road,Dushu Lake Higher Education Town, Suzhou Industrial Park, Suzhou, Jiangsu 215123, China

Correspondence should be addressed to Zili Wu, [email protected]



Copyright q 2010 Zili Wu. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

With a recent result of Suzuki (2001) we extend Caristi-Kirk’s fixed point theorem, Ekeland’svariational principle, and Takahashi’s minimization theorem in a complete metric space byreplacing the distance with a τ-distance. In addition, these extensions are shown to be equivalent.When the τ-distance is l.s.c. in its second variable, they are applicable to establish more equivalentresults about the generalized weak sharp minima and error bounds, which are in turn useful forextending some existing results such as the petal theorem.

1. Introduction

Let (X, d) be a complete metric space and f : X → (−∞,+∞] a proper lower semicontinuous(l.s.c.) bounded below function. Caristi-Kirk fixed point theorem [1, Theorem (2.1)′] statesthat there exists x0 ∈ Tx0 for a relation or multivalued mapping T : X → X if for each x ∈ Xwith infXf < f(x) there exists x ∈ Tx such that

d(x, x) + f(x) ≤ f(x), (1.1)

(see also [2, Theorem 4.12] or [3, Theorem C]) while Ekeland’s variational principle (EVP)[4, 5] asserts that for each ε ∈ (0,+∞) and u ∈ X with f(u) ≤ infXf + ε, there exists v ∈ Xsuch that f(v) ≤ f(u) and

f(x) + εd(v, x) > f(v) ∀x ∈ X with x /=v. (1.2)

EVP has been shown to have many equivalent formulations such as Caristi-Kirkfixed point theorem, the drop theorem [6], the petal theorem [3, Theorem F], Takahashi


minimization theorem [7, Theorem 1], and two results about weak sharp minima and errorbounds [8, Theorems 3.1 and 3.2]. Moreover, in a Banach space, it is equivalent to the Bishop-Phelps theorem (see [9]). EVP has played an important role in the study of nonlinear analysis,convex analysis, and optimization theory. For more applications, EVP and several equivalentresults stated above have been extended by introducing more general distances. For example,Kada et al. have presented the concept of a w-distance in [10] to extend EVP, Caristi’s fixedpoint theorem, and Takahashi minimization theorem. Suzuki has extended these three resultsby replacing a w-distance with a τ-distance in [11]. For more extensions of these theorems,with a w-distance being replaced by a τ-function and a Q-function, respectively, the reader isreferred to [12, 13].

Theoretically, it is interesting to reveal the relationships among the above existingresults (or their extensions). In this paper, while further extending the above theorems ina complete metric space with a τ-distance, we show that these extensions are equivalent. Forthe case where the τ-distance is l.s.c. in its second variable, we apply our generalizationsto extend several existing results about the weak sharp minima and error bounds and thendemonstrate their equivalent relationship. In particular, when the τ-distance reduces to thecomplete metric, our results turn out to be equivalent to EVP and hence to its existingequivalent formulations.

2. w-Distance and τ-Distance

For convenience, we recall the concepts of w-distance and τ-distance and some propertieswhich will be used in the paper.

Definition 2.1 (see [10]). Let (X, d) be a metric space. A function p : X×X → [0,+∞) is calleda w-distance on X if the following are satisfied:

(ω1) p(x, z) ≤ p(x, y) + p(y, z) for all (x, y, z) ∈ X ×X ×X;

(ω2) for each x ∈ X, p(x, ·) : X → [0,+∞) is l.s.c.;

(ω3) for each ε > 0 there exists δ > 0 such that

p(z, x) ≤ δ, p(z, y) ≤ δ =⇒ d

(x, y) ≤ ε. (2.1)

From the definition, we see that the metric d is a w-distance on X. If X is a normedlinear space with norm ‖ · ‖, then both p1 and p2 defined by

p1(x, y)=∥∥y∥∥, p2

(x, y)= ‖x‖ + ∥∥y∥∥ ∀(x, y) ∈ X ×X (2.2)

are w-distances on X. Note that p1(x, x)/= 0/= p2(x, x) for each x ∈ X with x /= 0. For moreexamples, we see [10].

It is easy to see that for any α ∈ (0, 1) and w-distance p, the function αp is also aw-distance. For any positive M and w-distance p on X, the function pM defined by

pM(x, y)

:= min{p(x, y),M} ∀(x, y) ∈ X ×X (2.3)

is a bounded w-distance on X.


The following proposition shows that we can construct another w-distance from agiven w-distance under certain conditions.

Proposition 2.2. Let x0 ∈ X, p a w-distance on X, and h : [0,+∞) → [0,+∞) a nondecreasingfunction. If, for each r > 0,

infx∈X

∫p(x0,x)+r

p(x0,x)

dt

1 + h(t)> 0, (2.4)

then the function q defined by

q(x, y)

:=∫p(x0,x)+p(x,y)

p(x0,x)

dt

1 + h(t)for(x, y) ∈ X ×X (2.5)

is a w-distance. In particular, if p is bounded on X ×X, then q is a w-distance.

Proof. Since h is nondecreasing, for (x, z) ∈ X ×X,

q(x, z) =∫p(x0,x)+p(x,z)

p(x0,x)

dt

1 + h(t)≤∫p(x0,x)+p(x,y)+p(y,z)

p(x0,x)

dt

1 + h(t)

=∫p(x0,x)+p(x,y)

p(x0,x)

dt

1 + h(t)+∫p(x0,x)+p(x,y)+p(y,z)

p(x0,x)+p(x,y)

dt

1 + h(t)

≤∫p(x0,x)+p(x,y)

p(x0,x)

dt

1 + h(t)+∫p(x0,y)+p(y,z)

p(x0,y)

dt

1 + h(t)

= q(x, y)+ q(y, z).

(2.6)

In addition, q is obviously lower semicontinuous in its second variable.Now, for each ε > 0, there exists δ1 > 0 such that

p(z, x) ≤ δ1, p(z, y) ≤ δ1 =⇒ d

(x, y) ≤ ε. (2.7)

Taking δ such that

0 < δ < infx∈X

∫p(x0,x)+δ1

p(x0,x)

dt

1 + h(t), (2.8)

we obtain that, for x, y, z in X with q(z, x) ≤ δ and q(z, y) ≤ δ,

q(z, x) =∫p(x0,z)+p(z,x)

p(x0,z)

dt

1 + h(t)≤ δ <

∫p(x0,z)+δ1

p(x0,z)

dt

1 + h(t), (2.9)


from which it follows that p(z, x) ≤ δ1. Similarly, we have p(z, y) ≤ δ1. Thus d(x, y) ≤ ε.Therefore, q is a w-distance on X.

Next, if p is bounded on X ×X, then there exists M > 0 such that

∫p(x0,x)+r

p(x0,x)

dt

1 + h(t)≥ r

1 + h(M + r)> 0 ∀x ∈ X. (2.10)

Thus q is also a w-distance on X.

When p is unbounded on X ×X, the condition in Proposition 2.2 may not be satisfied.However, if h is a nondecreasing function satisfying

∫+∞

0

dt

1 + h(t)= +∞, (2.11)

then the function q in Proposition 2.2 is a τ-distance (see [11, Proposition 4]), a more generaldistance introduced by Suzuki in [11] as below.

Definition 2.3 (see [11]). p : X ×X → [0,+∞) is said to be a τ-distance on X provided that

(τ1) p(x, z) ≤ p(x, y) + p(y, z) for all (x, y, z) ∈ X × X × X and there exists a functionη : X × [0,+∞) → [0,+∞) such that

(τ2) η(x, 0) = 0 and η(x, t) ≥ t for all (x, t) ∈ X×[0,+∞), and η is concave and continuousin its second variable;

(τ3) limn→+∞xn = x and limn→+∞ sup{η(zn, p(zn, xm)) : n ≤ m} = 0 imply

p(w,x) ≤ lim infn→+∞

p(w,xn) ∀w ∈ X; (2.12)

(τ4) limn→+∞ sup{p(xn, ym) : n ≤ m} = 0 and limn→+∞η(xn, tn) = 0 imply

limn→+∞

η(yn, tn

)= 0; (2.13)

(τ5) limn→+∞η(zn, p(zn, xn)) = 0 and limn→+∞η(zn, p(zn, yn)) = 0 imply

limn→+∞

d(xn, yn

)= 0. (2.14)

Suzuki has proved that a w-distance is a τ-distance [11, Proposition 4]. If a τ-distancep satisfies p(z, x) = 0 and p(z, y) = 0 for (x, y, z) ∈ X ×X ×X, then x = y (see [11, Lemma 2]).For more properties of a τ-distance, the reader is referred to [11].

3. Fixed Point Theorems

From now on, we assume that (X, d) is a complete metric space and f : X → (−∞,+∞] is aproper l.s.c. and bounded below function unless specified otherwise. In this section, mainly


motivated by fixed point theorems (for a single-valued mapping) in [10, 11, 14–16], wepresent two similar results which are applicable to multivalued mapping cases. The followingtheorem established by Suzuki’s in [11] plays an important role in extending existing resultsfrom a single-valued mapping to a multivalued mapping.

Theorem 3.1 (see [11, Proposition 8]). Let p be a τ-distance on X. Denote

M(x) :={y ∈ X : p

(x, y)+ f(y) ≤ f(x)} ∀x ∈ X. (3.1)

Then for each u ∈ X withM(u)/= ∅, there exists x0 ∈M(u) such thatM(x0) ⊆ {x0}. In particular,there exists y0 ∈ X such thatM(y0) ⊆ {y0}.

Based on Theorem 3.1, [11, Theorem 3] asserts that a single-valued mapping T : X →X has a fixed point x0 in X when Tx ∈ M(x) holds for all x ∈ X (which generalizes [10,Theorem 2] by replacing a w-distance with a τ-distance). We show that the conclusion can bestrengthened under a slightly weaker condition (in which Tx ∩M(x)/= ∅ holds on a subset ofX instead) for a multivalued mapping T.

Theorem 3.2. Let p be a τ-distance on X and T : X → X a multivalued mapping. Suppose that forsome ε ∈ (0,+∞] there holds Tx ∩M(x)/= ∅ for each x ∈ X with infXf ≤ f(x) < infXf + ε. Thenthere exists x0 ∈ X such that

{x0} =M(x0) ={x ∈M(x0) : x ∈ Tx, p(x, x) = 0, inf

Xf ≤ f(x) < inf

Xf + ε

}, (3.2)

whereM(x0) := {y ∈ X : p(x0, y) + f(y) ≤ f(x0)}.

Proof. For each x ∈ X with infXf ≤ f(x) < infXf + ε, the set

Mx :={y ∈ X : f

(y) ≤ f(x)} (3.3)

is a nonempty closed subset of X since f is lower semicontinuous and

x ∈M(x) :={y ∈ X : p

(x, y)+ f(y) ≤ f(x)} ⊆Mx (3.4)

for some x ∈ Tx. Thus (Mx, d) is a complete metric space. By Theorem 3.1, there exists x0 ∈M(x) such that M(x0) ⊆ {x0}. Since

infXf ≤ f(x0) ≤ f(x) < inf

Xf + ε, (3.5)

there exists x0 ∈ Tx0 such that x0 ∈M(x0). Thus M(x0) = {x0}, x0 = x0 ∈ Tx0, and

0 ≤ p(x0, x0) = p(x0, x0) ≤ f(x0) − f(x0) = 0. (3.6)


Clearly, [8, Thoerem 4.1] follows as a special case of Theorem 3.2 with p = d. Inaddition, when ε = +∞ and T is a single-valued mapping, Theorem 3.2 contains [11, Theorem3]. The following simple example further shows that Theorem 3.2 is applicable to more cases.

Example 3.3. Consider the mapping T : [0,+∞) → [0,+∞) defined by

Tx =

⎧⎪⎨⎪⎩

[x − x2, x − 1

2x2)

for x ∈ [0, 1);{x + x2} for x ∈ [1,+∞)

(3.7)

and the function f(x) = 2√x for x ∈ [0,+∞). Obviously f(0) = inf[0,+∞)f . For any ε ∈ (0, 1],

x ∈ [0, ε), and y ∈ [0, x], we have

∣∣x − y∣∣ = x − y =(√

x +√y)(√

x −√y) ≤ f(x) − f(y), (3.8)

so, applying Theorem 3.2 to the above T and f with p(x, y) = |x − y| for x, y ∈ X := [0,+∞),we obtain x0 ∈ X as in Theorem 3.2.

Motivated by [16, Theorem 7] and [14, Theorem 2.3], we further extend Theorem 3.2as follows.

Theorem 3.4. Let p be a τ-distance on X and T : X → X a multivalued mapping. Let ε ∈ (0,+∞]and ϕ : f−1(−∞, infXf + ε] → [0,+∞) satisfy

γ := sup{ϕ(x) : x ∈ f−1

(−∞, inf

Xf + min

{ε, η}]}

< +∞, (3.9)

for some η > 0. If for each x ∈ X with infXf ≤ f(x) < infXf + ε, there exists x ∈ Tx such that

f(x) ≤ f(x), p(x, x) ≤ ϕ(x)[f(x) − f(x)], (3.10)

then there exists x0 ∈ X such that

{x0} =Mγ(x0) ={x ∈Mγ(x0) : x ∈ Tx, p(x, x) = 0, inf

Xf ≤ f(x) < inf

Xf + ε

}, (3.11)

whereMγ(x0) := {y ∈ X : p(x0, y) ≤ (γ + 1)[f(x0) − f(y)]}.

Proof. For each x ∈ X with infXf ≤ f(x) < infXf + min{ε, η}, by assumption, there existsx ∈ Tx such that

p(x, x) ≤ ϕ(x)[f(x) − f(x)] ≤ (γ + 1)[f(x) − f(x)], (3.12)


based on the inequalities 0 ≤ ϕ(x) and f(x) ≤ f(x). Upon applying Theorem 3.2 to the lowersemicontinuous function (γ + 1)f on f−1(−∞, infXf + ε] which is complete, we arrive at theconclusion.

Next result is immediate from Theorem 3.4.

Theorem 3.5. Let p be a τ-distance on X, g : [infXf, infXf + ε] → [0,+∞) either nondecreasingor upper semicontinuous (u.s.c.), and T : X → X a multivalued mapping. If for some ε ∈ (0,+∞]and each x ∈ X with infXf ≤ f(x) < infXf + ε, there exists x ∈ Tx such that

f(x) ≤ f(x), p(x, x) ≤ g(f(x))[f(x) − f(x)], (3.13)

then there exists x0 ∈ X such that

{x0} =Mγ(x0) ={x ∈Mγ(x0) : x ∈ Tx, p(x, x) = 0, inf

Xf ≤ f(x) < inf

Xf + ε

}, (3.14)

whereMγ(x0) := {y ∈ X : p(x0, y) ≤ (γ + 1)[f(x0) − f(y)]} with

γ := sup{g(s) : inf

Xf ≤ s ≤ inf

Xf + min{ε, 1}

}. (3.15)

Proof. For x ∈ f−1(−∞, infXf + ε], define ϕ(x) = g(f(x)). Then for the case where g isnondecreasing we have

sup{ϕ(x) : x ∈ f−1

(−∞, inf

Xf + min{ε, 1}

]}≤ g(

infXf + min{ε, 1}

)< +∞. (3.16)

Thus the conclusion follows from Theorem 3.4.For the case where g is u.s.c., we define c : [infXf, infXf + ε] → [0,+∞) by c(t) :=

sup{g(s) : infXf ≤ s ≤ t}. Since g is u.s.c., c is well defined and nondecreasing. Now, forsome ε ∈ (0,+∞] and each x ∈ X with infXf ≤ f(x) < infXf + ε there exists x ∈ Tx satisfying

f(x) ≤ f(x), p(x, x) ≤ g(f(x))[f(x) − f(x)] ≤ c(f(x))[f(x) − f(x)], (3.17)

so we can apply the conclusion in the previous paragraph to c to get the same conclusion.

Remark 3.6. When ε = +∞ and T is a single-valued mapping, Theorem 3.4 reduces to [16,Theorem 7] while Theorem 3.5 to [16, Theorems 8 and 9]. If also p(x, y) = d(x, y) for all(x, y) ∈ X × X, then Theorem 3.5 reduces to [14, Theorem 2.3] (when g is nondecreasing)and [15, Theorem 3] (when g is upper semicontinuous). In the later case, it also extends [14,Theorem 2.4].

Furthermore, we will see that the relaxation of T from a single-valued mapping (as inseveral existing results stated before) to a multivalued one (as in Theorems 3.2–3.5) is morehelpful for us to obtain more results in the next section.


4. Extensions of Ekeland’s Variational Principle

As applications of Theorems 3.4 and 3.5, several generalizations of EVP will be presented inthis section.

Theorem 4.1. Let p be a τ-distance on X, ε ∈ (0,+∞], u ∈ X satisfy f(u) ≤ infXf + ε, andϕ : f−1(−∞, infXf + ε] → (0,+∞) satisfy

sup{ϕ(x) : x ∈ f−1

(−∞, inf

Xf + min

{ε, η}]}

< +∞, (4.1)

for some η > 0. Then there exists v ∈ X such that f(v) ≤ f(u) and

p(v, x) > ϕ(v)[f(v) − f(x)] ∀x ∈ X with x /=v. (4.2)

Proof. Take Mu := {x ∈ X : f(x) ≤ f(u)}. Then (Mu, d) is a nonempty complete metric space.We claim that there must exist v ∈Mu such that

p(v, x) > ϕ(v)[f(v) − f(x)] ∀x ∈Mu with x /=v. (4.3)

Otherwise for each x ∈Mu the set

Tx :=

⎧⎨⎩{y ∈Mu : y /=x, p

(x, y) ≤ ϕ(x)[f(x) − f(y)]} if f(x) < +∞;

Mu \ {x} if f(x) = +∞(4.4)

would be nonempty and x /∈ Tx. As a mapping from Mu to Mu, T satisfies the conditions inTheorem 3.4, so there exists x0 ∈Mu such that x0 ∈ Tx0. This is a contradiction.

Now, for each x ∈ X \Mu, since f(x) > f(u) ≥ f(v) and p(v, x) ≥ 0, inequality (4.3)still holds.

It is worth noting that T in the above proof is a multivalued mapping to whichTheorem 3.4 is directly applicable, in contrast to [11, Theorem 3] and [16, Theorem 7].

From the proof of Theorem 3.5, we see that the function ϕ defined by

ϕ(x) := sup{g(s) : inf

Xf ≤ s ≤ f(x)

}(4.5)

satisfies the condition in Theorem 4.1 when g : [infXf, infXf + ε] → (0,+∞) is anondecreasing or u.s.c. function. So, based on Theorem 4.1 or Theorem 3.5, we obtain nextresult (from which [11, Theorem 4] follows by taking g = 1).

Theorem 4.2. Let p be a τ-distance on X, ε ∈ (0,+∞], u ∈ X satisfy f(u) ≤ infXf + ε, andg : [infXf, infXf + ε] → (0,+∞) either nondecreasing or u.s.c.. Denote


Xf ≤ s ≤ f(x)

}for x ∈ f−1

(−∞, inf

Xf + ε

]. (4.6)


Then there exists v ∈ X such that f(v) ≤ f(u) and

p(v, x) > g(f(v)

)[f(v) − f(x)] ∀x ∈ X with x /=v. (4.7)

If also p(u, u) = 0 and p, is l.s.c. in its second variable, then there exists v ∈ X satisfying the aboveproperty and the following inequality:

p(u, v) ≤ ϕ(u)[f(u) − f(v)]. (4.8)

Proof. Similar to the proof of Theorem 4.1, the first part of the conclusion can be derived fromTheorem 3.5.

Now, let p(u, u) = 0 and p l.s.c. in its second variable. Then the set

M(u) :={x ∈ X : p(u, x) + ϕ(u)f(x) ≤ ϕ(u)f(u)} (4.9)

is nonempty and complete. Note that c(t) := sup{g(s) : infXf ≤ s ≤ t} is nondecreasing andϕ(x) = c(f(x)). Applying the conclusion of the first part to the function f onM(u), we obtainv ∈M(u) such that

p(v, x) > ϕ(v)[f(v) − f(x)] (4.10)

for all x ∈M(u) with x /=v. For x ∈ X \M(u), we still have the inequality. Otherwise, therewould exist x ∈ X \M(u) such that f(x) ≤ f(v) and

p(v, x) ≤ ϕ(v)[f(v) − f(x)]. (4.11)

This with v ∈M(u) and the triangle inequality yield

p(u, x) ≤ ϕ(u)[f(u) − f(v)] + ϕ(v)[f(v) − f(x)]≤ ϕ(u)[f(u) − f(x)], (4.12)

that is, x ∈M(u), which is a contradiction.

Remark 4.3. (i) For the case where g is nondecreasing, the function ϕ(x) in the proof ofTheorem 4.2 reduces to g(f(x)). From the proof we can further see that the nonemptinessand the closedness of M(u) imply the existence of v in M(u) such that M(v) ⊆ {v}.

(ii) If we apply Theorem 4.1 directly, then the factor g(f(v)) on the right-hand side ofthe inequality

p(v, x) > g(f(v)

)[f(v) − f(x)] (4.13)

in Theorem 4.2 can be replaced with ϕ(v).


(iii) When x0 ∈ X, p is a w-distance on X, and h is a nondecreasing function such that

∫+∞

0

dt

1 + h(t)= +∞, (4.14)

applying Theorem 4.2 to the τ-distance

∫p(x0,x)+p(x,y)

p(x0,x)

dt

1 + h(t)for(x, y) ∈ X ×X (4.15)

and g(t) = λ/ε, we arrive at the following conclusion, from which (by taking p = d) we canobtain [17, Theorem 1.1], a generalization of EVP.

Corollary 4.4. Let x0 ∈ X, p a w-distance on X, ε > 0 and u ∈ X satisfy p(u, u) = 0 and f(u) ≤infXf + ε. Let h : [0,+∞) → [0,+∞) be a nondecreasing function such that

∫+∞

0

dt

1 + h(t)= +∞. (4.16)

Then for each λ > 0, there exists v ∈ X such that f(v) ≤ f(u),

∫p(x0,u)+p(u,v)

p(x0,u)

dt

1 + h(t)≤ λ,

f(x) +ε

λ· p(v, x)

1 + h(p(x0, v)

) > f(v) ∀x ∈ X with x /=v.

(4.17)

Note that there exist nondecreasing functions h satisfying

∫+∞

0

dt

1 + h(t)< +∞. (4.18)

For example, h(t) = t2 and h(t) = et. Clearly, Corollary 4.4 is not applicable to theseexamples. For these cases, we present another extension of EVP by using Theorem 4.1 andProposition 2.2.

Theorem 4.5. Let p be a w-distance on X, ε ∈ (0,+∞], u ∈ X satisfy f(u) ≤ infXf + ε, andϕ : f−1(−∞, infXf + ε] → (0,+∞) satisfying

sup{ϕ(x) : x ∈ f−1

(−∞, inf

Xf + min

{ε, η}]}

< +∞, (4.19)


for some η > 0. If h : [0,+∞) → [0,+∞) is a nondecreasing function and for some x0 ∈ X and eachr > 0 there holds

infx∈X

∫p(x0,x)+r

p(x0,x)

dt

1 + h(t)> 0, (4.20)

then there exists v ∈ X such that f(v) ≤ f(u) and

p(v, x)1 + h

(p(x0, v)

) > ϕ(v)[f(v) − f(x)] ∀x ∈ X with x /=v. (4.21)

Proof. Proposition 2.2 shows that the function q defined by

q(x, y)

:=∫p(x0,x)+p(x,y)

p(x0,x)

dt

1 + h(t)for(x, y) ∈ X ×X (4.22)

is a w-distance. Applying Theorem 4.1 to the w-distance, the desired conclusion follows.

Remark 4.6. We have obtained Theorem 4.5 from Theorem 4.1. Conversely, when p is a w-distance, Theorem 4.1 follows from Theorem 4.5 by taking h(t) = 0 for all t ∈ [0,+∞). In thiscase they are equivalent results. If also p(x, y) ≤ M holds for some M > 0 and all (x, y) ∈X × X, Theorem 4.5 is obviously applicable. In particular, when we take x0 = u for certainpoint u ∈ X, the condition in Theorem 4.5 about h can be deleted.

Theorem 4.7. Let p be a w-distance on X, ε ∈ (0,+∞], g : [infXf, infXf + ε] → (0,+∞) eithernondecreasing or u.s.c., and h : [0,+∞) → [0,+∞) nondecreasing. Denote


Xf ≤ s ≤ f(x)

}for x ∈ f−1

(−∞, inf

Xf + ε

]. (4.23)

Then for u ∈ X with p(u, u) = 0 and

ϕ(u)[f(u) − inf

Xf

]< min

{ε,

∫+∞

0

dt

1 + h(t)

}, (4.24)

there exists v ∈ X such that

∫p(u,v)0

dt

1 + h(t)≤ ϕ(u)[f(u) − f(v)],

p(v, x)1 + h

(p(u, v)

) > ϕ(v)[f(v) − f(x)] ∀x ∈ X with x /=v.

(4.25)


Proof. Let a ≥ 0 satisfy

∫a0

dt

1 + h(t)= ϕ(u)

[f(u) − inf

Xf

],

p1(x, y)

:= min{p(x, y), ϕ(u)

[f(u) − inf

Xf

]+ 1 + a

}.

(4.26)

It is easy to see that p1 is a bounded w-distance on X and hence

q1(x, y)

:=∫p1(u,x)+p1(x,y)

p1(u,x)

dt

1 + h(t)(4.27)

is a w-distance. By Theorem 4.2, there exists v ∈ X such that

p1(v, x)1 + h

(p1(u, v)

) ≥ q1(v, x) > ϕ(v)[f(v) − f(x)], (4.28)

for all x ∈ X with x /=v and

∫p1(u,v)

0

11 + h(t)

dt = q1(u, v) ≤ ϕ(u)[f(u) − f(v)] ≤ ϕ(u)

[f(u) − inf

Xf

], (4.29)

from which we obtain p1(u, v) ≤ a and hence p1(u, v) = p(u, v). Thus the desired conclusionfollows.

Upon taking g = 1 and h = 0 in Theorem 4.7 and replacing p with εp, we obtain (ii) of[10, Theorem 3], which is also an extension to EVP.

5. Nonconvex Minimization Theorems

In this section we mainly apply the extensions of EVP obtained in Section 4 to establishminimization theorems which generalize [11, Theorem 5] (an extension to [10, Theorem 1]and [7, Theorem 1]). From these results we also derive Theorem 3.2. Consequently, seventheorems established in Sections 3–5 are shown to be equivalent.

Firstly, we use Theorem 4.1 to prove the following result.

Theorem 5.1. Let p be a τ-distance on X, ε ∈ (0,+∞], and ϕ : f−1(−∞, infXf + ε] → (0,+∞)satisfy

sup{ϕ(x) : x ∈ f−1

(−∞, inf

Xf + min

{ε, η}]}

< +∞, (5.1)


for some η > 0. If for each x ∈ X with infXf < f(x) < infXf + ε there exists y ∈ X such that y /=xand

p(x, y) ≤ ϕ(x)[f(x) − f(y)], (5.2)

then there exists x0 ∈ X such that f(x0) = infXf .

Proof. Denote

Mx :={y ∈ X : f

(y) ≤ f(x)}, for x ∈ X. (5.3)

Let x ∈ X (with infXf < f(x) < infXf + ε) be fixed. Since f is l.s.c., the set (Mx, d) isnonempty and complete. Thus, by Theorem 4.1, there exists v ∈Mx such that

p(v, y)> ϕ(v)

[f(v) − f(y)] ∀y ∈Mx with y /=v. (5.4)

The point v must satisfy f(v) = infXf . Otherwise, we suppose that

infXf < f(v) ≤ f(x) < inf

Xf + ε. (5.5)

By the assumption, there exists a point v ∈ X with v /=v such that

p(v, v) ≤ ϕ(v)[f(v) − f(v)], (5.6)

which implies v ∈Mx and hence contradicts the inequality

p(v, v) > ϕ(v)[f(v) − f(v)]. (5.7)

Similarly, we can use Theorem 4.2 to establish the following result.

Theorem 5.2. Let p be a τ-distance onX, ε ∈ (0,+∞], and g : [infXf, infXf +ε] → (0,+∞) eithernondecreasing or u.s.c.. If for each x ∈ X with infXf < f(x) < infXf + ε there exists y ∈ X such thaty /=x and

p(x, y) ≤ g(f(x))[f(x) − f(y)], (5.8)

then there exists x0 ∈ X such that f(x0) = infXf .

Example 5.3. Consider the function f(x) =√x for x ∈ [0,+∞). Obviously, f attains its

minimum at x = 0. For this simple example, we can also apply Theorem 5.2 to concludethat there exists x0 ∈ [0,+∞) such that f(x0) = inf[0,+∞)f since for any ε ∈ (0,+∞) and each


x ∈ (0, ε] we have y ∈ (0, x) such that

d(x, y)=∣∣x − y∣∣ < 2

√x(√

x −√y) = g(f(x))[f(x) − f(y)], (5.9)

where g(x) = 2x for x ∈ (0, ε] and g(0) = 1.

Remark 5.4. Up to now, beginning with Theorem 3.1, we have established the followingresults with the proof routes:

Theorem 3.2 =⇒ Theorem 3.4 =⇒ Theorem 3.5;

Theorem 3.4 =⇒ Theorem 4.1 =⇒ Theorem 5.1;

Theorem 3.5 =⇒ Theorem 4.2 =⇒ Theorem 5.2.

(5.10)

As a conclusion in this paper, the following result states that these seven theorems areequivalent.

Theorem 5.5. Theorems 3.2–3.5, 4.1-4.2, and 5.1-5.2 are all equivalent.

Proof. By Remark 5.4, it suffices to show that Theorems 5.1-5.2 both imply Theorem 3.2.Suppose that for some ε ∈ (0,+∞] and for each x ∈ X with infXf ≤ f(x) < infXf + ε

there exists x ∈ Tx such that x ∈M(x), that is,

p(x, x) ≤ f(x) − f(x). (5.11)

If there exists x0 ∈ X with f(x0) < infXf + ε such that M(x0) = {x0}, then, since there existsx0 ∈ Tx0 such that x0 ∈M(x0), x0 = x0, p(x0, x0) = 0. In this case, Theorem 3.2 follows.

Next we claim that there must exist x0 ∈ X such that

M(x0) = {x0}, f(x0) < infXf + ε. (5.12)

Otherwise, suppose that M(x)/= {x} for each x ∈ X with f(x) < infXf + ε. By Theorem 5.1or Theorem 5.2 there exists x1 ∈ X such that f(x1) = infXf . Since p(x1, x) = 0 for x ∈M(x1),according to the property that p(x1, x) = 0 and p(x1, y) = 0 imply x = y, M(x1) is a singleton.This implies that there exists x0 such that M(x1) = {x0} and f(x0) = infXf = f(x1), fromwhich and the triangle inequality we obtain

∅/=M(x0) ⊆M(x1) ⊆ {x0}. (5.13)

This gives M(x0) = {x0} and hence a contradiction to the assumption.

6. Generalized ε-Conditions of Takahashi and Hamel

The condition in Theorem 5.2 is sufficient for f to attain minimum on X. In this section weshow that such a condition implies more when the τ-distance p (onX×X) is l.s.c. in its secondvariable. For convenience we introduce the following notions.


Definition 6.1. A function f : X → (−∞,+∞] is said to satisfy the generalized ε-condition ofTakahashi (Hamel) if for some ε ∈ (0,+∞], some nondecreasing function g : [infXf, infXf +ε] → (0,+∞), and each x ∈ X with infXf < f(x) < infXf + ε there exists y ∈ X (y ∈ Z) suchthat y /=x and

p(x, y) ≤ g(f(x))[f(x) − f(y)], (6.1)

where Z = {z ∈ X : f(z) = infXf}. In particular, for the case ε = +∞ the generalized ε-condition of Takahashi (Hamel) is called the generalized condition of Takahashi (Hamel ).

When g = 1, the above concepts, respectively, reduce to ε-condition of Takahashi(Hamel) and the condition of Takahashi (Hamel) in [8].

It is clear that for any 0 < ε1 < ε2 the generalized ε2-condition of Takahashi impliesthe generalized ε1-condition of Takahashi and the generalized ε2-condition of Hamel impliesthe generalized ε1-condition of Hamel. For any ε ∈ (0,+∞] the generalized ε-condition ofTakahashi and the generalized ε-condition of Hamel are, respectively, weaker than that ofTakahashi and of Hamel. For example, whenX = [0,+∞), the function f(x) =

√x satisfies the

generalized ε-conditions of Takahashi and Hamel for any ε ∈ (0,+∞) but it does not satisfythat of Takahashi nor of Hamel. Furthermore, the generalized ε-condition of Hamel alwaysimplies that of Takahashi. Next result asserts that the converse is also true in a completemetric space.

Theorem 6.2. Let p be a τ-distance on X such that p(x, ·) is l.s.c. on X for each x ∈ X. For ε ∈(0,+∞], f satisfies the generalized ε-condition of Takahashi if and only if f satisfies the generalizedε-condition of Hamel.

Proof. The sufficiency is obvious, so it suffices to prove the necessity. Let f satisfy thegeneralized ε-condition of Takahashi and let g be the corresponding nondecreasing functionin the definition. Denote

M(x) :={y ∈ X : p

(x, y)+ g(f(x)

)f(y) ≤ g(f(x))f(x)}, for x ∈ X. (6.2)

Then for the case 0 < ε < +∞, it suffices to prove that the set M(x) ∩ Z is nonempty for eachx ∈ X with infXf < f(x) < infXf + ε, where

Z ={z ∈ X : f(z) = inf

Xf

}. (6.3)

Let x ∈ X with infXf < f(x) < infXf + ε be fixed. Since f and p(x, ·) are both l.s.c.,the set M(x) is nonempty and complete. Thus, by Theorem 4.1 or Theorem 4.2, there existsx ∈M(x) such that

p(x, y)> g(f(x)

)[f(x) − f(y)] ∀y ∈M(x) with y /=x. (6.4)

The point x must be in Z. Otherwise, if x were not in Z, then

infXf < f(x) ≤ f(x) < inf

Xf + ε. (6.5)


By the assumption, there exists a point y ∈ X with y /=x such that

p(x, y) ≤ g(f(x))[f(x) − f(y)], (6.6)

from which and the inequality g(f(x)) ≤ g(f(x)) we obtain

p(x, y) ≤ p(x, x) + p(x, y) ≤ g(f(x))[f(x) − f(y)], (6.7)

that is, y ∈M(x). And hence p(x, y) > g(f(x))[f(x)−f(y)]. This is a contradiction. Therefore,x ∈M(x) ∩ Z.

Next, we suppose that f satisfies the generalized condition of Takahashi. For each0 < ε < +∞, the function f satisfies the generalized ε-condition of Takahashi, so f satisfiesthe generalized ε-condition of Hamel. This implies that Z is nonempty. For each x ∈ X withinfXf < f(x), if f(x) < +∞, then infXf < f(x) < infXf + ε for some 0 < ε < +∞. In this casewe can find z ∈ Z such that

p(x, z) ≤ g(f(x))[f(x) − f(z)]. (6.8)

If f(x) = +∞, then this inequality holds for each z ∈ Z. Therefore f satisfies the generalizedcondition of Hamel.

7. Generalized Weak Sharp Minima and Error Bounds

As stated in [8], the ε-condition of Takahashi is one of sufficient conditions for an inequalitysystem to have weak sharp minima and error bounds. With Theorem 6.2 being established,the generalized ε-condition of Takahashi plays a similar role for the generalized weak sharpminima and error bounds introduced below.

For a proper l.s.c. and bounded below function f : X → (−∞,+∞], we say that f hasgeneralized local (global) weak sharp minima if the set Z of minimizers of f on X is nonemptyand if for some ε ∈ (0,+∞)(ε = +∞) and some nondecreasing function g : [infXf, infXf+ε] →(0,+∞) and each x ∈ X with infXf < f(x) < infXf + ε there holds

pZ(x) ≤ g(f(x)

)[f(x) − inf

Xf

], (7.1)

where pZ(x) = inf{p(x, z) : z ∈ Z}.Due to the equivalence stated in Theorem 6.2, the generalized ε-condition of Takahashi

is sufficient for f to have generalized local (global) weak sharp minima.

Theorem 7.1. Let p be a τ-distance on X such that p(x, ·) is l.s.c. on X for each x ∈ X. If, for someε ∈ (0,+∞], f satisfies the generalized ε-condition of Takahashi, then the set Z of minimizers of f onX is nonempty and for every x ∈ X with infXf < f(x) < infXf + ε and each z ∈ Z there holds

pZ(x) ≤ g(f(x)

)[f(x) − f(z)]. (7.2)


Proof. The proof is immediate from Theorem 6.2.

For an l.s.c. function f : X → (−∞,+∞], denote

S :={x ∈ X : f(x) ≤ 0

}, pS(x) := inf

{p(x, s) : s ∈ S}. (7.3)

We say that f (or S) has a generalized local error bound if there exist ε ∈ (0,+∞) and anondecreasing function g : [0, ε) → (0,+∞) such that

pS(x) ≤ g(f(x)+

)f(x)+ ∀x ∈ X with f(x) < ε, (7.4)

where f(x)+ = max{0, f(x)}. The function f is said to have a generalized global error bound ifthe above statement is true for ε = +∞.

When p = d and g = 1, the study of generalized error bounds has received growingattention in the mathematical programming (see [18] and the references therein). Now, usingTheorem 7.1, we present the following sufficient condition for an l.s.c. inequality system tohave generalized error bounds.

Theorem 7.2. Let p be a τ-distance on X such that p(x, ·) is l.s.c. on X for each x ∈ X and f : X →(−∞,+∞] be a proper l.s.c. function. Let ε1 ∈ (0,+∞] and g : [0, ε1) → (0,+∞) be a nondecreasingfunction. Suppose for each ε ∈ (0, ε1], the set f−1(−∞, ε) is nonempty and for each x ∈ f−1(0, ε)there exists a point y ∈ f−1[0, ε) such that y /=x and

p(x, y) ≤ g(f(x))[f(x) − f(y)]. (7.5)

Then S := {x ∈ X : f(x) ≤ 0} is nonempty and

pS(x) ≤ g(f(x)+

)f(x)+ ∀x ∈ f−1(−∞, ε1). (7.6)

Proof. Let ε1 ∈ (0,+∞] be given. Since f(·)+ is l.s.c. and bounded below with S = {x ∈ X :f(x)+ = 0} and infXf+ ≥ 0, by Theorem 7.1, it suffices to prove

S = Z :={z ∈ X : f(z)+ = inf

Xf+

}, (7.7)

that is, infXf+ = 0. This must be true. Otherwise, if infXf+ > 0, then for 0 < ε < min{ε1, infXf+}the set f−1(−∞, ε) would be empty. This contradicts the assumption.

Remark 7.3. Note that the nonemptiness of S in Theorem 7.2 is not a part of assumption buta part of conclusion. In addition, the condition in Theorem 7.2 implies that f+ satisfies thegeneralized ε-condition of Takahashi, that is,

Mg(x) :={y ∈ X : p

(x, y) ≤ g(f(x))[f(x) − f(y)]}/⊆{x}, (7.8)

for each x ∈ X with infXf+ < f(x) < infXf+ + ε. However, once Mg(x) is nonempty, thereexists x0 ∈Mg(x) such that Mg(x0) ⊆ {x0} as stated below.


Theorem 7.4. Let p be a τ-distance such that p(x, ·) is l.s.c. on X for each x ∈ X and g : [0,+∞) →(0,+∞) be a nondecreasing function. Denote

Mg(x) :={y ∈ X : p

(x, y)+ g(f(x)

)f(y) ≤ g(f(x))f(x)} ∀x ∈ X. (7.9)

Then for each u ∈ X with Mg(u)/= ∅, there exists x0 ∈ Mg(u) such that Mg(x0) ⊆ {x0}. Inparticular, there exists y0 ∈ X such thatMg(y0) ⊆ {y0}.

Proof. Since both p and f are l.s.c., for u ∈ X with Mg(u)/= ∅, (Mg(u), d) is nonemptycomplete metric space. Suppose that for each x ∈ Mg(u) there held Mg(x)/⊆{x}. Then foreach x ∈Mg(u) there exists x ∈Mg(x) such that x /=x. Define

F(x) := f(x) − infMg(u)

f for x ∈Mg(u) (7.10)

and denote S := {x ∈Mg(u) : F(x) = 0}. Then

S =

{x ∈Mg(u) : f(x) = inf

Mg(u)f

}. (7.11)

By Theorem 7.2, the set S is nonempty.Now for x ∈ S, since f(x) < +∞ (no matter whether f(u) < +∞ or f(u) = +∞), there

exists x ∈Mg(x) such that x /=x and

0 ≤ p(x, x) ≤ g(f(x))[f(x) − f(x)] ≤ 0 (7.12)

from which we obtain p(x, x) = 0 and f(x) = f(x). Similarly, we have x ∈ Mg(x) such thatx /=x and p(x, x) = 0. This, with p(x, x) = 0, implies p(x, x) = 0. Thus x = x, which is acontradiction.

Remark 7.5. When g = 1 and p is a τ-distance such that p(x, ·) is l.s.c. on X for each x ∈X, we can obtain Theorem 3.1 by applying Theorem 7.4 to the function f − infXf . As moreapplications, the following two propositions are immediate from Theorem 7.4 by taking g = 1,f(·) = p(b, ·)/γ, and f(·) = p(·, b)/γ , respectively, on (X, d).

Proposition 7.6. Let X be a complete nonempty subset of a metric space (E, d), a ∈ X, b ∈ E \ X,and let p be a τ-distance on E such that p(x, ·) is l.s.c. on X for each x ∈ X. Denote

Pγ(a, b) :={x ∈ E : γp(a, x) + p(b, x) ≤ p(b, a)}, for γ ∈ (0,+∞). (7.13)

Suppose that X ∩ Pγ(a, b) is nonempty for some γ ∈ (0,+∞). If p(x, x) = 0 for all x ∈ X ∩ Pγ(a, b),then there exists x0 ∈ X ∩ Pγ(a, b) such that

X ∩ Pγ(x0, b) = {x0}. (7.14)


Proposition 7.7. Let X be a complete nonempty subset of a metric space (E, d), a ∈ X, b ∈ E \ X,and let p be a τ-distance on E. Denote

Qγ(a, b) :={x ∈ E : γp(a, x) + p(x, b) ≤ p(a, b)}, for γ ∈ (0,+∞). (7.15)

Suppose that p is l.s.c. in its both variables and X ∩ Qγ(a, b) is nonempty for some γ ∈ (0,+∞). Ifp(x, x) = 0 for all x ∈ X ∩Qγ(a, b), then there exists x0 ∈ X ∩Qγ(a, b) such that X ∩Qγ(x0, b) ={x0}. In particular, if p(a, a) = 0 and p(x, x) = 0 for all x ∈ X ∩Q1(a, b), then there exists x0 ∈ Xsuch that p(a, b) = p(a, x0) + p(x0, b) and

{x ∈ X : p(x0, b) = p(x0, x) + p(x, b)

}= {x0}. (7.16)

Remark 7.8. Upon taking p(x, y) = d(x, y) in Propositions 7.6 and 7.7, we obtain [3,Theorem F] which is equivalent to EVP in a complete metric space. In this case EVP impliesTheorem 3.1.

Finally, following the statement in Theorem 5.5, on the condition that the τ-distancep(x, ·) is l.s.c. on X for each x ∈ X, Theorems 3.1–3.5, 4.1-4.2, 5.1-5.2, 6.2, and 7.1–7.4 turn outto be equivalent since we have further shown that

Theorem 4.2 =⇒ Theorem 6.2 =⇒ Theorem 7.1

=⇒Theorem 7.2 =⇒ Theorem 7.4 =⇒ Theorem 3.1(7.17)

in Sections 6 and 7. In particular, each theorem stated above is equivalent to Theorem 4.5(as stated in Remark 4.6) when p is a w-distance on X, to [3, Theorem F] and EVP whenp = d (see Remark 7.8), and to the Bishop-Phelps Theorem in a Banach space when p is thecorresponding norm. Therefore, we can conclude our paper as below.

Theorem 7.9. Let (X, d) be a complete metric space and p a τ-distance on X such that p(x, ·) is l.s.c.for each x ∈ X. Then

(i) Theorems 3.1–3.5, 4.1-4.2, 5.1-5.2, 6.2, and 7.1-7.4 are all equivalent;

(ii) when p is a w-distance on X, each theorem in (i) is equivalent to Theorem 4.5;

(iii) when p = d, each theorem in (i) is equivalent to EVP.

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Research ArticleRegularization and Iterative Methods forMonotone Variational Inequalities

Xiubin Xu1 and Hong-Kun Xu2

1 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China2 Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan

Correspondence should be addressed to Xiubin Xu, [email protected]



Copyright q 2010 X. Xu and H.-K. Xu. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We provide a general regularization method for monotone variational inequalities, where theregularizer is a Lipschitz continuous and strongly monotone operator. We also introduce aniterative method as discretization of the regularization method. We prove that both regularizationand iterative methods converge in norm.

1. Introduction

Variational inequalities (VIs) have widely been studied (see the monographs [1–3]). Amonotone variational inequality problem (VIP) is stated as finding a point x∗ with thefollowing property:

x∗ ∈ C, 〈Ax∗, x − x∗〉 ≥ 0, ∀x ∈ C, (1.1)

whereC is a nonempty closed convex subset of a real Hilbert spaceH with inner product 〈·, ·〉and norm ‖ · ‖, respectively, and A is a monotone operator in H with domain dom(A) ⊃ C.

Recall that A is monotone if

⟨Ax −Ay, x − y⟩ ≥ 0, ∀x, y ∈ dom(A). (1.2)

A typical example of monotone operators is the subdifferential of a proper convexlower semicontinuous function.


Variational inequality problems are equivalent to fixed point problems. As a matter offact, x∗ solves VIP (1.1) if and only if x∗ solves the following fixed point problem (FPP), forany γ > 0,

x∗ = PC(I − γA)

x∗, (1.3)

where PC is the metric (or nearest point) projection from H onto C; namely, for each x ∈ H,PCx is the unique point in C with the property

‖x − PCx‖ = min{∥∥x − y∥∥ : y ∈ C}. (1.4)

The equivalence between VIP (1.1) and FPP (1.3) is an immediate consequence of thefollowing characterization of PC:

Givenx ∈ H and z ∈ C; then z = PCx ⇐⇒⟨x − z, y − z⟩ ≤ 0, ∀y ∈ C. (1.5)

The dual VIP of (1.1) is the following VIP:

x∗ ∈ C, 〈Ax, x − x∗〉 ≥ 0, x ∈ C. (1.6)

The following equivalence between the dual VIP (1.6) and the primal VIP (1.1) playsa useful role in our regularization in Section 2.

Lemma 1.1 (cf. [4]). Assume that A : C → H is monotone and weakly continuous along segments(i.e., A((1 − t)x + ty) → Ax weakly as t → 0 for x, y ∈ C), then the dual VIP (1.6) is equivalent tothe primal VIP (1.1).

To guarantee the existence and uniqueness of a solution of VIP (1.1), one has to imposeconditions on the operator A. The following existence and uniqueness result is well known.

Theorem 1.2. IfA is Lipschitz continuous and strongly monotone, then there exists one and only onesolution to VIP (1.1).

However, if A fails to be Lipschitz continuous or strongly monotone, then the resultof the above theorem is false in general. We will assume that A is Lipschitz continuous, butdo not assume strong monotonicity of A. Thus, VIP (1.1) is ill-posed and regularization isneeded; moreover, a solution is often sought through iteration methods.

In the special case where A is of the form A = I − T , with T being a nonexpansivemapping, regularization and iterative methods for VIP (1.1) have been investigated inliterature; see, for example, [5–19]; work related to variational inequalities of monotoneoperators can be found in [20–25], and work related to iterative methods for nonexpansivemappings can be found in [26–33].

The aim of this paper is to provide a regularization and its induced iteration methodfor VIP (1.1) in the general case. The paper is structured as follows. In the next section wepresent a general regularization method for VI (1.1) with the regularizer being a Lipschitzcontinuous and strongly monotone operator. In Section 3, by discretizing the implicit method


of the regularization obtained in Section 2, we introduce an iteration process and prove itsstrong convergence. In the final section, Section 4, we apply the results obtained in Sections 2and 3 to a convex minimization problem.

2. Regularization

Since VIP (1.1) is usually ill-posed, regularization is necessary, towards which we let B :H → H be a Lipschitz continuous, everywhere defined, strongly monotone, and single-valued operator. Consider the following regularized variational inequality problem:

xε ∈ C, 〈Axε + εBxε, x − xε〉 ≥ 0, x ∈ C. (2.1)

Since A+ εB is strongly monotone, VI (2.1) has a unique solution which is denoted by xε ∈ C.Indeed, VI (2.1) is equivalent to the fixed point equation

xε = PC(I − γ(A + εB)

)xε ≡ Tεxε, (2.2)

where Tε = PC(I − γ(A + εB)) ≡ PC(I − γFε), with Fε = A + εB.To analyze more details of VI (2.1) (or its equivalent fixed point equation (2.2)), we

need to impose more assumptions on the operators A and B. Assume that A and B areLipschitz continuous with Lipschiz constants L1, L2, respectively. We also assume that B isβ-strongly monotone; namely, there is a constant β > 0 satisfying the property

〈Bx1 − Bx2, x1 − x2〉 ≥ β‖x1 − x2‖2, x1, x2 ∈ H. (2.3)

Lemma 2.1. If γ is chosen in such a way that

0 < γ <2εβ

(L1 + εL2)2, (2.4)

then Tε is a contraction with contraction coefficient

√1 − γ

[2εβ − γ(L1 + εL2)2

]< 1. (2.5)

Moreover, if

0 < γ <2εβ

(L1 + εL2)2 + (ε2/4), (2.6)


then

√1 − γ

[2εβ − γ(L1 + εL2)2

]≤ 1 − 1

2βεγ ; (2.7)

hence, Tε is a (1 − (1/2)βεγ)-contraction.

Proof. Noticing that Fε is (L1 + εL2)-Lipschitzian and εβ-strongly monotone, we deduce that,for x, y ∈ H,

∥∥Tεx − Tεy∥∥2 =∥∥PC(I − γFε)x − PC(I − γFε)y∥∥2

≤ ∥∥(I − γFε)x − (I − γFε)y∥∥2

=∥∥(x − y) − γ(Fεx − Fεy)∥∥2

=∥∥x − y∥∥2 − 2γ〈x − y, Fεx − Fεy〉 + γ2∥∥Fεx − Fεy∥∥2

≤(

1 − γ[2εβ − γ(L1 + εL2)2

])∥∥x − y∥∥2.

(2.8)

It turns out that if γ satisfies (2.4), then Tε is a contraction with coefficient given by the leftside of (2.5).

Finally, it is straightforward that (2.7) holds provided that γ satisfies (2.6).

Below we always assume that γ satisfies (2.6) so that Tε is a (1 − (1/2)βεγ)-contractionfrom C into itself. Therefore, for such a choice of γ , Tε has a unique fixed point in C which isdenoted as xε whose asymptotic behavior when ε → 0 is given in the following result.

Theorem 2.2. Assume that

(a) A : C → H is monotone on C and weakly continuous along segments in C (i.e., A((1 −t)x + ty) → Ax weakly as t → 0 for x, y ∈ C),

(b) B is β-monotone onH,

(c) the solution set S of VI (1.1) is nonempty.

For ε ∈ (0, 1), let xε be the unique solution of the regularized VIP (2.1). Then, as ε → 0, xε convergesin norm to a point ξ in S which is the unique solution of the VIP

ξ ∈ S, 〈Bξ, x − ξ〉 ≥ 0, ∀x ∈ S. (2.9)

Therefore, if one takes B to be the identity operator, then the regularized solution (xε) of thecorresponding regularized VIP (2.1) converges in norm to the minimal norm point of the solutionset S.

To prove Theorem 2.2, we first prove the boundedness of the net (xε).


Lemma 2.3. Assume that A is monotone on C. Assume conditions (b) and (c) in Theorem 2.2. Then(xε) is bounded; indeed, for any x∗ ∈ S,

‖x∗ − xε‖ ≤ 1β‖Bx∗‖, ∀ε ∈ (0, 1). (2.10)

Proof. We have (2.1) holds for all x ∈ C. In particular, for x∗ ∈ S, we have

〈Axε + εBxε, x∗ − xε〉 ≥ 0. (2.11)

It turns out that

〈Axε, x∗ − xε〉 + ε〈Bxε, x∗ − xε〉 ≥ 0. (2.12)

Since A is monotone and B is β-strongly monotone, we have

〈Ax∗, x∗ − xε〉 ≥ 〈Axε, x∗ − xε〉,

〈Bx∗, x∗ − xε〉 ≥ 〈Bxε, x∗ − xε〉 + β‖x∗ − xε‖2.(2.13)

Substituting them into (2.12) we obtain

εβ‖x∗ − xε‖2 ≤ 〈Ax∗, x∗ − xε〉 + ε〈Bx∗, x∗ − xε〉. (2.14)

However, since x∗ ∈ S, 〈Ax∗, x∗ − xε〉 ≤ 0. We therefore get from (2.14) that

‖x∗ − xε‖2 ≤ 1β〈Bx∗, x∗ − xε〉. (2.15)

Now (2.10) follows immediately from (2.15).

Proof of Theorem 2.2. Since (xε) is bounded by Lemma 2.3, the set of weak limit points as ε →0 of the net (xε), ωw(xε), is nonempty. Pick a ξ ∈ ωw(xε) and let (εn) be a null sequence in theinterval (0, 1) such that xεn → ξ weakly as n → ∞. We first show that ξ ∈ S. To see this weuse the equivalent dual VI of (2.1):

xε ∈ C, 〈Ax + εBx, x − xε〉 ≥ 0, x ∈ C. (2.16)

Thus, we have, for all x ∈ C and n,

〈Ax + εnBx, x − xεn〉 ≥ 0. (2.17)


Taking the limit as n → ∞ yields that

〈Ax, x − ξ〉 ≥ 0, ∀x ∈ C. (2.18)

It turns out that ξ ∈ S.We next prove that the sequence {xεn} actually converges to ξ strongly. Replacing in

(2.15) x∗ with ξ gives

‖ξ − xεn‖2 ≤ 1β〈Bξ, ξ − xεn〉, x ∈ C. (2.19)

Now it is straightforward from (2.19) that the weak convergence to ξ of {xεn} implies strongconvergence to ξ of {xεn}.

The relation (2.15) particularly implies that, for ε > 0,

〈Bx∗, x∗ − xε〉, x∗ ∈ S, (2.20)

which in turns implies that every point ξ ∈ ωw(xε) ⊂ S solves the VIP

ξ ∈ S, 〈Bx∗, x∗ − ξ〉 ≥ 0, ∀x∗ ∈ S, (2.21)

or equivalently, the VIP

ξ ∈ S, 〈Bξ, x∗ − ξ〉 ≥ 0, ∀x∗ ∈ S. (2.22)

However, since B is strongly monotone, the solution to VIP (2.22) is unique. This has shownthat the unique solution ξ of VIP (2.22) is the strong limit of the net {xε}.

Finally, if B is the identity operator, then VIP (2.22) is reduced to

〈ξ, x∗ − ξ〉 ≥ 0, ∀x∗ ∈ S. (2.23)

This is equivalent to

‖ξ‖2 ≤ 〈x∗, ξ〉, ∀x∗ ∈ S, (2.24)

which immediately implies that ‖ξ‖ ≤ ‖x∗‖ for all x∗ ∈ S and hence ξ is the minimal norm ofS.

Remark 2.4. In Theorem 2.2, we have proved that if the solution set S of VIP (1.1) is nonempty,then the net (xε) of the solutions of the regularized VIPs (2.1) is bounded (and henceconverges in norm). The converse is indeed also true; that is, the boundedness of the net(xε) implies that the solution set S of VIP (1.1) is nonempty. As a matter of fact, suppose that(xε) is bounded and M > 0 is a constant such that ‖xε‖ ≤M for all ε ∈ (0, 1).


By Lemma 1.1, we have

xε ∈ C, 〈Ax + εBxε, x − xε〉 ≥ 0, x ∈ C. (2.25)

Since (xε) is bounded, we can easily see that every weak cluster point ξ of the net (xε) solvesthe VIP

ξ ∈ C, 〈Ax, x − ξ〉 ≥ 0, x ∈ C. (2.26)

This is the dual VI to the primal VI (2.1); hence ξ is a solution of VI (2.1) by Lemma 1.1.

3. Iterative Method

From the fixed point equation (2.2), it is natural to consider the following iteration methodthat generates a sequence {xn} according to the recursion:

xn+1 = PC(xn − γn(Axn + εnBxn)

), n = 0, 1, . . . , (3.1)

where the initial guess x0 ∈ C is selected arbitrarily, and {γn} and {εn} are two sequences ofpositive numbers in (0, 1). Put in another way, xn+1 ∈ C is the unique solution in C of thefollowing VIP:

〈xn − γn(Axn + εnBxn) − xn+1, x − xn+1〉 ≤ 0, x ∈ C. (3.2)

Theorem 3.1. Assume that

(a) A is L1-Lipschitz continuous and monotone on C,

(b) B is L2-Lipschitz continuous and β-monotone onH,

(c) the solution set S of VI (1.1) is nonempty.

Assume in addition that

(i) 0 < γn < βεn/((L1 + εnL2)2 + (ε2

n/4)),

(ii) εn → 0 as n → ∞,

(iii)∑∞

n=1 εnγn =∞,

(iv) limn→∞(|γn − γn−1| + |εnγn − εn−1γn−1|)/(εnγn)2 = 0,

then the sequence {xn} generated by the algorithm (3.1) converges in norm to the unique solution ofVI (2.9).

To prove Theorem 3.1, we need a lemma below.

Lemma 3.2 (cf. [20]). Assume that {an} is a sequence of nonnegative real numbers such that

an+1 ≤(1 − βn

)an + βnσn, n ≥ 0, (3.3)


where {βn} and {σn} are real sequences such that(i) βn ∈ (0, 1) for all n, and

∑∞n=1 βn =∞;

(ii) lim supn→∞σn ≤ 0,

then limn→∞an = 0.

Proof of Theorem 3.1. Let Tn = PC(I − γnFn), where Fn = A + εnB. By assumption (i) andLemma 2.1, Tn is a contraction and has a unique fixed point which is denoted by zn. Moreover,by Theorem 2.2, {zn} converges in norm to the unique solution ξ of VI (2.9). Therefore, itsuffices to prove that ‖xn+1 − zn‖ → 0 as n → ∞.

To see this, observing that Tn is a (1 − (1/2)βεnγn)-contraction, we obtain

‖xn+1 − zn‖ = ‖Tnxn − Tnzn‖

≤(

1 − 12βεnγn

)‖xn − zn‖

≤(

1 − 12βεnγn

)‖xn − zn−1‖ + ‖zn − zn−1‖.

(3.4)

However, we have

‖zn − zn−1‖ = ‖Tnzn − Tn−1zn−1‖≤ ‖Tnzn − Tnzn−1‖ + ‖Tnzn−1 − Tn−1zn−1‖

≤(

1 − 12βεnγn

)‖zn − zn−1‖ +

∥∥(I − γnFn)zn−1 −(I − γn−1Fn−1

)zn−1

∥∥

=(

1 − 12βεnγn

)‖zn − zn−1‖ +

∥∥(γn − γn−1)Azn−1 +

(εnγn − εn−1γn−1

)Bzn−1

∥∥.

(3.5)

Since {zn} is bounded, it turns out that, for an appropriate constant M > 0,

‖zn − zn−1‖ ≤∣∣γn − γn−1

∣∣ + ∣∣εnγn − εn−1γn−1∣∣

εnγnM. (3.6)

Substituting (3.6) into (3.4) and setting βn = (1/2)βεnγn, we get

‖xn+1 − zn‖ ≤(1 − βn

)‖xn − zn−1‖ + βnσn, (3.7)

where

σn =

∣∣γn − γn−1∣∣ + ∣∣εnγn − εn−1γn−1

∣∣(εnγn

)2M′, (3.8)

with M′ = 2M/β. Assumptions (iii) and (iv) assure that∑∞

n=1 βn =∞ and σn → 0 as n → ∞,respectively. Therefore, we can apply lemma to (3.7) to conclude that ‖xn+1 −zn‖ → 0; hence,xn → ξ in norm.


Remark 3.3. Assume 0 < ε ≤ γ < 1 satisfy 2ε + γ < 1, then it is not hard to see that for anappropriate constant a > 0,

εn :=1

(n + 1)ε, γn :=

a

(n + 1)γ, n ≥ 0 (3.9)

satisfy the assumptions (i)–(iv) of Theorem 3.1.

4. Application

Consider the constrained convex minimization problem:

minx∈C

ϕ(x), (4.1)

where C is a closed convex subset of a real Hilbert space H and ϕ : H → R is a real-valuedconvex function. Assume that ϕ is continuously differentiable with a Lipschitz continuousgradient:

∥∥∇ϕ(x) − ∇ϕ(y)∥∥ ≤ L∥∥x − y∥∥, ∀x, y ∈ H, (4.2)

where L is a constant.It is known that the minimization (4.1) is equivalent to the variational inequality

problem:

x∗ ∈ C, 〈∇ϕ(x∗), x − x∗〉 ≥ 0, ∀x ∈ C. (4.3)

Therefore, applying Theorems 2.2 and 3.1, we get the following result.

Theorem 4.1. Assume the Lipschitz continuity (4.2) for the gradient ∇ϕ.(a) For ε ∈ (0, 1), let xε ∈ C be the unique solution of the regularized VIP

xε ∈ C, 〈∇ϕ(xε) + εxε, x − xε〉 ≥ 0, ∀x ∈ C. (4.4)

Equivalently, xε ∈ C is the unique solution in C of the regularized minimization problem:

minx∈C

{ϕ(x) +

12ε‖x‖2

}. (4.5)

Then, as ε → 0, xε remains bounded if and only if (4.1) has a solution, and in this case, xε convergesin norm to the minimal norm solution of (4.1).

(b) Assume that (4.1) has a solution. Assume in addition that

(i) 0 < γn < εn/((L + εn)2 + (ε2

n/4)),

(ii) εn → 0 as n → ∞,


(iii)∑∞

n=1 εnγn =∞,

(iv) limn→∞(|γn − γn−1| + |εnγn − εn−1γn−1|)/(εnγn)2 = 0.

Starting x0 ∈ C, one defines {xn} by the iterative algorithm

xn+1 = PC(xn − γn

(∇ϕ(xn) + εnxn)). (4.6)

Then {xn} converges in norm to the minimum-norm solution of the constrained minimization problem(4.1).

Proof. Apply Theorems 2.2 and 3.1 to the case whereA = ∇ϕ and B = I is the identity operatorto get the conclusions in (a) and (b).

Acknowledgments

The authors are grateful to the anonymous referees for their comments and suggestionswhich improved the presentation of this paper. This paper is dedicated to Professor WilliamArt Kirk for his significant contributions to fixed point theory. The first author was supportedin part by a fund (Grant no. 2008ZG052) from Zhejiang Administration of Foreign ExpertsAffairs. The second author was supported in part by NSC 97-2628-M-110-003-MY3, and byDGES MTM2006-13997-C02-01.

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Research ArticleFixed Points for DiscontinuousMonotone Operators

Yujun Cui1 and Xingqiu Zhang2

1 Department of Applied Mathematics, Shandong University of Science and Technology,Qingdao 266510, China

2 School of Mathematics, Liaocheng University, Liaocheng 252059, China

Correspondence should be addressed to Yujun Cui, [email protected]



Copyright q 2010 Y. Cui and X. Zhang. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We obtain some new existence theorems of the maximal and minimal fixed points fordiscontinuous monotone operator on an order interval in an ordered normed space. Moreover,the maximal and minimal fixed points can be achieved by monotone iterative method under someconditions. As an example of the application of our results, we show the existence of extremalsolutions to a class of discontinuous initial value problems.

1. Introduction

LetX be a Banach space. A nonempty convex closed set P ⊂ X is said to be a cone if it satisfiesthe following two conditions: (i) x ∈ P , λ ≥ 0 implies λx ∈ P ; (ii) x ∈ P , −x ∈ P implies x = θ,where θ denotes the zero element. The cone P defines an ordering in E given by x ≤ y if andonly if y − x ∈ P . Let D = [u0, v0] be an ordering interval in X, and A : D → X an increasingoperator such that u0 ≤ Au0, Av0 ≤ v0. It is a common knowledge that fixed point theoremson increasing operators are used widely in nonlinear differential equations and other fieldsin mathematics ([1–7]).

But in most well-known documents, it is assumed generally that increasing operatorspossess stronger continuity and compactness. Recently, there have been some papersthat considered the existence of fixed points of discontinuous operators. For example,Krasnosel’skii and Lusnikov [8] and Chen [9] discussed the fixed point problems fordiscontinuous monotonically compact operator. They called an operator A to be amonotonically compact operator if x1 ≤ · · · ≤ xn ≤ · · · ≤ w (x1 ≥ · · · ≥ xn ≥ · · · ≥ w)implies that Axn converges to some x∗ ∈ X in norm and that x∗ = sup{Axn} (x = inf{Axn}).


A monotonically compact operator is referred to as an MMC-operator. A is said to be h-monotone if x < y implies Ax < Ay − α(x, y)h, where h ∈ P , h/= θ, and α(x, y) > 0. Theyproved the following theorem.

Theorem 1.1 (see [8]). Let A : E → E be an h-monotone MMC-operator with u < Au ≤ Av < v.Then A has at least one fixed point x∗ ∈ [u, v] possessing the property of h-continuity.

Motivated by the results of [3, 8, 9], in this paper we study the existence of the minimaland maximal fixed points of a discontinuous operator A, which is expressed as the formCB. We do not assume any continuity on A. It is only required that C (or B) is an MMC-operator and B(D) (or A(D)) possesses the quasiseparability, which are satisfied naturallyin some spaces. As an example for application, we applied our theorem to study first orderdiscontinuous nonlinear differential equation to conclude our paper.

We give the following definitions.

Definition 1.2 (see [3]). Let Y be an Hausdorff topological space with an ordering structure.Y is called an ordered topological space if for any two sequences {xn} and {yn} in Y , xn ≤yn (n = 1, 2, . . .) and xn → x, yn → y (n → ∞) imply x ≤ y.

Definition 1.3 (see [3]). Let Y be an ordered topological space, S is said to be a quasi-separableset in Y if for any totally ordered set M in S, there exists a countable set {yn} ⊂M such that{yn} is dense inM (i.e., for any y ∈M, there exists {ynj} ⊂ {yn} such that ynj → y (n → ∞)).

Obviously, the separability implies the quasi-separability.

Definition 1.4 (see [3]). LetX,Y be two ordered topological spaces. An operatorA : X → Y issaid to be a monotonically compact operator if x1 ≤ · · · ≤ xn ≤ · · · ≤ w (x1 ≥ · · · ≥ xn ≥ · · · ≥ w)implies that Axn converges to some y∗ ∈ Y in norm and that y∗ = sup{Axn} (y∗ = inf{Axn}).

Remark 1.5. The definition of the MMC-operator is slightly different from that of [8, 9].

2. Main Results

Theorem 2.1. Let X be an ordered topological space, and D = [u0, v0] an order interval in X. LetA : D → X be an operator. Assume that

(i) there exist ordered topological space Y , increasing operator C : D → Y , and increasingoperator B : [Cu0, Cv0] = {y ∈ Y | Cu0 ≤ y ≤ Cv0} → X such that A = BC;

(ii) A(D) is quasiseparable and C is an MMC-operator;(iii) u0 ≤ Au0, Av0 ≤ v0.

Then A has at least one fixed point in D.

Proof. It follows from the monotonicity of A and condition (iii) that A : D → D. Set R = {x ∈A(D) | x ≤ Ax}. Since Au0 ∈ R, R is nonempty. Suppose that M is a totally ordered set in R.We now show that M has an upper bound in R.

Since M ⊂ A(D), by condition (ii) there exists a countable subset {xi} of M such that{xi} is dense in M. Consider the sequence

z1 = x1, zi = max{zi−1, xi}, i = 1, 2, . . . . (2.1)


Since M is a totally ordered set, zi makes sense and

z1 ≤ z2 ≤ · · · ≤ zi ≤ · · · . (2.2)

By condition (ii), M ⊂ D = [u0, v0] and Definition 1.4, there exists y∗ ∈ Y such that

Czi −→ y∗ = sup{Czi}, (i −→ ∞), (2.3)

Cu0 ≤ y∗ ≤ Cv0, (2.4)

and hence By∗ make sense.Set

x∗ = By∗. (2.5)

Using (2.1) and (2.2), we have

xi ≤ Axi = BCxi ≤ BCzi ≤ By∗ = x∗. (2.6)

Since {xi} is dense in M, for any x ∈ M there exists a subsequence {xij} of {xi} such thatxij → x (j → ∞). By (2.6) and Definition 1.2, we get

x ≤ x∗, ∀x ∈M. (2.7)

Hence x ≤ Ax ≤ Ax∗, therefore Ax∗ is an upper bound of M.Now we show Ax∗ ∈ R. By virtue of (2.4) and condition (iii)

u0 ≤ Au0 = BCu0 ≤ By∗ = x∗ ≤ BCv0 ≤ v0. (2.8)

Thus x∗ ∈ [u0, v0] = D and hence Ax∗ ∈ D. By (2.7) and condition (ii), we get zi ≤ x∗ andhence Czi ≤ Cx∗. By (2.3) and Definition 1.2, we get y∗ ≤ Cx∗ and

x∗ = By∗ ≤ BCx∗ = Ax∗. (2.9)

Hence Ax∗ ≤ A(Ax∗), and therefore Ax∗ ∈ R.This shows that Ax∗ is an upper bound of M in R. It follows from Zorn’s lemma that

R has maximal element x. Thus x ≤ Ax. And so Ax ≤ A(Ax), which implies that Ax ∈ R andx ≤ Ax. As x is a maximal element of R, x = Ax; that is, x is a fixed point of A.




(ii) [Cu0, Cv0] is quasiseparable and B is an MMC-operator;

(iii) u0 ≤ Au0, Av0 ≤ v0.

Then A has at least one fixed point in D.

Proof. Let y1 = Cu0, y2 = Cv0. By the conditions (i) and (iii), we have

y1 = Cu0 ≤ CAu0 = CBCu0 = CBy1, CBy2 = CBCv0 = CAv0 ≤ Cv0 = y2. (2.10)

Since CB is increasing, for any y ∈ [y1, y2], we get

y1 ≤ CBy1 ≤ CBy ≤ CBy2 ≤ y2, (2.11)

that is, CB : [y1, y2] → [y1, y2]; therefore the quasiseparability of [Cu0, Cv0] implies thatCB ([y1, y2]) is quasiseparable. Applying Theorem 2.1, the operator CB has at least one fixedpoint y∗ in [y1, y2], that is,

y∗ = CBy∗, y∗ ∈ [y1, y2

]. (2.12)

Set x∗ = By∗. Since B is increasing, by (2.12), we have

u0 ≤ Au0 = BCu0 ≤ By∗ = x∗ ≤ Bcv0 = Av0 ≤ v0,

x∗ = By∗ = B(CBy∗

)= BC

(By∗

)= Ax∗;

(2.13)

that is, x∗ is a fixed point of the operator A in [u0, v0].

Theorem 2.3. If the conditions in Theorem 2.1 are satisfied, then A has the minimal fixed point u∗

and the maximal fixed point v∗ inD; that is, u∗ and v∗ are fixed points ofA, and for any fixed point xof A in D, one has u∗ ≤ x ≤ v∗.

Proof. Set

FixA ={x ∈ Dx is a fixed point of A

}. (2.14)

By Theorem 2.1, FixA/= ∅. Set

S = {[u, v] | [u, v] is an order interval in X, u, v ∈ A(D), u ≤ Au, Av ≤ v, FixA ⊂ [u, v]}.(2.15)

Since A is increasing, for any x ∈ FixA, we have

u0 ≤ Au0 ≤ Ax = x ≤ Av0 ≤ v0, (2.16)


and hence

Au0 ≤ A2u0 ≤ Ax = x ≤ A2v0 ≤ Av0, (2.17)

therefore [Au0, Av0] ∈ S, and thus S/= ∅. An order of S is defined by the inclusion relation,that is, for any I1 ∈ S, I2 ∈ S, and if I1 ⊂ I2, then we define I1 ≤ I2. We show that S hasa minimal element. Let {[uα, vα] | α ∈ T} be a totally subset of S and M′ = {uα | α ∈ T}.Obviously, M′ is a totally ordered set in X. Since A(D) is quasiseparable, it follows fromM′ ⊂ A(D) that there exists a countable subset {yi} of M′ such that {yi} is dense in M′. Let

w1 = y1, wi = max{wi−1, yi

}, i = 2, 3, . . . . (2.18)

Since M′ is a totally ordered set, wi makes sense and

w1 ≤ w2 ≤ · · · ≤ wi ≤ · · · . (2.19)

Then there exists w ∈ Y such that

Cwi −→ w = sup{Cwi}. (2.20)

Using the same method as in Theorem 2.1, we can prove that w makes sense, Au (whereu = Bw) is an upper bound of M′, and

Au ≤ A(Au). (2.21)

Since FixA ⊂ [uα, vα] (for all α ∈ T), for any x ∈ FixA, we have uα ≤ x, for all α ∈ T . Sincewi ∈M′, wi ≤ x. By (2.20), w ≤ Cx, and hence u = Bw ≤ BCx = Ax = x, for all x ∈ FixA, andtherefore

Au ≤ Ax = x, ∀x ∈ FixA. (2.22)

Consider N = {vα | α ∈ T}. Similarly, we can prove that there exists v ∈ D such thatAv is a lower bound of N and

A(Av) ≤ Av, Av ≥ x, ∀x ∈ FixA. (2.23)

By (2.22) and (2.23), Au ≤ Av. Set I = [Au,Av]. By virtue of (2.21), (2.22), and (2.23), I ∈ S.It is easy to see that I is a lower bound of {[uα, vα] | α ∈ T} in S. It follows from Zorn’s lemmathat S has a minimal element.

Let [u∗, v∗] be a minimal element of S. Therefore, u∗ ≤ Au∗, Av∗ ≤ v∗, and FixA ⊂[u∗, v∗]. Obviously, u∗ is a fixed point of A. In fact, on the contrary, u∗ /=Au∗ and u∗ ≤ Au∗.Hence

Au∗ ≤ A(Au∗), Au∗ ≤ Ax = x, ∀x ∈ FixA. (2.24)


Since A is an increasing operator, this implies that FixA ⊂ [Au∗, v∗] and [u∗, v∗] includesproperly [Au∗, v∗]. This contradicts that [u∗, v∗] is the minimal element of S. Similarly, v∗ isa fixed point of A. Since FixA ⊂ [u∗, v∗], u∗ is the minimal fixed point of A and v∗ is themaximal fixed point of A.

Theorem 2.4. If the conditions in Theorem 2.2 are satisfied, then A has the minimal fixed point u∗

and the maximal fixed point v∗ inD; that is, u∗ and v∗ are fixed points ofA, and for any fixed point xof A in D, one has u∗ ≤ x ≤ v∗.

Proof. It is similar to the proof of Theorem 2.4; so we omit it.



(ii) B is an continuous operator;

(iii) C is a demicontinuous MMC-operator;

(iv) u0 ≤ Au0, Av0 ≤ v0.

ThenA has both the minimal fixed point u∗ and the maximal fixed point v∗ in [u0, v0], and u∗ and v∗

can be obtained via monotone iterates:

u0 ≤ Au0 ≤ · · · ≤ Anu0 ≤ · · · ≤ Anv0 ≤ · · · ≤ Av0 ≤ v0 (2.25)

with limn→∞Anu0 = u∗, and limn→∞Anv0 = v∗.

Proof. We define the sequences

un = Anu0, vn = Anv0, n = 1, 2, . . . (2.26)

and conclude from the monotonicity of operator A and the condition (iv) that

u0 ≤ u1 ≤ · · · ≤ un ≤ · · ·vn ≤ · · · ≤ v1 ≤ v0. (2.27)

Let

yn = Cun, n = 1, 2, . . . . (2.28)

Since C is increasing, y0 ≤ y1 ≤ · · · ≤ yn ≤ · · · ≤ Cv0 by (2.27). By the condition (iii), we get

yn −→ y∗ = sup{yn

}, n −→ ∞. (2.29)

By (2.29) and Definition 1.2, we have

y∗ ∈ [Cu0, Cv0], (2.30)


and hence By∗ makes sense. Set u∗ = By∗, then u∗ ∈ [u0, v0]. Since B is continuous,

un = Aun = BCun = Byn −→ By∗ = u∗. (2.31)

By the condition (iii),Cunw−−−→ Cu∗, that is, yn

w−−−→ Cu∗. Note that yn → y∗; we have y∗ = Cu∗;hence u∗ = By∗ = BCu∗ = Au∗; that is, u∗ is a fixed point of A. Similarly, there exists v∗ ∈ Dsuch that vn → v∗ and v∗ is a fixed point of A. By the routine standard proof, it is easy toprove that u∗ is the minimal fixed point of A and v∗ is the maximal fixed point of A in D.

3. Applications

As some simple applications of Theorem 2.5, we consider the existence of extremal solutionsfor a class of discontinuous scalar differential equations.

In the following, R stands for the set of real numbers and J = [0, a] a compact realinterval. Let C[J, R] be the class of continuous functions on J . C[J, R] is a normed linear spacewith the maximum norm and partially ordered by the cone K = {x ∈ C[J, R] : x(t) ≥ 0}. K isa normal cone in C[J, R].

For any 1 ≤ p < +∞, set

Lp[J, R] =

{x(t) : J → R | x(t) is measurable and

∫J

|x(t)|pdt <∞}. (3.1)

Then Lp[J, R] is a Banach space by the norm ‖x‖p = (∫J |x(t)|pdt)

1/p.A function f : J ×R → R is said to be a Caratheodory function if f(x, y) is measurable

as a function of x for each fixed y and continuous as a function of y for a.a. (almost all) x ∈ J .We list for convenience the following assumptions.

(H1) u0, v0 ∈ AC[J, R], u0 ≤ v0,

u′0(t) ≤ f(t, u0(t)), v′0(t) ≥ f(t, v0(t)) for a.a. t ∈ J. (3.2)

(H2) f : J × R → R is a Caratheodory function.

(H3) There exists p > 1 such that

f(t, u0(t)) ∈ LP [J, R], f(t, v0(t)) ∈ LP [J, R]. (3.3)

(H4) There exists M ≥ 0 such that f(t, x) +Mx is nondecreasing for a.a. t ∈ J .

Consider the differential equation

x′ = f(t, x), x(0) = x0, (3.4)


where f : J × R → R. It is a common knowledge that the initial value problem (3.4) isequivalent to the equation

x(t) = x0 +∫ t

0f(s, x(s))ds (3.5)

if f(t, x) is continuous. Therefore, when f(t, x) is not continuous, we define the solution ofthe integral equation (3.5) as the solution of the equation (3.4).

Theorem 3.1. Under the hypotheses (H1)–(H4), the IVP (3.4) has the minimal solution u∗ and max-imal solution v∗ in [u0, v0]. Moreover, there exist monotone iteration sequences {un(t)}, {vn(t)} ⊂[u0, v0] such that

un(t) −→ u∗(t), vn(t) −→ v∗(t) as n −→ ∞ uniformly on t ∈ J, (3.6)

where {un(t)} and {vn(t)} satisfy

u′n(t) = f(t, un−1(t)) −M(t)(un(t) − un−1(t)), un(0) = x0,

v′n(t) = f(t, vn−1(t)) −M(t)(vn(t) − vn−1(t)), vn(0) = x0,

u0 ≤ u1 ≤ · · · ≤ un ≤ · · · ≤ u∗ ≤ v∗ ≤ · · · ≤ vn ≤ · · · ≤ v1 ≤ v0.

(3.7)

Proof. For any h ∈ C[J, R], we consider the linear integral equation:

x(t) = h(t) − (Tx)(t), (3.8)

where (Tx)(t) Δ=∫ t

0Mu(s)ds. Obviously, T : C[J, R] → C[J, R] is a linear completelycontinuous operator. By direct computation, the operator equation x + Tx = θ has only zerosolution; then by Fredholm theorem, for any h ∈ C[J, R], the operator equation (3.8) has aunique solution in C[J, R]. We definition the mapping N : C[J, R] → C[J, R] by

Nh = uh, (3.9)

where uh is the unique solution of (3.8) corresponding to h. Obviously N is a linearcontinuous operator; now we show that N is increasing. Suppose that h1, h2 ∈ C[J, R],


h1 ≤ h2. Set m(t) = (Nh2)(t) − (Nh1)(t). By the definition of the operator N we get

m(t) = (Nh2)(t) − (Nh1)(t)

= h2(t) −M∫ t

0(Nh2)(s)ds −

[h1(t) −

∫ t

0(Nh1)(s)ds

]

= h2(t) − h1(t) −M∫ t

0[(Nh2)(s)ds − (Nh1)(s)]ds

≥ −M∫ t

0m(s)ds.

(3.10)

This integral inequality implies m(t) ≥ 0 (for all t ∈ J); that is, N is an increasing operator.Set

Qv = x0 +∫ t

0v(s)ds. (3.11)

Obviously, Q : Lp[J, R] → C[J, R] is an increasing continuous operator. Set

(Cx)(t) = f(t, x(t)) +Mx(t), x ∈ C[J, R]. (3.12)

By (H2), C maps element of C[J, R] into measurable functions. For any u ∈ [u0, v0], by (H3)and (H4) we get

Cu0 ≤ Cu ≤ Cv0. (3.13)

This implies Cu ∈ Lp[J, R]. Hence C maps [u0, v0] into Lp[J, R] and C is an increasingoperator. Set

C[J, R] = X, Lp[J, R] = Y, B =NQ, A = BC, D = [u0, v0]. (3.14)

By above discussions we know that C : D → Y and B : Y → X are all increasing. Thusconditions (i) and (ii) in Theorem 2.5 are satisfied.

Let hn, h∗ ∈ D such that hn → h∗ in C[J, R]; by (H2) we have

limn→∞

f(t, hn(t)) +Mhn(t) = f(t, h∗(t)) +Mh∗(t), for a.a. t ∈ J. (3.15)

For any ϕ(t) ∈ Lq[J, R] (p−1 + q−1 = 1), by (2.29), we have

0 ≤ f(t, hn(t)) +Mhn(t) −[f(t, u0(t)) +Mu0(t)

]

≤ f(t, v0(t)) +Mv0(t) −[f(t, u0(t)) +Mu0(t)

],

(3.16)


and hence

∣∣f(t, hn(t)) +Mhn(t)∣∣ ≤ H(t), (3.17)

where H(t) = |f(t, v0(t)) +Mv0(t)| + 2|f(t, u0(t)) +Mu0(t)|. By (H3), H(t) ∈ Lp[J, R]; thus

ϕ(t)∣∣f(t, hn(t)) +Mhn(t)

∣∣ ≤ ϕ(t)H(t), (3.18)

where ϕ(t)H(t) ∈ L1[J, R]. Applying the Lebesgue dominated convergence theorem, we have

limn→∞

∫J

ϕ(t)(f(t, hn(t)) +Mhn(t)

)dt =

∫J

ϕ(t)(f(t, h∗(t)) +Mh∗(t)

)dt. (3.19)

This implies that Chnw−−−→ Ch∗ in Lp[J, R]; that is, C is a demicontinuous operator. Since the

cone in Lp[J, R] is regular, it is easy to see that C is an MMC-operator. Thus condition (iii) inTheorem 2.5 is satisfied.

We now show that condition (iv) in Theorem 2.5 is fulfilled. By (H1) and (3.5), andnoting the definition of operator N, we get

(Au0)(t) − u0(t) = (NQC)u0(t) − u0(t)

=N

(x0 +

∫ t

0

[f(s, u0(s)) +Mu0(s)

]ds

)− u0(t)

= x0 +∫ t

0

[f(s, u0(s)) +Mu0(s)

]ds −M

∫ t

0(Au0)(s)ds − u0(t)

≥ −M∫ t

0[(Au0)(s) − u0(s)]ds.

(3.20)

This implies that (Au0)(t) − u0(t) ≥ 0, for all t ∈ J , that is, u0 ≤ Au0. Similarly we can showthat Av0 ≤ v0.

Since all conditions in Theorem 2.5 are satisfied, by Theorem 2.5, A has the maximalfixed point and the minimal fixed point in D. Observing that fixed point of A is equivalent tosolutions of (3.5), and (3.5) is equivalent to (3.4), the conclusions of Theorem 3.1 hold.

Remark 3.2. In the proof of Theorem 3.1, we obtain the uniformly convergence of themonotone sequences without the compactness condition.

Acknowledgment

The project supported by the National Science Foundation of China (10971179).


References

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[2] H. Amann, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,”SIAM Review, vol. 18, no. 4, pp. 620–709, 1976.

[3] J. Sun and Z. Zhao, “Fixed point theorems of increasing operators and applications to nonlinearintegro-differential equations with discontinuous terms,” Journal of Mathematical Analysis andApplications, vol. 175, no. 1, pp. 33–45, 1993.

[4] S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous NonlinearDifferential Equations, vol. 181 of Monographs and Textbooks in Pure and Applied Mathematics, MarcelDekker, New York, NY, USA, 1994.

[5] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Academic Press, New York, NY,USA, 1969.

[6] C. Klin-eam and S. Suantai, “Strong convergence of monotone hybrid method for maximal monotoneoperators and hemirelatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2009,Article ID 261932, 14 pages, 2009.

[7] S. Plubtieng and W. Sriprad, “An extragradient method and proximal point algorithm for inversestrongly monotone operators and maximal monotone operators in Banach spaces,” Fixed Point Theoryand Applications, vol. 2009, Article ID 591874, 16 pages, 2009.

[8] M. A. Krasnosel’skii and A. B. Lusnikov, “Regular fixed points and stable invariant sets of monotoneoperators,” Applied Functional Analysis, vol. 30, no. 3, pp. 174–183, 1996.

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Research ArticlePeriodic Point, Endpoint, and ConvergenceTheorems for Dissipative Set-Valued DynamicSystems with Generalized Pseudodistances inCone Uniform and Uniform Spaces

Kazimierz Włodarczyk and Robert Plebaniak

Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science,University of Łodz, Banacha 22, 90-238 Łodz, Poland

Correspondence should be addressed to Kazimierz Włodarczyk, [email protected]



Copyright q 2010 K. Włodarczyk and R. Plebaniak. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

In cone uniform and uniform spaces, we introduce the three kinds of dissipative set-valueddynamic systems with generalized pseudodistances and not necessarily lower semicontinuousentropies, we study the convergence of dynamic processes and generalized sequences of iterationsof these dissipative dynamic systems, and we establish conditions guaranteeing the existence ofperiodic points and endpoints of these dissipative dynamic systems and the convergence to theseperiodic points and endpoints of dynamic processes and generalized sequences of iterations ofthese dissipative dynamic systems. The paper includes examples.

1. Introduction

A set-valued dynamic system is defined as a pair (X, T), where X is a certain space and T is aset-valued map T : X → 2X ; in particular, a set-valued dynamic system includes the usualdynamic system where T is a single-valued map. Here 2X denotes the family of all nonemptysubsets of a space X.

Let (X, T) be a dynamic system. By Fix(T), Per(T), and End(T) we denote the sets ofall fixed points, periodic points, and endpoints of T , respectively, that is, Fix(T) = {w ∈ X : w ∈T(w)}, Per(T) = {w ∈ X : w ∈ T [q](w) for some q ∈ N} and End(T) = {w ∈ X : {w} = T(w)}.For each x ∈ X, a sequence (wm : m ∈ {0} ∪ N) such that

∀m∈{0}∪N{wm+1 ∈ T(wm)}, w0 = x, (1.1)


is called a dynamic process or a trajectory starting at w0 = x of the system (X, T) (for detailssee Aubin and Siegel [1], Aubin and Ekeland [2], and Aubin and Frankowska [3]). For eachx ∈ X, a sequence (wm : m ∈ {0} ∪ N) such that

∀m∈{0}∪N{wm+1 ∈ T [m+1](x)

}, w0 = x, (1.2)

T [m] = T ◦ T ◦ · · · ◦ T (m-times), m ∈ N, is called a generalized sequence of iterations starting atw0 = x of the system (X, T) (for details see Yuan [4, page 557], Tarafdar and Vyborny [5] andTarafdar and Yuan [6]). Each dynamic process starting from w0 is a generalized sequence ofiterations starting from w0, but the converse may not be true; the set T [m](w0) is, in general,bigger than T(wm−1). If (X, T) is single valued, then, for each x ∈ X, a sequence (wm : m ∈{0} ∪ N) such that

∀m∈{0}∪N{wm+1 = T [m+1](x)

}, w0 = x, (1.3)

is called a Picard iteration starting atw0 = x of the system (X, T). If (X, T) is single valued, then(1.1)–(1.3) are identical.

The notion of Banach’s contraction belongs to the most fundamental mathematicalideas. Caristi [7], Ekeland [8], Aubin and Siegel [1], Yuan [4], and Kirk [9] extended thisnotion to several directions (dissipative single-valued maps with lower semicontinuousentropies, variational inequlities for lower semicontinuous maps, dissipative set-valueddynamic systems with not necessarily lower semicontinuous entropies, generalized contrac-tions and asymptotic contractions, resp.). It is not our purpose to give a complete list ofrelated papers here.

LetX be a metric space with metric d and let (X, T) be a single-valued dynamic system.Racall that if

∃λ∈[0,1)∀x,y∈X{d(T(x), T

(y))

� λd(x, y)}, (1.4)

then (X, T) is called a Banach’s contraction (Banach [10]). (X, T) is called contractive if∀x,y∈X{0 < d(x, y)⇒ d(T(x), T(y)) < d(x, y)}. If ∃ε>0∀x,y∈X{0 < d(x, y) < ε ⇒ d(T(x), T(y)) <d(x, y)}, then (X, T) is called ε-contractive (Edelstein [11]). Contractive and ε-contractivemaps are some modifications of Banach’s contractions.

If (X, T) is single valued and

∀x∈X{d(x, T(x)) � ω(x) −ω(T(x))} (1.5)

for some ω : X → [0,+∞), then T is called Caristi’s map (Caristi [7]). Caristi’s maps(1.5) generalize Banach’s contractions (1.4) (for details see Kirk and Saliga [12, page 2766]).Banach’s contraction principle and Caristi’s fixed point theorem are essentially different:in complete metric space, Banach’s contraction is continuous, each Picard iteration of thiscontraction is convergent to a fixed point and this fixed point is unique (Banach [10]) whileCaristi’s map is not necessarily continuous and if ω in (1.5) is lower semicontinuous, theneach Picard iteration of this map is convergent to a fixed point and this fixed point is not


necessarily unique (Caristi [7]). Recall that Ekeland’s [8] variational principle concerninglower semicontinuous maps and Caristi’s fixed point theorem are equivalent.

A map ω : X → [0,+∞) is called a weak entropy or entropy of a set-valued dynamicsystem (X, T) if

∀x∈X∃y∈T(x){d(x, y)

� ω(x) −ω(y)} (1.6)

or

∀x∈X∀y∈T(x){d(x, y)

� ω(x) −ω(y)}, (1.7)

respectively, and (X, T) is called weak dissipative or dissipative if it has a weak entropy or anentropy, respectively; here ω is not necessarily lower semicontinuous. These two kinds ofdissipative maps were introduced and studied by Aubin and Siegel [1]. If (X, T) is singlevalued, then (1.5)–(1.7) are identical.

Various periodic, fixed point, convergence, and invariant set theorems for contractiveand ε-contractive single-valued and set-valued dynamic systems have been obtained byEdelstein [11], Ding and Nadler [13], and Nadler [14]. Investigations concerning theexistence of fixed points and endpoints and convergence of dynamic processes or generalizedsequences of iterations to fixed points or endpoints of single-valued and set-valuedgeneralized contractions (Yuan [4], Tarafdar and Yuan [6, 15], Tarafdar and Chowdhury[16], Tarafdar and Vyborny [5]) and dissipative dynamic systems when entropy ω is notnecessarily lower semicontinuous (Aubin and Siegel [1]) have been conducted by a numberof authors in different contexts; for example, see Kirk and Saliga [12], Willems [17], Zangwill[18], Justman [19], Maschler and Peleg [20] and Petrusel, Sıntamarian [21].

In this paper, inspired by these results, we introduce in cone uniform and uniformspaces the three kinds of dissipative set-valued dynamic systems with generalizedpseudodistances and with not necessarily lower semicontinuous entropies and we presentthe methods which are useful for establishing general conditions guaranteeing the existenceof periodic points and endpoints of these set-valued dynamic systems and conditions thatfor each starting point the dynamic processes or generalized sequences of iterations convergeand the limit is a periodic point or endpoint (see Sections 3–6). The presented definitions andresults are more general and different from those given in the literature and are new evenfor single-valued and set-valued dynamic systems in metric spaces. For details, see Section 7where examples, remarks, and some comparisons are included. This paper is a continuationof [22, 23].

2. Dissipative Set-Valued Dynamic Systems withGeneralized Pseudodistances in Cone Uniform Spaces

We define a real normed space to be a pair (L, ‖ · ‖), with the understanding that a vector spaceL over R carries the topology generated by the metric (a, b) → ‖a − b‖, a, b ∈ L.

A nonempty closed convex set H ⊂ L is called a cone in L if it satisfies: (H1)∀s∈(0,∞){sH ⊂ H}; (H2)H ∩ (−H) = {0}; (H3)H /= {0}.


It is clear that each cone H ⊂ L defines, by virtue of “a Hb if and only if b − a ∈ H,”an order of L under which L is an ordered normed space with cone H. We will write a≺Hb toindicate that a Hb but a/= b.

The following terminologies will be much used.

Definition 2.1 (see [22]). Let X be a nonempty set and let L be an ordered normed space with coneH.

(i) The family P = {pα : X×X → L, α ∈ A} is said to be a P-family of cone pseudometricson X(P-family, for short) if the following three conditions hold:

(P1) ∀α∈A∀x,y∈X{0 Hpα(x, y) ∧ x = y ⇒ pα(x, y) = 0};(P2) ∀α∈A∀x,y∈X{pα(x, y) = pα(y, x)};(P3) ∀α∈A∀x,y,z∈X{pα(x, z) Hpα(x, y) + pα(y, z)}.

(ii) If P is P-family, then the pair (X,P) is called a cone uniform space.

(iii) A P-family P is said to be separating if

(P4) ∀x,y∈X{x /=y ⇒ ∃α∈A{0≺Hpα(x, y)}}.

(iv) If a P-family P is separating, then the pair (X,P) is called a Hausdorff cone uniformspace.

A cone H is said to be solid if int(H)/= ∅; int(H) denotes the interior of H. We willwrite a�Hb to indicate that b − a ∈ int(H).

Definition 2.2. Let L be an ordered normed space with solid cone H and let (X,P) be a coneuniform space with cone H.

(i) We say that a sequence (wm : m ∈ N) in X is a P-convergent in X, if there existsw ∈ X such that ∀α∈A∀c∈L,0�Hc∃n0∈N∀m∈N;n0�m{pα(wm,w)�Hc}.

(ii) We say that a sequence (wm : m ∈ N) in X is a P-Cauchy sequence in X, if∀α∈A∀c∈L,0�Hc∃n0∈N∀m,n∈N;n0�m<n{pα(wm,wn)�Hc}.

(iii) If every P-Cauchy sequence in X is P-convergent in X, then (X,P) is called a P-sequentially complete cone uniform space.

(iv) The set-valued dynamic system (X, T) is called a cone closed set-valued dynamicsystem in X if whenever (wm : m ∈ N) is a sequence in X P-converging to w ∈ Xand (vm : m ∈ N) is a sequence P-converging to v ∈ X such that vm ∈ T(wm) for allm ∈ N, then v ∈ T(w).

(v) Let (X,P) be aP-sequentially complete cone uniform space. For an arbitrary subsetE of X, the cone closure of E, denoted by cl(E), is defined as the set cl(E) = {w ∈ X :∃(wm:m∈N)⊂E∀α∈A∀c∈L,0�Hc∃n0∈N∀m∈N;n0�m{pα(wm,w)�Hc}}. The subset E of X is saidto be a cone closed subset in X if cl(E) = E.

The cone H is normal if a real number M > 0 exists such that for each a, b ∈ H,0 Ha Hb implies ‖a‖ � M‖b‖. The number M satisfying the above is called the normalconstant of H.

The following holds.


Theorem 2.3 (see [22]). Let L be an ordered normed space with normal solid coneH and let (X,P)be a cone uniform space with coneH.

(a) Let (wm : m ∈ N) be a sequence in X and let w ∈ X. Then the sequence (wm : m ∈ N) isP-convergent to w if and only if

∀α∈A∀ε>0∃n0∈N∀m∈N;n0�m

{∥∥pα(wm,w)∥∥ < ε}. (2.1)

(b) Let (wm : m ∈ N) be a sequence in X. Then the sequence (wm : m ∈ N) is a P-Cauchysequence if and only if

∀α∈A∀ε>0∃n0∈N∀m,n∈N;n0�m<n

{∥∥pα(wm,wn)∥∥ < ε}. (2.2)

(c) Each P-convergent sequence is a P-Cauchy sequence.

Definition 2.4. Let L be an ordered normed space with normal solid cone H and let (X,P) bea cone uniform space with cone H.

(i) The family J = {Jα : X × X → L, α ∈ A} is said to be a J-family of conepseudodistances onX (J-family onX, for short) if the following three conditions hold:

(J1) ∀α∈A∀x,y∈X{0 HJα(x, y)};(J2) ∀α∈A∀x,y,z∈X{Jα(x, z) HJα(x, y) + Jα(y, z)};(J3) for any sequence (wm : m ∈ N) in X such that

∀α∈A∀ε>0∃n0∈N∀m,n∈N;n0�m�n{‖Jα(wm,wn)‖ < ε}, (2.3)

if there exists a sequence (vm : m ∈ N) in X satisfying

∀α∈A∀ε>0∃n0∈N∀m∈N;n0�m{‖Jα(wm, vm)‖ < ε}, (2.4)

then

∀α∈A∀ε>0∃n0∈N∀m∈N;n0�m

{∥∥pα(wm, vm)∥∥ < ε}. (2.5)

(ii) Let the family J = {Jα : X × X → L, α ∈ A} be a J-family on X. We say that asequence (wm : m ∈ N) in X is a J-Cauchy sequence in X if (2.3) holds.

For other families of cone pseudodistances in cone uniform spaces and variousapplications, see [22, 23]. The following is a consequence of Definition 2.4(i).

Proposition 2.5. Let (X,P) be a Hausdorff cone uniform space with coneH. Let the familyJ = {Jα :X ×X → L, α ∈ A} be a J-family. If ∀α∈A{Jα(x, y) = 0 ∧ Jα(y, x) = 0}, then x = y.


Proof. Let x, y ∈ X be such that ∀α∈A{Jα(x, y) = 0 ∧ Jα(y, x) = 0}. By (J2),∀α∈A{Jα(x, x) HJα(x, y) + Jα(y, x)}. By (J1), this gives ∀α∈A{Jα(x, x) = 0}. Thus, we get∀α∈A∀ε>0∃n0∈N∀m,n∈N;n0�m�n{‖Jα(wm,wn)‖ < ε} and ∀α∈A∀ε>0∃n0∈N∀m∈N;n0�m{‖Jα(wm, vm)‖ <ε} where wm = x, vm = y, m ∈ N, and, by (J3), ∀α∈A∀ε>0∃n0∈N∀m∈N;n0�m{‖pα(wm, vm)‖ < ε},that is, ∀α∈A∀ε>0{‖pα(x, y)‖ < ε}. Hence, ∀α∈A{pα(x, y) = 0}which, according to (P4), impliesthat x = y.

Now we introduce the following three kinds of dissipative set-valued dynamicsystems with generalized pseudodistances in cone uniform spaces (conditions (ii)–(iv)below).

Definition 2.6. Let (X,P) be a Hausdorff cone uniform space and let (X, T) be a set-valued dynamicsystem. Let J = {Jα : X × X → L, α ∈ A} be a J-family on X and let Ω = {ωα : X → L, α ∈ A}be a family of maps such that

∀α∈A∀x∈X{0 Hωα(x)}. (2.6)

(i) We say that a sequence (wm : m ∈ {0} ∪ N) in X is (J,Ω)-admissible if

∀α∈A∀m∈{0}∪N{Jα(wm,wm+1) Hωα(wm) −ωα(wm+1)}. (2.7)

(ii) If the following conditions are satisfied:

(C1) ∅/=X0 ⊂ X; and

(C2) x ∈ X0 if and only if there exists a (J,Ω)-admissible dynamic process (wm :m ∈ {0} ∪ N) starting at w0 = x of the system (X, T),

then we say that T is weak (J,Ω;X0)-dissipative on X

(iii) We say that T is (J,Ω)-dissipative on X if, for each x ∈ X, each dynamic process(wm : m ∈ {0} ∪ N) starting at w0 = x of the system (X, T) is (J,Ω)-admissible.

(iv) We say that T is strictly (J,Ω)-dissipative on X if, for each x ∈ X, each generalizedsequence of iterations (wm : m ∈ {0} ∪ N) starting at w0 = x of the system (X, T) is(J,Ω)-admissible.

If one of the conditions (ii)–(iv) holds, then we say that (X, T) is a dissipative set-valueddynamic system with respect to (J,Ω) (dissipative set-valued dynamic system, for short).

Remark 2.7. It is worth noticing that if a sequence (wm : m ∈ {0}∪N) inX is (J,Ω)-admissible,then, for each k ∈ N, a sequence (wm+k : m ∈ {0} ∪ N) is (J,Ω)-admissible. Consequently, ifT is weak (J,Ω;X0)-dissipative on X, x ∈ X0, and (wm : m ∈ {0} ∪ N) is a dynamic processstarting at w0 = x of the system (X, T) which is (J,Ω)-admissible, then ∀m∈N{wm ∈ X0}; ingeneral, T(X0)/=X0 (see Example 7.3).

Now we can give the following conclusion.


Proposition 2.8. Let (X,P) be a Hausdorff cone uniform space and let (X, T) be a set-valued dynamicsystem.

(a) If T is weak (J,Ω;X0)-dissipative on X, then (X0,KJ;T ) is a set-valued dynamic systemwhere, for each x ∈ X0,

KJ;T (x) =⋃{{w0, w1, w2, . . .} : (wm : m ∈ {0} ∪ N) ∈ KJ(T, x)}, (2.8)

KJ(T, x) = {(wm : m ∈ {0} ∪ N),

w0=x : ∀m∈{0}∪N{wm+1 ∈ T(wm)∧∀α∈A{Jα(wm,wm+1) Hωα(wm) −ωα(wm+1)}}}.

(2.9)

(b) If T is (J,Ω)-dissipative on X, then (X,WJ;T) is a set-valued dynamic system where, foreach x ∈ X,

WJ;T (x) =⋃{{w0, w1, w2, . . .} : (wm : m ∈ {0} ∪ N) ∈ WJ(T, x)}, (2.10)

WJ(T, x) ={(wm : m ∈ {0} ∪ N), w0 = x : ∀m∈{0}∪N{wm+1 ∈ T(wm)}

}. (2.11)

(c) If T is strictly (J,Ω)-dissipative onX, then (X,SJ;T ) is a set-valued dynamic system where,for each x ∈ X,

SJ;T (x) =⋃{{w0, w1, w2, . . .} : (wm : m ∈ {0} ∪ N) ∈ SJ(T, x)}, (2.12)

SJ(T, x) ={(wm : m ∈ {0} ∪ N), w0 = x : ∀m∈{0}∪N

{wm+1 ∈ T [m+1](w0)

}}. (2.13)

Proof. The fact that

KJ;T : X0 −→ 2X0 , WJ;T : X −→ 2X, SJ;T : X −→ 2X (2.14)

follows from (1.1), (1.2), Definition 2.6, Remark 2.7, and (2.8)–(2.13).

Remark 2.9. By Proposition 2.8 and Definition 2.6, we get:

(i) If T is (J,Ω)-dissipative onX, then T is weak (J,Ω;X0)-dissipative onX forX0 = Xand ∀x∈X0{KJ;T (x) = WJ;T (x)}.

(ii) If T is strictly (J,Ω)-dissipative on X, then T is (J,Ω)-dissipative on X and∀x∈X{WJ;T (x) ⊂ SJ;T (x)}.

Definition 2.10. Let L be an ordered normed space with solid cone H. The cone His called regular if for every incresing (decresing) sequence which is bounded fromabove (below), that is, if for each sequence (cm : m ∈ N) in L such thatc1 Hc2 H · · · Hcm H · · · Hb (b H · · · Hcm H · · · Hc2 Hc1) for some b ∈ L, there existsc ∈ L such that limm→∞‖cm − c‖ = 0.

Remark 2.11. Every regular cone is normal; see [24].


3. Periodic Point and Convergence Theorem for Weak(J,Ω;X0)-Dissipative, (J,Ω)-Dissipative, andStrictly (J,Ω)-Dissipative Set-Valued DynamicSystems in Cone Uniform Spaces

Our main result of this section is the following.

Theorem 3.1. Let L be an ordered Banach space with a regular solid cone H and let (X,P) be aHausdorff sequentially complete cone uniform space with coneH. Let J = {Jα : X ×X → L, α ∈ A}be a J-family on X and let Ω = {ωα : X → L, α ∈ A} be a family of maps such that∀α∈A∀x∈X{0 Hωα(x)}. Let (X, T) be a set-valued dynamic system. The following hold.

(a) If T is weak (J,Ω;X0)-dissipative onX, then, for each x ∈ X0 and for each dynamic process(wm : m ∈ {0} ∪ N) ∈ KJ(T, x), there exists w ∈ cl(X0) such that (wm : m ∈ {0} ∪ N)is P-convergent to w. If, in addition, the map T [q] is cone closed in X for some q ∈ N, thenw ∈ T [q](w).

(b) If T is (J,Ω)-dissipative on X, then, for each x ∈ X and for each dynamic process (wm :m ∈ {0} ∪ N) ∈ WJ(T, x), there exists w ∈ X such that (wm : m ∈ {0} ∪ N) is P-convergent to w. If, in addition, the map T [q] is cone closed in X for some q ∈ N, thenw ∈ T [q](w).

(c) If T is strictly (J,Ω)-dissipative on X, then, for each x ∈ X and for each generalizedsequence of iterations (wm : m ∈ {0} ∪ N) ∈ SJ(T, x), there exists w ∈ X such that(wm : m ∈ {0} ∪ N) is P-convergent to w. If, in addition, the map T [q] is cone closed inX for some q ∈ N, then, for each x ∈ X, there exists a generalized sequence of iterations(wm : m ∈ {0}∪N) ∈ SJ(T, x) andw ∈ X such that (wm : m ∈ {0}∪N) isP-convergentto w and w ∈ T [q](w).

Proof. The proof will be broken into three steps.

Step 1. Let (i) x ∈ X0 and (wm : m ∈ {0} ∪ N) ∈ KJ(T, x); or (ii) x ∈ X and (wm : m ∈{0} ∪N) ∈ WJ(T, x) ∪ SJ(T, x). We show that (wm : m ∈ {0} ∪N) is J-Cauchy and P-Cauchy,that is,

∀α∈A∀ε>0∃n0∈N∀m,n∈N;n0�m�n{‖Jα(wm,wn)‖ < ε}, (3.1)

∀α∈A∀ε>0∃n0∈N∀m,n∈N;n0�m<n

{∥∥pα(wm,wn)∥∥ < ε}, (3.2)

respectively; see Definitions 2.4(ii) and 2.2(ii) and Theorem 2.3(b).

Indeed, since L is transitive, by (2.9), (2.11), (2.13), Definition 2.6(ii)–(iv) and (J1),we get that ∀α∈A∀m∈{0}∪N{ωα(wm+1) Hωα(wm)}. According to (2.6), for each α ∈ A, thesequence (ωα(wm) : m ∈ {0} ∪ N) is contained in H, bounded from below and, by the above,nonincreasing. Since H is a closed and regular cone, it follows that

∀α∈A∃uα∈H{

limm→∞

‖ωα(wm) − uα‖ = 0}. (3.3)


Let now α0 ∈ A and ε0 > 0 be arbitrary and fixed. By (3.3),

∃n0∈N∀m;n0�m

{‖ωα0(wm) − uα0‖ <

ε0

2M

}, (3.4)

where M is a normal constant of H (see Remark 2.11). Furthermore, for n0 < m � n, using(J1), (J2), and (2.7), 0 HJα0(wm,wn) H

∑n−1k=m Jα0(wk,wk+1 Hωα0(wm)−ωα0(wn) and next, by

(3.4) and the fact thatH is normal (see Remark 2.11), ‖Jα0(wm,wn)‖ � M‖ωα0(wm)−ωα0(wn)‖=M‖ωα0(wm)−uα0−ωα0(wn)+uα0‖ � M‖ωα0(wm)−uα0‖+M‖ωα0(wn)−uα0‖ � ε0/2+ε0/2 = ε0.Therefore, (3.1) holds.

Also we can show that (3.2) holds. Indeed, by (3.1), ∀α∈A∀ε>0∃n0∈N∀m�n0

∀k∈{0}∪N{‖Jα(wm,wk+m)‖ < ε}. Hence, if i0 ∈ N and j0 ∈ {0} ∪ N where i0 > j0 and

um = wi0+m, vm = wj0+m for m ∈ N, (3.5)

then we obtain

∀α∈A∀ε>0∃n0∈N∀m�n0{‖Jα(wm, um)‖ < ε ∧ ‖Jα(wm, vm)‖ < ε}. (3.6)

We obtain according to (3.1), (3.6), and (J3) that

∀α∈A∀ε>0∃n0∈N∀m�n0

{∥∥pα(wm, um)∥∥ < ε ∧ ∥∥pα(wm, vm)

∥∥ < ε}. (3.7)

By (3.5), from (3.7) it follows that

∀α∈A∀ε>0∃n0∈N∀m�n0

{∥∥pα(wm,wi0+m)∥∥ < ε

2M∧ ∥∥pα(wm,wj0+m

)∥∥ < ε

2M

}. (3.8)

Next, if n0 � m < n, then n = i0+n0 andm = j0+n0 for some i0 ∈ N and j0 ∈ {0}∪N suchthat i0 > j0. Thus, by (P1)–(P3), ∀α∈A{0 Hpα(wm,wn) = pα(wi0+n0 , wj0+n0) Hpα(wn0 , wi0+n0) +pα(wn0 , wj0+n0)}. Using (3.8), this gives ∀α∈A{‖pα(wm,wn)‖ � M‖pα(wn0 , wi0+n0)‖ +M‖pα(wn0 , wj0+n0)‖ < ε/2 + ε/2 = ε}. The proof of (3.2) is complete.

Step 2. Assertions (a) and (b) hold.

Indeed, let (i) x ∈ X0 and (wm : m ∈ {0} ∪ N) ∈ KJ(T, x); or (ii) x ∈ X and (wm : m ∈{0} ∪ N) ∈ WJ(T, x).

Since ∀m∈{0}∪N{wm ∈ KJ;T (x)} or ∀m∈{0}∪N{wm ∈ WJ;T(x)}, X is a Hausdorffand sequentially complete cone space and (2.14) holds, therefore, by virtue of Step 1,Proposition 2.8 and Theorem 2.3(b) and (c), we claim that (wm : m ∈ {0} ∪ N) is a P-Cauchysequence and there exists a unique w ∈ cl(KJ;T(x)) or w ∈ cl(WJ;T (x)), respectively, whereKJ;T (x) ⊂ X0, WJ;T (x) ⊂ X, and cl(X) = X, such that the sequence (wm : m ∈ {0} ∪ N) isP-convergent to w, that is, ∀α∈A{limm→∞‖pα(wm,w)‖ = 0}.

Now we see that if T [q] is cone closed for some q ∈ N, then the point w satisfiesw ∈ T [q](w). Indeed, by (2.9) or (2.11), we conclude that

∀m∈N{wm ∈ T(wm−1) ⊂ T [2](wm−2) ⊂ · · · ⊂ T [m−1](w1) ⊂ T [m](w0)

}, (3.9)


which gives

wmq+k ∈ T [q](w(m−1)q+k)

for k = 1, 2, . . . , q, m ∈ N. (3.10)

Since T [q] is cone closed in X and limm→∞wm = w, therefore, by (3.10) and Definition 2.2(iv),we get w ∈ T [q](w).

Step 3. Assertion (c) holds.

Indeed, if x ∈ X and (wm : m ∈ {0} ∪ N) ∈ SJ(T, x), then, by virtue of Step 1,Proposition 2.8 and Theorem 2.3(b) and (c), we claim that ∀m∈{0}∪N{wm ∈ T [m](x) ⊂ SJ;T (x) ⊂X}, (wm : m ∈ {0} ∪ N) is a P-Cauchy sequence, and there exists a unique w ∈ cl(SJ;T(x))such that the sequence (wm : m ∈ {0} ∪ N) P-converges to w.

If T [q] is cone closed in X for some q ∈ N, then, for each x ∈ X, by Remark 2.9(b)and Step 2 (part (b)), there exists a generalized sequence of iterations (wm : m ∈ {0} ∪ N) ∈SJ(T, x) ∩ WJ(T, x) ⊂ SJ(T, x) satisfying (3.9) and (3.10) and there exists w ∈ cl(WJ;T (x)) ⊂cl(SJ;T (x)) ⊂ X such that (wm : m ∈ {0} ∪ N) is P-convergent to w and w ∈ T [q](w).

4. Dissipative Set-Valued Dynamic Systems withGeneralized Pseudodistances in Uniform Spaces

Let (X,D) be a Hausdorff uniform space with uniformity defined by a saturated family D ={dα : α ∈ A} of pseudometrics dα, α ∈ A, uniformly continuous on X2.

Definition 4.1. Let (X,D) be a Hausdorff uniform space. The familyU = {Uα : X×X → [0,∞),α ∈ A} is said to be a U-family of generalized pseudodistances on X (U-family, for short) if thefollowing two conditions hold:

(U1) ∀α∈A∀x,y,z∈X{Uα(x, z) � Uα(x, y) +Uα(y, z)};(U2) for any sequence (wm : m ∈ N) in X such that

∀α∈A{

limm→∞

supn>m

Uα(wm,wn) = 0}, (4.1)

if there exists a sequence (vm : m ∈ N) in X satisfying

∀α∈A{

limm→∞

Uα(wm, vm) = 0}, (4.2)

then

∀α∈A{

limm→∞

dα(wm, vm) = 0}. (4.3)

Definition 4.2. Let (X,D) be a Hausdorff uniform space and let (X, T) be a set-valued dynamicsystem. Let the family U = {Uα : X ×X → [0,∞), α ∈ A}be a U-family and let Γ = {γα : X →[0,+∞), α ∈ A} be a family of maps.


(i) We say that a sequence (wm : m ∈ {0} ∪ N) in X is (U,Γ)-admissible if

∀α∈A∀m∈{0}∪N{Uα(wm,wm+1) � γα(wm) − γα(wm+1)

}. (4.4)

(ii) If the following conditions are satisfied:

(C1) ∅/=X0 ⊂ X;(C2) x ∈ X0 if and only if there exists a (U,Γ)-admissible dynamic process (wm :

m ∈ {0} ∪ N) starting at w0 = x of the system (X, T),

then we say that T is weak (U,Γ;X0)-dissipative on X.

(iii) We say that T is (U,Γ)-dissipative on X if, for each x ∈ X, each dynamic process(wm : m ∈ {0} ∪ N) starting at w0 = x of the system (X, T) is (U,Γ)-admissible.

(iv) We say that T is strictly (U,Γ)-dissipative on X if, for each x ∈ X, each generalizedsequence of iterations (wm : m ∈ {0} ∪ N) starting at w0 = x of the system (X, T) is(U,Γ)-admissible.

If one of the conditions (ii)–(iv) holds, then we say that (X, T) is a dissipative set-valueddynamic system with respect to (U,Γ)(dissipative set-valued dynamic system, for short).

Proposition 4.3. Let (X,D) be a Hausdorff uniform space and let (X, T) be a set-valued dynamicsystem.

(a) If T is weak (U,Γ;X0)-dissipative on X, then (X0,KU;T ) is a set-valued dynamic systemwhere, for each x ∈ X0, KU;T (x) =

⋃{{w0, w1, w2, . . .} : (wm : m ∈ {0} ∪ N) ∈KU(T, x)} and KU(T, x) = {(wm : m ∈ {0} ∪ N), w0 = x : ∀m∈{0}∪N{wm+1 ∈T(wm) ∧ ∀α∈A{Uα(wm,wm+1) � γα(wm) − γα(wm+1)}}}.

(b) If T is (U,Γ)-dissipative on X, then (X,WU;T ) is a set-valued dynamic system where, foreach x ∈ X, WU;T (x) =

⋃{{w0, w1, w2, . . .} : (wm : m ∈ {0} ∪ N) ∈ WU(T, x)}, andWU(T, x) = {(wm : m ∈ {0} ∪ N), w0 = x : ∀m∈{0}∪N{wm+1 ∈ T(wm)}}.

(c) If T is strictly (U,Γ)-dissipative onX, then (X,SU;T ) is a set-valued dynamic system where,for each x ∈ X, SU;T (x) =

⋃{{w0, w1, w2, . . .} : (wm : m ∈ {0} ∪ N) ∈ SU(T, x)} andSU(T, x) = {(wm : m ∈ {0} ∪ N), w0 = x : ∀m∈{0}∪N{wm+1 ∈ T [m+1](w0)}}.

5. Periodic Point and Convergence Theorem for Weak(U,Γ;X0)-Dissipative, (U,Γ)-Dissipative, and Strictly(U,Γ)-Dissipative Set-Valued Dynamic Systems in Uniform Spaces

Let (Λ,≥Λ) denote a directed set whose elements will be indicated by the letters λ, η, μ. LetT : X → 2Y where X and Y are topological spaces.

The following are equivalent: (a) the map T is closed, that is, the graph of T is closedin X × X; (b) whenever (wλ : λ ∈ Λ) is a net converging to w and (vλ : λ ∈ Λ) is a netconverging to v such that vλ ∈ T(wλ) for all λ ∈ Λ, then v ∈ T(w). Recall that the graph of T is{(x, y) : x ∈ X, y ∈ T(x)}.

The map T is called upper semicontinuous at w ∈ X if for each open set G containingT(w) there exists a neighbourhood N(w) of w such that T(v) ⊂ G for each v ∈ N(w) and


upper semicontinuous in X if it is upper semicontinuous at each point w of X and T(w) iscompact for each w ∈ X; see Berge [25, page 111].

Remark 5.1. It is known that

(i) if the map T is closed, then, for each x ∈ X, the set T(x) is closed [25, page 111] but,generally, the converse is not possible [26, Example 7.19, page 75].

(ii) every upper semicontinuous map is closed [25, Theorem 6, page 112] and, if X is acompact space, then the map is closed if and only if it is upper semicontinuous [25,Corollary, page 112].

Definition 5.2. Let (X,D) be a Hausdorff uniform space.

(i) We say that a sequence (wm : m ∈ N) in X is a Cauchy sequence in X if∀α∈A{limm→∞ supn>m dα(wm,wn) = 0}.

(ii) We say that a sequence (wm : m ∈ N) in X converges in X, if there exists w ∈ Xsuch that ∀α∈A{limm→∞ dα(wm,w) = 0}.

(iii) If every Cauchy sequence inX is convergent inX, then (X,D) is called a sequentiallycomplete uniform space.

(iv) Let the family U = {Uα : X × X → [0,∞), α ∈ A} be a U-family on X.We say that a sequence (wm : m ∈ N) in X is a U-Cauchy sequence in X if∀α∈A{limm→∞ supn>m Uα(wm,wn) = 0}.

Consequence of Theorem 3.1 is the following.

Theorem 5.3. Let (X,D) be a Hausdorff sequentially complete uniform space and let (X, T) be a set-valued dynamic system. Let the family U = {Uα : X × X → [0,∞), α ∈ A} be a U-family and letΓ = {γα : X → 0,+∞], α ∈ A} be a family of proper maps. The following hold.

(a) If T is weak (U,Γ;X0)-dissipative on X, then, for each x ∈ X0 and for each dynamic process(wm : m ∈ {0} ∪ N) ∈ KU(T, x), there exists w ∈ cl(X0) such that (wm : m ∈ {0} ∪ N)is convergent tow. If, in addition, the map T [q] is closed (or T [q] is upper semicontinuous )in X for some q ∈ N, then w ∈ T [q](w).

(b) If T is (U,Γ)-dissipative on X, then, for each x ∈ X and for each dynamic process (wm :m ∈ {0}∪N) ∈ WU(T, x), there existsw ∈ X such that (wm : m ∈ {0}∪N) is convergenttow. If, in addition, the map T [q] is closed (or T [q] is upper semicontinuous ) in X for someq ∈ N, then w ∈ T [q](w).

(c) If T is strictly (U,Γ)-dissipative on X, then, for each x ∈ X and for each generalizedsequence of iterations (wm : m ∈ {0} ∪ N) ∈ SU(T, x), there exists w ∈ X such that(wm : m ∈ {0} ∪ N) is convergent to w. If, in addition, the map T [q] is closed (or T [q]

is upper semicontinuous ) in X for some q ∈ N, then, for each x ∈ X, there exists ageneralized sequence of iterations (wm : m ∈ {0} ∪ N) ∈ SU(T, x) and w ∈ X such that(wm : m ∈ {0} ∪ N) is convergent to w and w ∈ T [q](w).


6. Endpoint and Convergence Theorem for(U,Γ)-Dissipative and Strictly (U,Γ)-Dissipative Set-ValuedDynamic Systems in Uniform Spaces

We recall the following.

Definition 6.1. Let (X, τ) be a topological space and let (X, T) be a set-valued dynamic system.The map T is called lower semicontinuous at w ∈ X (written: lsc at w ∈ X) if and only if for anyv ∈ T(w) and for any net (wλ : λ ∈ Λ) of elements wλ ∈ X, λ ∈ Λ, τ-converging to w, thereexists a net (vλ : λ ∈ Λ) of elements vλ ∈ T(wλ), λ ∈ Λ, τ-converging to v. The map T is calledlsc on X if it is lsc at every point of w ∈ X.

The main result of this section is what follows.

Theorem 6.2. Let (X,D) be a Hausdorff sequentially complete uniform space and let (X, T) be a set-valued dynamic system. Let the family U = {Uα : X × X → [0,∞), α ∈ A} be a U-family and letΓ = {γα : X → 0,∞), α ∈ A} be a family of maps.

Assume that T is (U,Γ)-dissipative on X. Then the following hold.

(a1) The map WU;T is (U,Γ)-dissipative on X and, for each x ∈ X, there exist (wm : m ∈{0} ∪ N) ∈ WU(WU;T , x) and w ∈ X such that (wm : m ∈ {0} ∪ N) converges to w and⋂m�0 cl(WU;T (wm)) = {w}.

(a2) If, for each x ∈ X, WU;T (x) is a closed set, then End(T)/= ∅ and, for each x ∈ X, there exist(wm : m ∈ {0} ∪ N) ∈ WU(WU;T , x) and w ∈ End(T) such that (wm : m ∈ {0} ∪ N)converges to w and

⋂m�0 WU;T (wm) = {w}.

(a3) If T is lsc on X, then End(T)/= ∅ and, for each x ∈ X, there exist (wm : m ∈ {0} ∪ N) ∈WU(WU;T , x) and w ∈ End(T) such that (wm : m ∈ {0} ∪ N) converges to w and⋂m�0 cl(WU;T (wm)) = {w}.

Assume that T is strictly (U,Γ)-dissipative on X. Then the following hold.

(b1) The map SU;T is strictly (U,Γ)-dissipative on X and, for each x ∈ X, there exist (wm : m ∈{0} ∪ N) ∈ SU(SU;T , x) and w ∈ X such that (wm : m ∈ {0} ∪ N) converges to w and⋂m�0 cl(SU;T (wm)) = {w}.

(b2) If, for each x ∈ X, SU;T (x) is a closed set, then End(T)/= ∅ and, for each x ∈ X, there exist(wm : m ∈ {0} ∪ N) ∈ SU(SU;T , x) and w ∈ End(T) such that (wm : m ∈ {0} ∪ N)converges to w and

⋂m�0 SU;T (wm) = {w}.

(b3) If T is lsc on X, then End(T)/= ∅ and, for each x ∈ X, there exist (wm : m ∈ {0} ∪N) ∈ SU(SU;T , x) and w ∈ End(T) such that (wm : m ∈ {0} ∪ N) converges to w and⋂m�0 cl(SU;T (wm)) = {w}.

Proof. (a1) The proof of (a1) will be broken into six steps.

Step 1. We show that ∀α∈A∀x∈X∀y∈WU;T (x){Uα(x, y) � γα(x) − γα(y)}.

Indeed, let x ∈ X and y ∈ WU;T (x) be arbitrary and fixed. By definition of WU;T (x),there exist a dynamic process (wm : m ∈ {0} ∪ N) ∈ WU(T, x) starting at w0 = x ofthe system (X, T) and m0 ∈ {0} ∪ N such that y = wm0 ; recall that then (1.1) and (4.4)hold (i.e., ∀m∈{0}∪N{wm+1 ∈ T(wm)} and ∀α∈A∀m∈{0}∪N{Uα(wm,wm+1) � γα(wm) − γα(wm+1)}


hold). By virtue of (U1), this gives ∀α∈A{Uα(x, y) � ∑m0−1m=0 Uα(wm,wm+1) � ∑m0−1

m=0 [γα(wm) −γα(wm+1)] = γα(x) − γα(y)}.

Step 2. We show that WU;T is (U,Γ)-dissipative on X.

If x ∈ X and (wm : m ∈ {0} ∪ N) is a dynamic process starting at w0

= x of the system (X,WU;T ), that is, ∀m∈{0}∪N{wm+1 ∈ WU;T (wm)}, then, by Step 1,∀α∈A∀m∈{0}∪N{Uα(wm,wm+1) � γα(wm) − γα(wm+1)}, that is, (wm : m ∈ {0} ∪ N) is (U,Γ)-admissible. This gives that WU;T is (U,Γ)-dissipative on X.

Step 3. We show that ∀x∈X{WU(T, x) ⊂ WU(WU;T , x)}.

Indeed, by Proposition 4.3(b), for x ∈ X, WU(T, x) = {(wm : m ∈ {0} ∪ N) :∀m∈{0}∪N{wm+1 ∈ T(wm)}, w0 = x}. Next, by Step 2, WU;T is (U,Γ)-dissipative on X.Consequently,

∀x∈X{WU(WU;T , x) =

{(cm : m ∈ {0} ∪ N) : ∀m∈{0}∪N{cm+1 ∈WU;T (cm)}, c0 = x

}}, (6.1)

where

∀m∈{0}∪N{WU;T (cm) =⋃{{s0, s1, s2, . . .} :

(sj : j ∈ {0} ∪ N

) ∈ WU(T, cm)}

=⋃{{s0, s1, s2, . . .} : s0 = cm, sj ∈ T

(sj−1), j ∈ N

}}.

(6.2)

Let now (wm : m ∈ {0} ∪ N) ∈ WU(T, x). Then ∀m∈{0}∪N{wm+1 ∈ T(wm)}.Hence ∀m∈{0}∪N∀k�m{wk+1 ∈ T(wk)}. Thus ∀m∈{0}∪N{(wm,wm+1, wm+2, . . .) ∈ WU(T,wm)}.Hence, by (6.2), ∀m∈{0}∪N{{wm,wm+1, wm+2, . . .} ⊂ WU;T (wm)}. In particular, ∀m∈{0}∪N{wm+1 ∈WU;T (wm)}. By (6.1), (wm : m ∈ {0} ∪ N) ∈ WU(WU;T , x).

Step 4. If x ∈ X and (wm : m ∈ {0} ∪ N) ∈ WU(WU;T , x), then

∀α∈A∀m∈{0}∪N{γα(wm+1) � γα(wm)

}. (6.3)

Indeed, by (6.1), ∀m∈{0}∪N{wm+1 ∈ WU;T (wm)}, w0 = x. Hence, by virtue of theStep 2 and Definition 4.2(iii), the sequence (wm : m ∈ {0} ∪ N) is (U,Γ)-admissible, that is,∀α∈A∀m∈{0}∪N{Uα(wm,wm+1) � γα(wm)− γα(wm+1)}. By definitions ofU and Γ, this gives (6.3).

Step 5. Let

∀α∈A∀x∈X{Δα(WU;T (x)) = sup{Uα(x, t) : t ∈WU;T (x)}

}(6.4)

and let

∀α∈A∀x∈X{α;WU;T (x) = inf

{γα(t) : t ∈WU;T (x)

}}. (6.5)


Then

∀α∈A∀x∈X{Δα(WU;T (x)) � γα(x) − α;WU;T (x)

}. (6.6)

Indeed, if α ∈ A and x ∈ X, then, by Step 1, we get Δα(WU;T (x)) = sup{Uα(x, t) : t ∈WU;T (x)} � sup{γα(x) − γα(t) : t ∈WU;T (x)} � γα(x) − ρα;WU;T (x).

Step 6. Let δα(E) = sup{dα(y1, y2) : y1, y2 ∈ E}, E ∈ 2X , α ∈ A. Then, for each x ∈ X, thereexist a dynamic process (wm : m ∈ {0} ∪ N) ∈ WU(WU;T , x) and a unique w ∈ X such that

∀α∈A{

limm→∞

supn>m

Uα(wm,wn) = 0}, (6.7)

∀α∈A{

limm→∞

supn>m

dα(wm,wn) = 0}, (6.8)

∀α∈A{

limm→∞

dα(wm,w) = 0}, (6.9)

∀α∈A{

limm→∞

δα(WU;T (wm)) = limm→∞

δα(cl(WU;T (wm))) = 0}, (6.10)

⋂m�0

cl(WU;T (wm)) = {w}. (6.11)

Indeed, first, let us observe that since T is (U,Γ)-dissipative on X and ∀x∈X{x ∈WU;T (x)} (i.e., ∀x∈X{WU;T (x)/= ∅}), thus there exists

∀α∈A∀x∈X{α;WU;T (x) = inf

{γα(t) : t ∈WU;T (x)

}}. (6.12)

Now let x ∈ X and α0 ∈ A be arbitrary and fixed. Definingw0 = x, since T is (U,Γ)-dissipativeon X and w0 ∈WU;T (w0), by (6.2), we have that

WU;T (w0) =⋃{{s0, s1, s2, . . .} : s0 = w0, sj ∈ T

(sj−1), j ∈ N

}/= ∅. (6.13)

Therefore, by (6.12) and (6.13), there existsw1 ∈WU;T (w0) such that γα0(w1) � α0;WU;T (w0)+1.Similarly, there exists w2 ∈WU;T (w1) such that γα0(w2) � α0;WU;T (w1) + 2−1. By induction, wemay construct a dynamic process (wm : m ∈ {0} ∪ N) ∈ WU(WU;T , x) satisfying (see (6.1),(6.2), and (6.3) and Step 1)

∀m∈N{wm ∈WU;T (wm−1)}, (6.14)

∀m∈{0}∪N{γα0(wm+1) � α0;WU;T (wm) + 2−m

}, (6.15)

where ∀m∈N{α0;WU;T (wm) = inf{γα0(t) : t ∈WU;T (wm)}}. Next, by (6.2),

∀m∈N{WU;T (wm) ⊂WU;T (wm−1)} (6.16)


which gives

∀m∈{0}∪N{α0;WU;T (wm) � α0;WU;T (wm+1)

}. (6.17)

By (6.6), (6.15), and (6.17), Δα0(WU;T (wm+1)) � γα0(wm+1) − α0;WU;T (wm+1) � α0;WU;T (wm) +2−m − α0;WU;T (wm)] � 2−m which implies

limm→∞

Δα0(WU;T (wm+1)) = 0. (6.18)

Moreover, by (6.14) and (6.16), for arbitrary and fixed m ∈ N, we have that

∀n>m{{wm,wn} ⊂WU;T (wm)}. (6.19)

Next, by (6.19), we have

∀n>m{Uα0(wm,wn) � sup{Uα0(wm, t) : t ∈WU;T (wm)} = Δα0 (WU;T (wm)}. (6.20)

Therefore, (6.20) and (6.18) imply (6.7).Using (6.7) and analogous argument as in Step 1 of the proof of Theorem 3.1, we

obtain (6.8). Indeed, from (6.7), ∀α∈A∀ε>0∃n1∈N∀m>n1{sup{Uα(wm,wn) : n > m} < ε} and,in particular, ∀α∈A∀ε>0∃n1∈N∀m>n1∀k∈N{Uα(wm,wk+m) < ε}. If i0, j0 ∈ N, i0 > j0, are arbitraryand fixed and um = wi0+m and vm = wj0+m, m ∈ N, this gives ∀α∈A{limm→∞Uα(wm, um) =limm→∞Uα(wm, vm) = 0}. By (6.7) and (U2), ∀α∈A{limm→∞ dα(wm, um) = limm→∞ dα(wm, vm)= 0}. Hence

∀α∈A∀ε>0∃n2∈N∀m>n2

{dα(wm,wi0+m) <

ε

2

},

∀α∈A∀ε>0∃n3∈N∀m>n3

{dα(wm,wj0+m

)<ε

2

}.

(6.21)

Therefore, if α0 ∈ A and ε0 > 0 are arbitrary and fixed, n0 = max{n2, n3} + 1 and k,l ∈ N be arbitrary and fixed and such that k > l > n0, then k = i0 + n0 and l = j0 + n0 for somei0, j0 ∈ N such that i0 > j0 and we get dα0(wk,wl) = dα0(wi0+n0 , wj0+n0) � dα0(wn0 , wi0+n0) +dα0(wn0 , wj0+n0) < ε0/2 + ε0/2 = ε0.

Consequently, ∀α∈A∀ε>0∃n∈N∀k,l∈N, k>l>n{dα(wk,wl) < ε}. The proof of (6.8) is complete.By (6.8), there exists a unique w ∈ X such that (6.9) holds.Now we prove (6.10). With the aim of this, let xm, ym ∈WU;T (wm), m ∈ N, be arbitrary

and fixed. Then, by (6.18) and definition of Δα(WU;T (wm)), we have ∀α∈A{limm→∞Uα(wm,xm) = limm→∞Uα(wm, ym) = 0}. Hence, by (6.7) and (U2), ∀α∈A{limm→∞ dα(wm, xm) =limm→∞ dα(wm, ym) = 0} which gives ∀α∈A{limm→∞ dα(xm, ym) = 0}, that is, formula (6.10)holds.

Finally, let us observe that X is sequentially complete and Hausdorff, inclusions (6.16)imply that the sequence of sets {WU;T (wm)} has the property of finite intersections and thatthe properties (6.9), (6.10), and (6.14) hold. Consequently, (6.11) holds.


(a2) From Proposition 4.3(b), Step 2 of the proof of (a1), we conclude that

∀x∈X{T(WU;T (x)) ⊂WU;T (x)} (6.22)

and if the sequence (wm : m ∈ {0} ∪ N) is such as in Step 6 of the proof of (a1), then, since,for each x ∈ X, the set WU;T (x) is closed, using (6.11) and (6.22), we conclude that T(w)= T(

⋂m�0 cl(WU;T (wm))) = T(

⋂m�0 WU;T (wm)) ⊂

⋂m�0 T(WU;T (wm)) ⊂

⋂m�0 WU;T (wm) =⋂

m�0 cl(WU;T (wm)) = {w}, that is, that w is an endpoint of T .(a3) Let x ∈ X be arbitrary and fixed and let the sequence (wm : m ∈ {0} ∪ N) ∈

WU(WU;T , x) be such as in Step 6 of the proof of (a1). Then

∀m∈N{T(cl(WU;T (wm))) ⊂ cl(WU;T (wm))}. (6.23)

Indeed, let m0 ∈ N be arbitrary and fixed. We prove that, if u ∈ cl(WU;T (wm0)) and if y ∈ T(u)is arbitrary and fixed, then y ∈ cl(WU;T (wm0)). With the aim of this, we consider two cases.

Case 1. Assume that u ∈ WU;T (wm0). Then, by (6.22), we have that y ∈ T(u) ⊂ WU;T (wm0) ⊂cl(WU;T (wm0)).

Case 2. Assume that u ∈ cl(WU;T (wm0)) \WU;T (wm0) and let (uλ : λ ∈ Λ) be a net of elementsuλ ∈ WU;T (wm0), λ ∈ Λ, which is convergent to u. Since T is lsc at u and y ∈ T(u), byDefinition 6.1 and the fact that u ∈ cl(WU;T (wm0)) ⊂ X (cf. Proposition 4.3(b)), then thereexists a net (yλ : λ ∈ Λ) of elements yλ ∈ T(uλ), λ ∈ Λ, which is convergent to y. However,since uλ ∈ WU;T (wm0), λ ∈ Λ, thus, by (6.22), we have T(uλ) ⊂ WU;T (wm0), λ ∈ Λ, and,consequently, yλ ∈ T(uλ) ⊂WU;T (wm0), λ ∈ Λ. Since (yλ : λ ∈ Λ) is convergent to y, this givesthat y ∈ cl(WU;T (wm0)).

Now, using (a1) and (6.23), we get T(w) = T(⋂m�0 cl(WU;T (wm))) ⊂ ⋂

m�0T(cl(WU;T (wm))) ⊂

⋂m�0 cl(WU;T (wm)) = {w}, that is, w is an endpoint of T .

(b1) The proof of (b1) will be broken into sixt steps.

Step 7. The map SU;T is strictly (U,Γ)-dissipative on X.

Indeed, since

∀x∈X

⎧⎨⎩SU;T (x) =

⋃m∈{0}∪N

T [m](x)

⎫⎬⎭, (6.24)

therefore

∀m∈N∀x∈X{(SU;T )[m](x) = SU;T (x)

}. (6.25)

On the other hand, by assumption that T is strictly (U,Γ)-dissipative on X, we have that ifx ∈ X is arbitrary and fixed and a generalized sequence of iterations (wm : m ∈ {0} ∪ N)is such that w0 = x and ∀m∈{0}∪N{wm+1 ∈ T [m+1](x)}, then ∀α∈A∀m∈{0}∪N{Uα(wm,wm+1) �γα(wm) − γα(wm+1)}. However, then, by Proposition 4.3(c), (wm : m ∈ {0} ∪N) ∈ SU(T, x) and


∀m∈{0}∪N{wm ∈ SU;T (x)}. Using (6.25), this gives ∀m∈{0}∪N{wm ∈ (SU;T )[m](x)}; remember that

the sequence (wm : m ∈ {0} ∪ N) satisfies ∀α∈A∀m∈{0}∪N{Uα(wm,wm+1) � γα(wm) − γα(wm+1)}.By virtue of Proposition 4.3(c), and Definition 4.2(iv), this implies that SU;T is strictly (U,Γ)-dissipative on X.

Step 8. We show that ∀x∈X{SU(T, x) ⊂ SU(SU;T , x)}.

Indeed, if (wm : m ∈ {0} ∪ N) ∈ SU(T, x), then ∀m∈{0}∪N{wm+1 ∈ T [m+1](x)} which,by (6.24), implies that ∀m∈{0}∪N{wm+1∈k∈{0}∪NT [k](x) = SU;T (x)}, w0 = x. Next, by Step 7,Definition 4.2(iv), Proposition 4.3(c) and (6.25), ∀x∈X{SU(SU;T , x) = {(wm : m ∈ {0} ∪ N) :∀m∈{0}∪N{wm ∈ (SU;T )

[m](x)} = {(wm : m ∈ {0} ∪ N) : ∀m∈{0}∪N{wm ∈ SU;T (x)}}} where w0 = xand (SU;T )

[0] = IX . Consequently, (wm : m ∈ {0} ∪ N) ∈ SU(SU;T , x).

Step 9. Let x ∈ X. If (wm : m ∈ {0}∪N) ∈ SU(SU;T , x), then ∀α∈A∀m∈{0}∪N{γα(wm+1) � γα(wm)}.

By (6.25) and Proposition 4.3(c), ∀m∈{0}∪N{wm+1 ∈ S[m+1]U;T (x) = SU;T (x)}, w0 = x, and

then, by Step 7 and Definition 4.2(iv), ∀α∈A∀m∈{0}∪N{Uα(wm,wm+1) � γα(wm) − γα(wm+1)}.

Step 10. We have ∀α∈A∀x∈X∀y∈SU;T (x){Uα(x, y) � γα(x) − γα(y)}.

Indeed, if x ∈ X and y ∈ SU;T (x), then there exist m0 ∈ {0} ∪ N, y ∈ T [m0+1](x) and(wm : m ∈ {0} ∪ N) ∈ SU(T, x) such that y = wm0+1 and w0 = x. However, ∀m∈{0}∪N{wm+1 ∈T [m+1](x)} and ∀α∈A∀m∈{0}∪N{Uα(wm,wm+1) � γα(wm) − γα(wm+1)}. Hence, ∀α∈A{Uα(x, y) �∑m0

m=0 Uα(wm,wm+1) �∑m0m=0[γα(wm) − γα(wm+1)] = γα(x) − γα(y)}.

Step 11. We have ∀α∈A∀x∈X{Δα(SU;T (x)) � γα(x) − α;SU;T (x)} where ∀α∈A∀x∈X{Δα(SU;T (x)) =sup{Uα(x, t) : t ∈ SU;T (x)} and ∀α∈A∀x∈X{α;SU;T (x) = inf{γα(t) : t ∈ SU;T (x)}}.

This is a consequence of the Step 10, (U1), and (U2).

Step 12. For each x ∈ X, there exist a generalized sequence of iterations (wm : m ∈ {0} ∪ N) ∈SU(SU;T , x) and a unique w ∈ X such that

∀α∈A{

limm→∞

supn>m

Uα(wn,wm) = 0},

∀α∈A{

limm→∞

supn>m

dα(wn,wm) = 0},

∃w∈X{

limm→∞

wm = w}, (6.26)

∀α∈A{

limm→∞

δα(SU;T (wm)) = limm→∞

δα(cl(SU;T (wm))) = 0},

⋂m�0

cl(SU;T (wm)) = {w}.

This can be obtained by an analogous argument as in Step 6, using Steps 7–11.(b2) We show that wis an endpoint of T .


Indeed, ∀x∈X{T(SU;T (x)) ⊂ SU;T (x)} and assuming that (wm : m ∈ {0} ∪ N) and w issuch as in Step 12, then we conclude that the following holds T(w) = T(

⋂m�0 cl(SU;T (wm))) =

T(⋂m�0 SU;T (wm)) ⊂

⋂m�0 T(SU;T(wm)) ⊂

⋂m�0 SU;T (wm) =

⋂m�0 cl(SU;T (wm)) = {w}, that

is, w is an endpoint of T .(b3) We show that wis an endpoint of T .Indeed, let (wm : m ∈ {0} ∪ N) ∈ SU(SU;T , x), x ∈ X, and w be such as in Step 12.

Analogously as in the proof of (a3), we get ∀m∈N{T(cl(SU;T (wm))) ⊂ cl(SU;T (wm))}. ThusT(w) = T(

⋂m�0 cl(SU;T (wm))) ⊂

⋂m�0 T(cl(SU;T (wm))) ⊂

⋂m�0 cl(SU;T (wm)) = {w}, that is, w

is an endpoint of T .

7. Examples, Remarks, and Comparisons ofOur Results with the Well-Known Ones

In this section we present some examples illustrating the concepts introduced so far.In Examples 7.1 and 7.2 we construct J-families and U-families, respectively.

Example 7.1. Let L be an ordered normed space with cone H ⊂ L, let the family P = {pα :X × X → L, α ∈ A} be a P-family, and let (X,P) be a Hausdorff cone uniform space withcone H.

(A) The family P is a J-family.

(B) Let both X and H contain at least two different points and let H be normal with anormal constant M. Let S1 = {v,w}, v /=w, be a subset of X and, for each α ∈ A, letcα, eα ∈ H be such that cα�Heα�H0 and

∀α∈A∀x,y∈S1

{pα(x, y)+ eα≺Hcα

}. (7.1)

Let J = {Jα : X × X → L : α ∈ A} be a family where, for each α ∈ A, Jα is defined by theformula

Jα(x, y)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, if x = y = w,

pα(x, y)+ eα, if

{x, y} ∩ S1 =

{x, y} ∧ ∃u∈{x,y}{u/=w},

cα, if{x, y} ∩ S1 /=

{x, y},

(7.2)

x, y ∈ X. We show that the family J is a J-family on X.

Of course, condition (J1) holds.Now, we show that condition (J2) holds. Indeed, let x, y, z ∈ X be arbitrary and fixed.

We consider three cases: (i) if Jα(x, z) = 0, then it is clear that Jα(x, z) HJα(x, y) + Jα(y, z);(ii) if Jα(x, z) = pα(x, z) + eα, then {x, z} ∩ S1 = {x, z} ∧ ∃u∈{x,z}{u/=w} which implies thatJα(x, y)/= 0 or Jα(y, z)/= 0 and, by (7.1), we get Jα(x, z) HJα(x, y)+Jα(y, z); (iii) if Jα(x, z) = cα,then {x, z} ∩ X \ S1 /= ∅ and, consequently, Jα(x, y) = cα or Jα(y, z) = cα which implies thatJα(x, z) HJα(x, y) + Jα(y, z). Therefore, (J2) holds.


Now, let us observe that ∀α∈A∀x,y∈X{pα(x, y) + eα − eα = pα(x, y) ∈ H}which gives that∀α∈A∀x,y∈X{eα Hpα(x, y) + eα}. Hence

∀α∈A∀x,y∈X{‖eα‖ � M

∥∥pα(x, y) + eα∥∥}. (7.3)

Next assume that the sequences {wm} and {vm} in X satisfy (2.3) and (2.4). Then, inparticular, from (2.4) we conclude that

∀α∈A∀0<ε<||eα||/M∃m0=m0(α,ε)∈N∀m�m0

{‖Jα(wm, vm)‖ < ε < ‖eα‖

M

}. (7.4)

Hence, by (7.3),

∀α∈A∀m�m0

{‖Jα(wm, vm)‖ <∥∥pα(wm, vm) + eα

∥∥}. (7.5)

Since ∀α∈A{‖eα‖/M � ‖cα‖}, condition (7.4) gives

∀α∈A∀m�m0{‖Jα(wm, vm)‖ < ‖cα‖}. (7.6)

By definition of a family J, from (7.5) and (7.6), denoting m′ = min{m0(α, ε) : α ∈ A},we conclude that ∀m�m′ {wm = vm = w}, which implies ∀α∈A∀m�m′ {||pα(wm, vm)|| = 0}.In consequence, we obtain ∀α∈A∀0<ε<||eα||/M∃m′∈N∀m�m′ {‖pα(wm, vm)‖ = 0 < ε}. Thus, thesequences {wm} and {vm} satisfy (2.5). Therefore, the condition (J3) holds.

Example 7.2. Let (X,D) be a Hausdorff uniform space with uniformity defined by a saturatedfamily D = {dα : α ∈ A} of pseudometrics dα : X ×X → [0,∞), α ∈ A, uniformly continuouson X2.

(A) The family D is a U-family.

(B) Let X contain at least two different points. Let S2 ⊂ X, containing at least twodifferent points, be arbitrary and fixed and let {cα}α∈A satisfy ∀α∈A{δα(S2) < cα}.Let U = {Uα : X × X → [0,∞), α ∈ A} be a family where, for each α ∈ A, Uα isdefined by the formula

Uα

(x, y)=

⎧⎨⎩dα(x, y), if S2 ∩

{x, y}={x, y},

cα, if S2 ∩{x, y}/={x, y},

x, y ∈ X. (7.7)

We show that the family U is a U-family on X.Indeed, we see that condition (U1) does not hold only if there exist some α ∈ A and

x, y, z ∈ X such that Uα(x, z) = cα, Uα(x, y) = dα(x, y), Uα(y, z) = dα(y, z), and dα(x, y) +dα(y, z) < cα. However, then we conclude that there exists v ∈ {x, z} such that v /∈S2 andx, y, z ∈ S2, which is impossible. Therefore, ∀α∈A∀x,y,z∈X{Uα(x, z) � Uα(x, y)+Uα(y, z)}, thatis, condition (U1) holds.


To prove that (U2) holds we assume that the sequences {wm} and {vm} in X satisfy(4.1) and (4.2). Then, in particular, (4.2) yields

∀α∈A∀0<ε<cα∃m0=m0(α,ε)∈N∀m�m0{Uα(wm, vm) < ε}. (7.8)

By (7.8) and definition of U, denoting m′ = min{m0(α, ε) : α ∈ A}, we conclude that

∀m�m′ {S2 ∩ {wm, vm} = {wm, vm}}. (7.9)

From (7.9), the definition of U and (7.8), we get

∀α∈A∀0<ε<cα∃m′∈N∀m�m′ {dα(wm, vm) = 0 < ε}. (7.10)

The result is that the sequences {wm} and {vm} satisfy (4.3). Therefore, condition (U2) holds.The following example illustrates Theorem 3.1(a) in cone metric space.

Example 7.3. Let (L, ‖ · ‖), L = R2, be a real normed space, let H be a regular solid cone of

the form H = {(x, y) ∈ L : x, y � 0} and let (X,P) be a cone metric space (see [27]) witha cone H where X = [0, 1] ⊂ R, P = {p}, and p : X × X → L is a cone metric of the formp(x, y) = (|x − y|, 2|x − y|), x, y ∈ X. Let T : X → 2X be defined by

T(x) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

[12, 1], if x = 0,

{1}, if x ∈(

0,12

)∪(

12, 1],

{0, 1}, if x =12.

(7.11)

We note that for q = 1, the map T [q] is closed in X.

Let S1 = {1/2, 1} and let J : X ×X → L be of the form

J(x, y)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(0, 0), if x = y = 1,

p(x, y)+(

14,

12

), if

{x, y} ∩ S1 =

{x, y} ∧ ∃u∈{x,y}{u/= 1},

(2, 2), if{x, y} ∩ S1 /=

{x, y}

(7.12)

for x, y ∈ X. By Example 7.1(B), the family J = {J} is a J-family. Now we define Ω = {ω},ω : X → L, as follows:

ω(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(1,

32

), for x ∈ [0, 1) \

{12

},

(2, 2), for x =12,

(0, 0), for x = 1.

(7.13)

Of course, ∀x∈X{0 Hω(x)} and ω is not lsc on X.


(a) The map T is weak (J,Ω;X0)-dissipative on X where Ω = {ω} and X0 = {1/2, 1}.Indeed, if x = 1, then, by definition T , a sequence (wm : m ∈ {0} ∪ N) satisfying (1.1)

is of the form wm = 1 where m ∈ {0} ∪N and satisfies ∀m∈{0}∪N{J(wm,wm+1) = (0, 0) Hω(1) −ω(1) = ω(wm) −ω(wm+1) = (0, 0)}, that is, (2.7) holds.

If x = 1/2, then there exists a sequence (wm : m ∈ {0} ∪ N) satisfying (1.1) of the formw0 = 1/2, wm = 1 where m ∈ N and satisfies J(w0, w1) = J(1/2, 1) = (3/4, 3/2) H(2, 2) −(0, 0) = ω(w0) − ω(w1) and ∀m∈N{J(wm,wm+1) = (0, 0) Hω(1) − ω(1) = ω(wm) − ω(wm+1) =(0, 0)}, that is, (2.7) holds.

Consequently, {1/2, 1} ⊂ X0.We see that X0 = {1/2, 1}. Otherwise, X0 \ {1/2, 1}/= ∅ and the following two cases

hold:

Case 1. If x = 0, then, by definition of T , for each sequence (wm : m ∈ {0} ∪N) satisfying (1.1)we have that w0 = 0, w1 ∈ [1/2, 1] and then, by definition of J ,

J(w0, w1) = J(0, w1) = (2, 2)�Hω(0) −ω(w1) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

(0, 0) if w1 ∈(

12, 1)

(1,

32

)if w1 = 1

−(

1,12

)if w1 =

12,

(7.14)

which means that (2.7) does not hold.

Case 2. If x /∈ {0, 1/2, 1}, then, by definition of T , each sequence (wm : m ∈ {0} ∪ N) satisfying(1.1) is of the form w0 = x and, for each m ∈ N, wm = 1. Next, by definition of J , since x /∈S1,we obtain

J(w0, w1) = J(x, 1) = (2, 2)�Hω(w0) −ω(w1) =(

1,32

), (7.15)

which means that (2.7) does not hold.

Consequently, X0 = {1/2, 1}.(b) All assumptions of Theorem 3.1(a) hold, and 1 ∈ X0 is the periodic point of T , that

is, 1 ∈ Fix(T).

Remark 7.4. Let L,H, (X,P), T and J-family be such as in Example 7.3.

(i) We see that for this J-family the map T is not (J,Ω)-dissipative on X for any Ω(consequently, by Remark 2.9(ii), for any Ω, T is not strictly (J,Ω)-dissipativeon X). Indeed, suppose that there exists Ω = {ω} such that ω : X → L is amap satisfying ∀x∈X{0 Hω(x)} and such that T is (J,Ω)-dissipative on X. Then,in particular, for a dynamic process w0 = 1/2, w1 = 0, w2 = 1/2, w3 = 0, and wm = 1for m � 4, by (2.7), we must have (0, 0)≺H(2, 2) = J(w0, w1) = J(1/2, 0) Hω(1/2) −ω(0) so ω(0)≺Hω(1/2) and (0, 0)≺H(2, 2) = J(w1, w2) = J(0, 1/2) Hω(0) − ω(1/2)so ω(1/2)≺Hω(0), which are contradictions.


(ii) We see that X0 = {1/2, 1} and T(X0) = {0, 1}/=X0.

In Examples 7.5 and 7.7 we illustrate Theorem 5.3(a) for dissipative set-valued andsingle-valued dynamic systems, respectively.

Example 7.5. Let (X, d) be a metric space, whereX = [0, 1/2]∪{3/4, 1} and d : X×X → [0,∞)is a metric of the form d(x, y) = |x − y|, x, y ∈ X. Let T : X → 2X be of the form

T(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{1}, for x = 0,

[0,

12x

], for x ∈

(0,

12

],

{0}, for x =34,{

34

}, for x = 1.

(7.16)

Let S2 = [0, 1/2]. By Example 7.2(B), the family U = {U : X ×X → [0,∞)}, where

U(x, y)=

⎧⎨⎩d(x, y), if S2 ∩

{x, y}={x, y},

2, if S2 ∩{x, y}/={x, y},

x, y ∈ X, (7.17)

is a U-family on X. We observe that

T [2](x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{34

}, for x = 0,

[0,

14x

]∪ {1}, for x ∈

(0,

12

],

{1}, for x =34,

{0}, for x = 1,

T [3](x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{0}, for x = 0,

[0,

18x

]∪{

34, 1}, for x ∈

(0,

12

],

{34

}, for x =

34,

{1}, for x = 1,


T [4](x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{1}, for x = 0,[

0,1

16x

]∪{

34, 1}, for x ∈

(0,

12

],

{0}, for x =34,{

34

}, for x = 1,

(7.18)

so, the map T [3] is closed in X. However, the map T [4] is not closed in X.

Let Γ = {γ}, γ : X → [0,∞), be of the form γ(x) = x, x ∈ X.

(a) T is weak (U,Γ;X0)-dissipative on X, where X0 = (0, 1/2]. Indeed, let x ∈ (0, 1/2]be arbitrary and fixed, then there exists a dynamic process (wm : m ∈ {0}∪N) givenby the formula w0 = x, wm = (1/2m)x, m ∈ N, such that U(w0, w1) = d(x, (1/2)x) =x/2 � x − x/2 = γ(w0) − γ(w1); ∀m∈N{U(wm,wm+1) = d(wm,wm+1) � (1/2m)x −(1/2m+1)x = γ(wm) − γ(wm+1)}. This implies that the dynamic process (wm : m ∈{0} ∪ N) satisfies (1.1) and (4.4). Consequently, (0, 1/2] ⊂ X0. The fact that X0 \(0, 1/2] = ∅ follows from considerations in the remark below.

(b) The 0 ∈ clX0 is the periodic point of T (q = 3).

Remark 7.6. Let X,D, T , andU-family be such as in Example 7.5. We see that for thisU-familythe map T is not (U,Γ)-dissipative on X for any Γ (consequently, by Remark 2.9(ii), for anyΓ, T is not strictly (U,Γ)-dissipative onX). Indeed, suppose that there exists Γ = {γ} such thatγ : X → [0,∞] and that T is (U, Γ)-dissipative onX. Then, we have a unique dynamic process(wm : m ∈ {0} ∪ N) starting at w0 = 3/4 which is defined by w1 = 0 ∈ T(w0), w2 = 1 ∈ T(w1),w3m+1 = 0 ∈ T(w3m), w3m+2 = 1 ∈ T(w3m+1), and w3m = 3/4 ∈ T(w3m−1) for m ∈ N and for thisprocess we have 0 < U(w0, w1) = 2 � γ(w0) − γ(w1), 0 < U(w1, w2) = 2 � γ(w1) − γ(w2), and0 < U(w2, w3) = 2 � γ(w2) − γ(w3) = γ(w2) − γ(w0). Hence γ(w0) < γ(w2) < γ(w1) < γ(w0),which is impossible.

Example 7.7. Let (X, d) be a metric space, where X = [0, 1] and d : X ×X → [0,∞) is a metricof the form d(x, y) = |x − y|, x, y ∈ X. Let T : X → X be of the form

T(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, for x ∈[

0,18

],

−2x +54, for x ∈

(18,

38

),

12, for x ∈

[38,

58

],

−2x +74, for x ∈

(58,

78

),

0, for x ∈[

78, 1].

(7.19)


By Example 7.2(A), the family U = {U : X ×X → [0,∞)}, where U(x, y) = d(x, y), x, y ∈ X,is U-family on X.

Let Γ = {γ}, γ : X → [0,∞), be of the form

γ(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1, for x ∈[

0,516

)∪(

1116, 1],

3, for x ∈[

516,

38

)∪(

58,

1116

],

2, for x ∈[

38,

12

)∪(

12,

58

],

0, for x =12,

x ∈ X. (7.20)

(a) We show that T is weak (U,Γ;X0)-dissipative on X, where X0 = [5/16, 11/16].Indeed, let x ∈ X0 be arbitrary and fixed. We consider the following four cases.

Case 1. If x = 1/2, then a dynamic process (wm : m ∈ {0} ∪ N) of (X, T) starting at x is ofthe form wm = 1/2, m ∈ {0} ∪ N. Consequently, ∀m∈{0}∪N{U(wm,wm+1) = d(wm,wm+1) = 0 �0 − 0 = γ(wm) − γ(wm+1)}, that is, (4.4) holds.

Case 2. If x ∈ [5/16, 3/8), then a dynamic process (wm : m ∈ {0} ∪N) of (X, T) starting at x isof the formw0 = x,w1 = −2x+5/4 ∈ (1/2, 5/8], andwm = 1/2,m � 2. Therefore,U(w0, w1) =d(w0, w1) � 1 = 3 − 2 = γ(w0) − γ(w1), U(w1, w2) = d(w1, w2) � 2 − 0 = γ(w1) − γ(w2), and∀m�2{U(wm,wm+1) = d(wm,wm+1) = 0 � 0 − 0 = γ(wm) − γ(wm+1)}, that is, (4.4) holds.

Case 3. If x ∈ [3/8, 1/2) ∪ (1/2, 5/8], then a dynamic process (wm : m ∈ {0} ∪ N) of (X, T)starting at x is of the form w0 = x and wm = 1/2 for m ∈ N. Moreover, (4.4) holds sinceU(w0, w1) = d(w0, w1) � 2 − 0 = γ(w0) − γ(w1) and ∀m∈N{U(wm,wm+1) = d(wm,wm+1) = 0 �0 − 0 = γ(wm) − γ(wm+1)}.

Case 4. If x ∈ (5/8, 11/16], then a dynamic process (wm : m ∈ {0} ∪ N) of (X, T) starting atx is of the form w0 = x, w1 = −2x + 7/4 ∈ [3/8, 1/2), and wm = 1/2, m � 2. We also getthat U(w0, w1) = d(w0, w1) � 1 = 3 − 2 = γ(w0) − γ(w1), U(w1, w2) = d(w1, w2) � 2 − 0 =γ(w1) − γ(w2), and ∀m�2{U(wm,wm+1) = d(wm,wm+1) = 0 � 0 − 0 = γ(wm) − γ(wm+1)}.

(b) Now, we show that if x ∈ [0, 5/16)∪(11/16, 1], then x /∈X0. Indeed, let x ∈ [0, 5/16)and let (wm : m ∈ {0}∪N) be a dynamic process of (X, T) starting at x. Then w0 = x,w1 ∈ (5/8, 1], and

U(w0, w1) = d(w0, w1) >

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−2 = 1 − 3 = γ(w0) −ω(w1), if w1 ∈(

58,

1116

],

0 = 1 − 1 = γ(w0) −ω(w1), if w1 ∈(

1116, 1].

(7.21)


Now, let x ∈ (11/16, 1] and let (wm : m ∈ {0} ∪ N) be a dynamic process of (X, T) starting atx. Then w0 = x, w1 ∈ [0, 3/8), and

U(w0, w1) = d(w0, w1) >

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 = 1 − 1 = γ(w0) −ω(w1), if w1 ∈[

0,5

16

),

−2 = 1 − 3 = γ(w0) −ω(w1), if w1 ∈[

516,

38

).

(7.22)

Consequently, T is weak (U,Γ;X0)-dissipative on X, where X0 = [5/16, 11/16].

(c) The map T is closed on X, 1/2 ∈ Fix(T), and 1/2 ∈ X0.

Remark 7.8. Let X, D, T andU-family be such as in Example 7.7. We see that for thisU-familythe map T is not (U,Γ)-dissipative on X for any Γ; see (b).

The following example shows that in Theorem 5.3 for the existence of periodic pointsthe assumption that the map T [q] is closed in X for some q ∈ N is essential.

Example 7.9. Let (X, d) be a metric space, where X = [0, 1] and d : X ×X → [0,∞) is a metricof the form d(x, y) = |x − y|, x, y ∈ X. Let T : X → 2X be of the form

T(x) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(12, 1], if x = 0,

(1

2m+1,

12m

], if x ∈

(1

2m,

12m−1

], m ∈ N,

(7.23)

let U = {d} and let Γ = {γ}, γ : X → [0,∞), be of the form

γ(x) =

⎧⎨⎩

4, if x ∈ 0,

x, if x ∈ (0, 1].(7.24)

We observe the following.

(a) T is (U,Γ)-dissipative and strictly (U,Γ)-dissipative on X.

(b) For each x ∈ X, WU(T, x) = SU(T, x), and {{0} = {w : limmd(wm,w) = 0 ∧ (wm :m ∈ {0} ∪ N) ∈ SU(T, x)}.

(c) For each q ∈ N, the map T [q] is not closed in X.

(d) The map T does not have periodic points in X.

The following example illustrates Theorem 6.2(a1).

Example 7.10. Let (X, d) be a metric space, where X = [0, 1] and d : X×X → [0,∞) is a metricof the form d(x, y) = |x − y|, x, y ∈ X.


Let S2 = [0, 1/2] and let U : X ×X → [0,∞) be of the form

U(x, y)=

⎧⎨⎩d(x, y), if S2 ∩

{x, y}={x, y},

2, if S2 ∩{x, y}/={x, y},

x, y ∈ X. (7.25)

By Example 7.2(B), the family U = {U} is a U-family. Define Γ = {γ}, γ : X → [0,∞], asfollows:

γ(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

x, if x ∈[

0,12

],

6, if x ∈(

12,

34

],

4, if x ∈(

34, 1),

8, if x = 1,

x ∈ X. (7.26)

Let T : X → 2X be of the form

T(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{0}, if x = 0,{1

n + 1

}, if x =

1n∧ n � 2,

{1

n + 1

}, if x ∈

(1

n + 1,

1n

)∧ n � 2,

{78

}, if x ∈

(12,

34

),

[78, 1), if x =

34,

{12

}, if x ∈

(34, 1),

(12,

34

), if x = 1.

(7.27)

We observe that

(a) T is (U,Γ)-dissipative on X. Indeed, let x ∈ X be arbitrary and fixed. We considerseven cases

Case 1. If x = 0, then each dynamic process starting at w0 = 0 is of the form ∀m∈N{wm = 0 ∈T(wm−1)} and ∀m∈{0}∪N{U(wm,wm+1) = 0 � γ(wm) − γ(wm+1)}.Case 2. If x ∈ (0, 1/2) \ {1/n : n � 3}, then there exists l0 � 2 such that x ∈ (1/(l0 + 1), 1/l0)and each dynamic process starting at w0 = x is of the form ∀m�1{wm = 1/(l0 +m)}. ThereforeU(w0, w1) = d(x, 1/(l0 + 1)) = x − 1/(l0 + 1) � γ(x) − γ(wl0+1) = γ(w0) − γ(w1)} and∀m∈N{U(wm,wm+1) = 1/(l0 +m) − 1/(l0 +m + 1) � γ(wm) − γ(wm+1)}.


Case 3. If x ∈ {1/n : n � 2}, then there exists l0 � 2 such that x = 1/l0 and eachdynamic process starting at w0 = x is of the form ∀m�1{wm = 1/(l0 + m)}. Therefore∀m∈{0}∪N{U(wm,wm+1) = 1/(l0 +m) − 1/(l0 +m + 1) � γ(wm) − γ(wm+1)}.Case 4. If x ∈ (1/2, 3/4), then each dynamic process starting at w0 = x is of the form w1 = 7/8and ∀m�2{wm = 1/m}. Therefore U(w0, w1) = 2 � 6 − 4 = γ(w0) − γ(w1), U(w1, w2) = 2 �4 − 1/2 = γ(w0) − γ(w1) and ∀m�2{U(wm,wm+1) = 1/m − 1/(m + 1) � γ(wm) − γ(wm+1)}.Case 5. If x = 3/4, then each dynamic process starting at w0 = x is of the form w1 ∈ [7/8, 1)and ∀m�2{wm = 1/m}. Therefore U(w0, w1) = 2 � 6 − 4 = γ(w0) − γ(w1), U(w1, w2) = 2 �4 − 1/2 = γ(w0) − γ(w1), and ∀m�2{U(wm,wm+1) = 1/m − 1/(m + 1) � γ(wm) − γ(wm+1)}.Case 6. If x ∈ (3/4, 1), then each dynamic process starting at w0 = x is of the form ∀m�1{wm =1/(m + 1)}. Therefore, U(w0, w1) = 2 � 4 − 1/2 = γ(w0) − γ(w1) and ∀m�2{U(wm−1, wm) =1/m − 1/(m + 1) � γ(wm−1) − γ(wm)}.Case 7. If x = 1, then each dynamic process starting at w0 = x is of the form w1 ∈ (1/2, 3/4),w2 = 7/8, w3 = 1/2, and ∀m�4{wm = 1/(m − 1)}. Therefore U(w0, w1) = 2 � 8 − 6 = γ(w0) −γ(w1), U(w1, w2) = 2 � 6 − 4 = γ(w0) − γ(w1) and by analogous argumentation as in Case 4we obtain that (1.1) and (4.4) hold in this case.

Consequently, T is (U,Γ)-dissipative on X.

(b) WU;T is of the form

WU;T (x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{0}, if x = 0,{1m

: m � n

}, if x =

1n∧ n � 2,

{1m

: m > n

}∪ {x}, if x ∈

(1

n + 1,

1n

)∧ n � 2,

{1n

: n � 2}∪{x,

78

}, if x ∈

(12,

34

),

{1n

: n � 2}∪{

34

}∪[

78, 1), if x =

34,

{1n

: n � 2}∪ {x}, if x ∈

(34, 1),

{1n

: n � 2}∪(

12,

34

)∪{

78, 1}, if x = 1.

(7.28)

(c) WU;T is (U,Γ)-dissipative on X.

(d) For each x ∈ X, there exist (wm : m ∈ {0} ∪ N) ∈ WU(WU;T , x) and w = 0 ∈ X suchthat (wm : m ∈ {0} ∪ N) converges to w and

⋂m�2 WU;T (wm) = {w} = {0}. We see

that w = 0 is an endpoint of T in X.

The assertions (a1) of the Theorem 6.2 hold.


Remark 7.11. It is worth noticing that in Example 7.10, there exists x ∈ X such that WU;T (x)is not closed. Indeed, WU;T (1/2) = {1/m : m � 2} is not closed. Moreover, T is not lsc on X.Consequently, the assumptions of Theorems 6.2(a2) and 6.2(a3) do not hold.

The following example illustrates Theorems 6.2(a2) and 6.2(a3).

Example 7.12. Let (X, d) be a metric space, where X = [0, 1/2], d(x, y) = |x − y|, x, y ∈ X, andU = {d}. Let T : X → 2X be of the form T(x) = [0, (1/2)x], x ∈ X, and let Γ = {γ : X →[0,∞)} be defined as follows γ(x) = x, x ∈ X.

(a) The map T is (U,Γ)-dissipative. Indeed, if x ∈ X is arbitrary and fixed, then eachdynamic process (wm : m ∈ {0}∪N) starting atw0 = x is of the formwm ∈ T(wm−1) =[0, (1/2)wm−1] for m ∈ N. Therefore, we have ∀m∈{0}∪N{0 < U(wm,wm+1) = wm −wm+1 = γ(wm) − γ(wm+1)}. Thus conditions (1.1) and (4.4) hold.

(b) We observe that

WU;T (x) =

⎧⎪⎨⎪⎩{0}, if x ∈ 0,[

0,12x

], if x ∈ (0, 1],

(7.29)

and WU;T is (U,Γ)-dissipative on X.

(c) For each x ∈ X, the set WU;T (x) is closed.

(d) The assertions of Theorem 6.2(a2) hold.

(e) The map WU;T is closed.

(f) We have that w = 0 ∈ End(T) and⋂m�0 cl(WU;T (wm)) = {w}.

(g) The map T is lsc on X.

(h) The assertions (a3) hold.

(i) The map T is not strictly (U,Γ)-dissipative. Indeed, if x = 1/2 ∈ X, then we havethat T [m](x) = [0, (1/2m)x] for m ∈ N and a generalized sequence of iterations(wm : m ∈ {0} ∪ N) starting at w0 = x, of the form w1 = 0, w2 = 1/8 and wm = 0 form � 3, not satisfies (4.4) since U(w1, w2) = 1/8 > −1/8 = w1 −w2 = γ(w1) − γ(w2).

The following example illustrates Theorem 6.2(a2).

Example 7.13. Let (X, d), X = [0, 1], and let U = {d}. Let T : X → 2X be of the form

T(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

{1}, if x = 0,[0,

12x

], if x ∈ (0, 1),

{1}, if x = 1,

(7.30)


and let Γ = {γ : X → [0,∞)}where γ is of the form

γ(x) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1, if x = 0,

x, if x ∈ (0, 1)

0, if x = 1.

, x ∈ X, (7.31)

(a) The map T is (U,Γ)-dissipative on X. Indeed, let x ∈ X be arbitrary and fixed. Weconsider three cases.

Case 1. If x = 0, then a dynamic process starting at w0 = 0 is of the form ∀m∈N{wm = 1 ∈T(wm−1)} and U(w0, w1) = 1 � 1 − 0 = γ(0) − γ(1) and ∀m∈N{U(wm,wm+1) = 0 � γ(wm) −γ(wm+1)}.

Case 2. If x ∈ (0, 1), then each dynamic process (wm : m ∈ {0} ∪ N) starting at w0 = x isof the form wm ∈ T(wm−1) = [0, (1/2)wm−1] for m ∈ N. Therefore, we have ∀m∈{0}∪N{0 <U(wm,wm+1) = wm −wm+1 = γ(wm) − γ(wm+1)}.

Case 3. If x = 1, then a dynamic process starting at w0 = 1 is of the form ∀m∈N{wm = 1 ∈T(wm−1)} and ∀m∈{0}∪N{U(wm,wm+1) = 0 � γ(wm) − γ(wm+1)}.

Thus conditions (1.1) and (4.4) hold.

(b) We observe that

WU;T (x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

{0, 1}, if x = 0,[0,

12x

]∪ {1}, if x ∈ (0, 1),

{1}, if x = 1,

(7.32)

and WU;T is (U,Γ)-dissipative on X.

(c) For each x ∈ X, the set WU;T (x) is closed.

(d) The assertions of Theorem 6.2(a2) hold.

(e) The map WU;T is not closed.

(f) We have that w = 1 ∈ End(T) and⋂m�0 cl(WU;T (wm)) = {w}.

(g) The map T is not lsc on X.

(h) The assumptions of Theorem 6.2(a3) do not hold.

Now, we present comparisons between our results and the well-known ones. Theresults of Kirk and Saliga [12] and Aubin and Siegel [1], concerning the existence offixed points and endpoints of dissipative single-valued and set-valued dynamic systems,respectively, we may read as follows.

Theorem 7.14 (Kirk and Saliga [12, Theorem 1.1]). Let (X, d) be a complete metric space and let(X, T) be a single-valued closed dynamic system satisfying

∀x∈X{d(x, T(x)) � ϕ(x) − ϕ(T(x))}, (7.33)


where ϕ : X → R is a map bounded from below. Then, for each x ∈ X, {T [m](x)} converges to a fixedpoint of T .

Here, we may assume, without loss of generality, that ϕ : X → [0,∞); in another case,we replace ϕ by ϕ0 = ϕ − infx∈Xϕ(x).

Theorem 7.15 (Aubin and Siegel [1, Theorem 2.4]). Let (X, d) be a complete metric space and let(X, T) be a set-valued closed dynamic system satisfying

∀x∈X∃y∈T(x){d(x, y)

� ϕ(x) − ϕ(y)}, (7.34)

where ϕ : X → [0,∞). Then, for each x ∈ X, there exists a dynamic process (wm : m ∈ {0} ∪ N)starting at w0 = x of the system (X, T) which converges to a fixed point of T .

Theorem 7.16 (Aubin-Siegel [1, Theorem 4.10]). Let (X, d) be a complete metric space and let(X, T) be a set-valued dynamic system satisfying

∀x∈X∀y∈T(x){d(x, y)

� ϕ(x) − ϕ(y)}, (7.35)

where ϕ : X → [0,∞). Assume that (a) T is lsc; or (b) Wd;T is closed. Then End(T)/= ∅.

Remark 7.17. It is worth noticing that

(i) by Example 7.2(A), Theorem 5.3(b) when q = 1 includes Theorems 7.14 and 7.15,

(ii) by Example 7.2(A) and Remark 5.1(i), Theorems 6.2(a2) and 6.2(a3) includeTheorem 7.16.

(iii) the map T defined in Example 7.7 satisfies the assumptions of Theorem 5.3(a) butdoes not satisfy the assumptions of Theorems 7.14, 7.15, and 7.16(a). Indeed, firstlet us observe that T is closed (and lsc as continuous). Next, suppose that the mapT satisfies the assumptions of Theorems 7.14, 7.15, and 7.16(a). Then there exists amap ϕ : X → [0,∞) such that the condition ∀x∈X{d(x, T(x)) � ϕ(x) − ϕ(T(x))}holds. Then, in particular, for x = 0, we have that 0 < d(x, T(x)) = d(0, 1) = 1 �ϕ(0) − ϕ(1), which means that ϕ(1) < ϕ(0). On the other hand, for x = 1, we get0 < d(x, T(x)) = d(1, 0) = 1 � ϕ(1) − ϕ(0), which means that ϕ(0) < ϕ(1). This isabsurd.

(iv) The map T defined in Example 7.13 satisfies the assumptions of Theorem 6.2(a2)but does not satisfy the assumptions of Theorems 7.16(a) and 7.16(b). This followsfrom (e) and (g).

References

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[8] I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, pp.324–353, 1974.


[10] S. Banach, “Sur les operations dans les ensembles abstraits et leurs applications aux equationsintegrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.

[11] M. Edelstein, “On fixed and periodic points under contractive mappings,” Journal of the LondonMathematical Society, vol. 37, pp. 74–79, 1962.

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Research ArticleOn Fixed Points of MaximalizingMappings in Posets

S. Heikkila

Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland

Correspondence should be addressed to S. Heikkila, [email protected]

Received 7 October 2009; Accepted 16 November 2009


Copyright q 2010 S. Heikkila. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We use chain methods to prove fixed point results for maximalizing mappings in posets. Concreteexamples are also presented.

1. Introduction

According to Bourbaki’s fixed point theorem (cf. [1, 2]) a mapping G from a partially orderedset X = (X,≤) into itself has a fixed point if G is extensive, that is, x ≤ G(x) for all x ∈ X, andif every nonempty chain of X has the supremum in X. In [3, Theorem 3] the existence ofa fixed point is proved for a mapping G : X → X which is ascending, that is, G(x) ≤ yimplies G(x) ≤ G(y). It is easy to verify that every extensive mapping is ascending. In [4] theexistence of a fixed point of G is proved if a ≤ G(a) for some a ∈ X, and if G is semi-increasingupward, that is, G(x) ≤ G(y) whenever x ≤ y and G(x) ≤ y. This property holds, for instance,if G is ascending or increasing, that is, G(x) ≤ G(y) whenever x ≤ y.

In this paper we prove further generalizations to Bourbaki’s fixed point theorem byassuming that a mapping G : X → X is maximalizing, that is, G(x) is a maximal elementof {x,G(x)} for all x ∈ X. Concrete examples of maximalizing mappings G which have ordo not have fixed points are presented. Chain methods introduced in [5, 6] are used in theproofs. These methods are also compared with three other chain methods.

2. Preliminaries

A nonempty set X, equipped with a reflexive, antisymmetric, and transitive relation ≤ inX × X, is called a partially ordered set (poset). An element b of a poset X is called an upper


bound of a subset A of X if x ≤ b for each x ∈ A. If b ∈ A, we say that b is the greatest elementof A, and denote b = maxA. A lower bound of A and the least element, minA, of A aredefined similarly, replacing x ≤ b above by b ≤ x. If the set of all upper bounds of A has theleast element, we call it the supremum of A and denote it by supA. We say that y is a maximalelement of A if y ∈ A, and if z ∈ A and y ≤ z imply that y = z. The infimum of A, infA, and aminimal element of A are defined similarly. A subset W of X is called a chain if x ≤ y or y ≤ xfor all x, y ∈W . We say that W is well ordered if nonempty subsets of W have least elements.Every well-ordered set is a chain.

Let X be a nonempty poset. A basis to our considerations is the following chainmethod (cf. [6, Lemma 2]).

Lemma 2.1. Given G : X → X and a ∈ X, there exists a unique well-ordered chain C in X, called aw-o chain of aG-iterations, satisfying

x ∈ C iff x = sup{a,G

[C<x]}, where C<x =

{y ∈ C : y < x

}. (2.1)

If x∗ = sup{a,G[C]} exists in X, then x∗ = max C, and G(x∗) ≤ x∗.

The following result helps to analyze the w-o chain of aG-iterations.

Lemma 2.2. Let A and B be nonempty subsets of X. If supA and supB exist, then the equation

sup(A ∪ B) = sup{

supA, supB}

(2.2)

is valid whenever either of its sides is defined.

Proof. The sets A ∪ B and {supA, supB} have same upper bounds, which implies theassertion.

A subset W of a chain C is called an initial segment of C if x ∈ W , y ∈ C, and y < ximply y ∈ W . If W is well ordered, then every element x of W which is not the possiblemaximum of W has a successor: Sx = min{y ∈ W : x < y}, in W . The next result gives acharacterization of elements of the w-o chain of aG-iterations.

Lemma 2.3. Given G : X → X and a ∈ X, let C be the w-o chain of aG-iterations. Then theelements of C have the following properties.

(a) minC = a.

(b) An element x of C has a successor in C if and only if sup{x,G(x)} exists and x <sup{x,G(x)}, and then Sx = sup{x,G(x)}.

(c) If W is an initial segment of C and y = supW exists, then y ∈ C.

(d) If a < y ∈ C and y is not a successor, then y = supC<y.

(e) If y = supC exists, then y = max C.


Proof. (a) minC = sup{a,G[C<minC]} = sup{a,G[∅]} = sup{a, ∅} = a.(b) Assume first that x ∈ C, and that Sx exists in C. Applying (2.1), Lemma 2.2, and

the definition of Sx we obtain

Sx = sup{a,G

[C<Sx

]}= sup

{a,G

[C<x] ∪ {G(x)}} = sup{x,G(x)}. (2.3)

Moreover, x < Sx, by definition, whence x < sup{x,G(x)}.Assume next that x ∈ C, that y = sup{x,G(x)} exists, and that x < sup{x,G(x)}. The

previous proof implies the following(i) There is no element w ∈ C which satisfies x < w < sup{x,G(x)}.

Then {z ∈ C : z ≤ x} = C<y, so that

x < sup{x,G(x)} = sup{

sup{a,G

[C<x]}, G(x)

}= sup

{{a} ∪G[C<x] ∪ {G(x)}}

= sup{a,G[{z ∈ C : z ≤ x}]}= sup

{a,G

[C<y]}.

(2.4)

Thus y = sup{x,G(x)} ∈ C by (2.1). This result and (i) imply that y = sup{x,G(x)} =min{z ∈ C : x < z} = Sx.

(c) Assume that W is an initial segment of C, and that y = supW exists. If there isx ∈W such that Sx/∈W , then x = max W = y, so that y ∈ C. Assume next that every elementx of W has the successor Sx in W . Since Sx = sup{x,G(x)} by (b), then G(x) ≤ Sx < y. Thisholds for all x ∈W . Since a = minC = minW < y, then y is an upper bound of {a}∪G[W]. Ifz is an upper bound of {a} ∪G[W], then x = sup{a,G[C<x]} = sup{a,G[W<x]} ≤ z for everyx ∈W . Thus z is an upper bound of W , whence y = supW ≤ z. But then y = sup{a,G[W]} =sup{a,G[C<y]}, so that y ∈ C by (2.1).

(d) Assume that a < y ∈ C, and that y is not a successor of any element of C.Obviously, y is an upper bound of C<y. Let z be an upper bound of C<y. If x ∈ C<y,then also Sx ∈ C<y since y is not a successor. Because Sx = sup{x,G(x)} by (b), thenG(x) ≤ Sx ∈ C<y. This holds for every x ∈ C<y. Since also a ∈ C<y, then z is an upperbound of {a} ∪G[C<y]. Thus y = sup{a,G[C<y]} ≤ z. This holds for every upper bound z ofC<y, whence y = supC<y.

(e) If y = supC exists, then y ∈ C by (c) when W = C, whence y = max C.

In the case when a ≤ G(a) we obtain the following result (cf. [7, Proposition 1]).

Lemma 2.4. Given G : X → X and a ∈ X, there exists a unique well-ordered chain C(a) in X,calleda w-o chain of G-iterations of a, satisfying

a = minC, x ∈ C \ {a} iff x = supG[C<x]. (2.5)

If a ≤ G(a), and if x∗ = supG[C(a)] exists, then a ≤ x∗ = max C(a), and G(x∗) ≤ x∗.


Lemma 2.4 is in fact a special case of Lemma 2.1, since the assumption a ≤ G(a) impliesthat C(a) equals to the w-o chain of aG-iterations. As for the use of C(a) in fixed point theoryand in the theory of discontinuous differential and integral equations, see, for example, [8, 9]and the references therein.

3. Main Results

Let X = (X,≤) be a nonempty poset. As an application of Lemma 2.1 we will prove our firstexistence result.

Theorem 3.1. A mapping G : X → X has a fixed point if G is maximalizing, that is, G(x) is amaximal element of {x,G(x)} for all x ∈ X, and if x∗ = sup{a,G[C]} exists in X for some a ∈ Xwhere C is the w-o chain of aG-iterations.

Proof. If C is the w-o chain of aG-iterations, and if x∗ = sup{a,G[C]} exists in X, then x∗ =max C and G(x∗) ≤ x∗ by Lemma 2.1. Since G is maximalizing, then G(x∗) = x∗, that is, x∗ isa fixed point of G.

The next result is a consequence of Theorem 3.1. and Lemma 2.3(e).

Proposition 3.2. Assume that G : X → X is maximalizing. Given a ∈ X, let C be the w-o chain ofaG-iterations. If z = supC exists, it is a fixed point of G if and only if x∗ = sup{z,G(z)} exists.

Proof. Assume that z = supC exists. It follows from Lemma 2.3(e) that z = max C. If z is afixed point of G, that is, z = G(z), then x∗ = sup{z,G(z)} = z, and x∗ = G(x∗).

Assume conversely that x∗ = sup{z,G(z)} exist. Applying (2.1) and Lemma 2.2 weobtain

x∗ = sup{z,G(z)} = sup{

sup{a,G

[C<z]}, sup{G(z)}}

= sup{{a} ∪G[

C<z] ∪ {G(z)}} = sup{a,G[C]}.(3.1)

Thus, by Theorem 3.1, x∗ = max C = z is a fixed point of G.

As a consequence of Proposition 3.2 we obtain the following result.

Corollary 3.3. If nonempty chains of X have supremums, if G : X → X is maximalizing, and ifsup{x,G(x)} exists for all x ∈ X, then for each a ∈ X the maximum of the w-o chain of aG-iterationsexists and is a fixed point of G.

Proof. Let C be the w-o chain of aG-iterations. The given hypotheses imply that both z =supC and x∗ = sup{z,G(z)} exist. Thus the hypotheses of Proposition 3.2 are valid.

The results of Lemma 2.3 are valid also when C is replaced by the w-o chain C(a) ofG-iterations of a. As a consequence of these results and Lemma 2.4 we obtain the followinggeneralizations to Bourbaki’s fixed point theorem.


Theorem 3.4. Assume that G : X → X is maximalizing, and that a ≤ G(a) for some a ∈ X, and letC(a) be the w-o chain of G-iterations of a.

(a) If x∗ = supG[C(a)] exists, then x∗ = maxC(a), and x∗ is a fixed point of G.

(b) If z = supC(a) exists, it is a fixed point of G if and only if x∗ = sup{z,G(z)} exists.(c) If nonempty chains of X have supremums, and if sup{x,G(x)} exists for all x ∈ X, then

x∗ = maxC(a) exists, and x∗ is a fixed point of G.

The previous results have obvious duals, which imply the following results.

Theorem 3.5. A mapping G : X → X has a fixed point if G is minimalizing, that is, G(x) isa minimal element of {x,G(x)} for all x ∈ X, and if inf{a,G[W]} exists in X for some a ∈ Xwhenever W is a nonempty chain in X.

Theorem 3.6. A minimalizing mapping G : X → X has a fixed point if inf G[W] exists wheneverW is a nonempty chain in X, and if G(a) ≤ a for some a ∈ X.

Proposition 3.7. A minimalizing mapping G : X → X has a fixed point if every nonempty chain Xhas the infimum in X, and if inf{x,G(x)} exists for all x ∈ X.

Remark 3.8. The hypothesis that G : X → X is maximalizing can be weakened in Theorems3.1 and 3.4 and in Proposition 3.2 to the form: G | {x∗} is maximalizing, that is, G(x∗) is amaximal element of {x∗, G(x∗)}.

4. Examples and Remarks

We will first present an example of a maximalizing mapping whose fixed point is obtained asthe maximum of the w-o chain of aG-iterations.

Example 4.1. Let X be a closed disc X = {(u, v) ∈ R2 : u2 + v2 ≤ 2}, ordered coordinate-wise.

Let [u] denote the greatest integer ≤ u when u ∈ R. Define a function G : X → R2 by

G(u, v) =(

min{1, 1 − [u] + [v]}, 12

([u] + v2

)), (u, v) ∈ X. (4.1)

It is easy to verify that G[X] ⊂ X, and that G is maximalizing. To find a fixed point ofG, choose a = (1, 0). It follows from Lemma 2.3(b) that the first elements of the w-o chain ofaG-iterations are successive approximations

x0 = a, xn+1 = Sxn = sup{xn,G(xn)}, n = 0, 1, . . . , (4.2)

as long as Sxn is defined. Denoting xn = (un, vn), these successive approximations can berewritten in the form

u0 = 1, un+1 = max{un,min{1, 1 − [un] + [vn]}},

v0 = 0, vn+1 = max{vn,

12

([un] + v2

n

)}, n = 0, 1, . . . ,

(4.3)


as long as un ≤ un+1 and vn ≤ vn+1, and at least one of these inequalities is strict. Elementarycalculations show that un = 1, for every n ∈ N0. Thus (4.3) can be rewritten as

un = 1, v0 = 0, vn+1 = max{vn,

12

(1 + v2

n

)}, n = 0, 1, . . . . (4.4)

Since the function g(v) = (1/2)(1+ v2) is increasing R+, then vn < g(vn) for every n = 0, 1, . . ..Thus (4.4) can be reduced to the form

un = 1, v0 = 0, vn+1 = g(vn) =12

(1 + v2

n

), n = 0, 1, . . . . (4.5)

The sequence (g(vn))∞n=0 is strictly increasing, whence also (vn)

∞n=0 is strictly increasing by

(4.5). Thus the set W = {(1, g(vn))}n∈N0is an initial segment of C. Moreover, v0 = 0 < 1,

and if 0 ≤ vn < 1, then 0 < g(vn) < 1. Since (g(vn))∞n=0 is bounded above by 1, then v∗ =

limng(vn) exists, and 0 < v∗ ≤ 1. Thus (1, v∗) = supW , and it belongs to X, whence (1, v∗) ∈C by Lemma 2.3(c). To determine v∗, notice that vn+1 → v∗ by (4.5). Thus v∗ = g(v∗), orequivalently, v2

∗ − 2v∗ + 1 = 0, so that v∗ = 1. Since supW = (1, v∗) = (1, 1), then (1, 1) ∈ Cby Lemma 2.3(c). Because (1, 1) is a maximal element of X, then (1, 1) = maxC. Moreover,G(1, 1) = (1, 1), so that (1, 1) is a fixed point of G.

The first m + 1 elements of the w-o chain C of aG-iterations can be estimated by thefollowing Maple program (floor(·) = [·]):

x := min(1,1-floor(u) + floor(v)): y := (floor(u) + v2)/2: (u,v) := (1, 0) : c[0] := (u,v):for n to m do (u,v) := (max(x,u), evalf(max(y,v)); c[n] := (u,v) end do;For instance, c[100000] = (1, 0.99998).The verification of the following properties are left to the reader.

(i) If c = (u, v) ∈ X, u < 1, and v < 1, then the elements of w-o chain C of aG-iterations,after two first terms if u < 1, are of the form (1, wn), n = 0, 1, . . ., where (wn)

∞n=0 is

increasing and converges to 1. Thus (1, 1) is the maximum of C and a fixed point ofG.

(ii) If a = (u, 1), u < 1, or a = (1,−1), then C = {a, (1, 1)}.(iii) If a = (1, 0), then G2ka = (1, zk) and G2k+1a = (0, yk), k ∈ N0, where the sequences

(zk) and (yk) are bounded and increasing. The limit z of (zk) is the smaller realroot of z4 − 8z + 4 = 0; z ≈ 0.50834742498666121699, and the limit y of (yk) is y =(1/2)z2 ≈ 0.12920855224528457650. Moreover G(1, y) = (0, z) and G(0, z) = (1, y),whence no subsequence of the iteration (Gna) converges to a fixed point of G.

(iv) For any choice of a = (u, v) ∈ P \ {(1, 1)} the iterations Gna and Gn+1a are not orderrelated when n ≥ 2. The sequence (Gnc) does not converge, and no subsequence ofit converges to a fixed point of G.

(v) Denote Y = {(u, v) ∈ R2+ : u2 + v2 ≤ 2, v > 0} ∪ {(1, 0)}. The function G, defined

by (4.1), satisfies G[Y ] ⊂ Y and is maximalizing. The maximum of the w-o chain ofaG-iterations with a = (1, 0) is x∗ = (1, 1), and x∗ is a fixed point of G. If x ∈ Y \{x∗},then x and G(x) are not comparable.

The following example shows that G need not to have a fixed point if either of thehypothesis of Theorem 3.1 is not valid.


Example 4.2. Denote a = (1, y) and b = (0, z), where y and z are as in Example 4.1. ChooseX = {a, b}, and let G : X → X be defined by (4.1). G is maximalizing, but G has no fixedpoints, since G(a) = b and G(b) = a. The last hypothesis of Theorem 3.1 is not satisfied.

Denoting c = (1, z), then the set X = {a, b, c} is a complete join lattice, that is, everynonempty subset of X has the supremum in X. Let G : X → X satisfy G(a) = b and G(b) =G(c) = a. G has no fixed points, but G is not maximalizing, since G(c) < c.

Example 4.3. The components u = 1, v = 1 of the fixed point of G in Example 4.1 form also asolution of the system

u = min{1, 1 − [u] + [v]}, v =[u] + v2

2. (4.6)

Moreover a Maple program introduced in Example 4.1 serves a method to estimate thissolution. When m = 100000, the estimate is u = 1, v = 0.99998.

Remark 4.4. The standard “solve” and “fsolve” commands of Maple 12 do not give a solutionor its approximation for the system of Example 4.3.

In Example 4.1 the mapping G is nonincreasing, nonextensive, nonascending, notsemiincreasing upward, and noncontinuous.

Chain C(a) is compared in [10] with three other chains which generalize the sequenceof ordinary iterations (Gn(a))∞n=0, and which are used to prove fixed point results for G.These chains are the generalized orbit O(a) defined in [10] (being identical with the set W(a)defined in [11]), the smallest admissible set I(a) containing a (cf. [12–14]), and the smallestcomplete G-chain B(a) containing a (cf. [10, 15]). If G is extensive, and if nonempty chainsof X have supremums, then C(a) = O(a) = I(a), and B(a) is their cofinal subchain (cf. [10,Corollary 7]). The common maximum x∗ of these four chains is a fixed point of G. This resultimplies Bourbaki’s Fixed Point Theorem.

On the other hand, if the hypotheses of Theorem 3.4 hold and x ∈ C(a)\{a, x∗}, then xand G(x) are not necessarily comparable. The successor of such an x in C(a) is sup{x,G(x)}by [14, Proposition 5]. In such a case the chains O(a), I(a) and B(a) attain neither x nor anyfixed point of G. For instance when a = (0, 0) in Example 4.1, then C(a) = {(0, 0)} ∪ C, whereC is the w-o chain of (1, 0)G-iterations. Since (Gn(0, 0))∞n=0 = {(0, 0)} ∪ (Gn(1, 0))∞n=0, then B(a)does not exist, O(a) = I(a) = {(0, 0), (1, 0)} (see [10]). Thus only C(a) attains a fixed pointof G as its maximum. As shown in Example 4.1, the consecutive elements of the iterationsequence (Gn(1, 0))∞n=0 are unordered, and their limits are not fixed points of G. Hence, inthese examples also finite combinations of chains W(ai) used in [16, Theorem 4.2] to prove afixed point result are insufficient to attain a fixed point of G.

Neither the above-mentioned four chains nor their duals are available to find fixedpoints of G if a and G(a) are unordered. For instance, they cannot be applied to proveTheorems 3.1 and 3.5 or Propositions 3.2 and 3.7.

References

[1] N. Bourbaki, Elements de Mathematique, I. Theorie des Ensembles, Fascicule de Resultats, ActualitesScientifiques et Industrielles, no. 846, Hermann, Paris, France, 1939.

[2] W. A. Kirk, Fixed Point Theory: A Brief Survey, Notas de Matematicas, no. 108, Universidas de LosAndes, Merida, Venezuela, 1990.


[3] J. Klimes, “A characterization of inductive posets,” Archivum Mathematicum, vol. 21, no. 1, pp. 39–42,1985.

[4] S. Heikkila, “Fixed point results for semi-increasing mappings,” to appear in Nonlinear Studies.[5] S. Heikkila, “Monotone methods with applications to nonlinear analysis,” in Proceedings of the 1st

World Congress of Nonlinear Analysts, vol. 1, pp. 2147–2158, Walter de Gruyter, Tampa, Fla, USA, 1996.[6] S. Heikkila, “A method to solve discontinuous boundary value problems,” Nonlinear Analysis, vol. 47,

pp. 2387–2394, 2001.[7] S. Heikkila, “On recursions, iterations and well-orderings,” Nonlinear Times and Digest, vol. 2, no. 1,

pp. 117–123, 1995.[8] S. Carl and S. Heikkila, Nonlinear Differential Equations in Ordered Spaces, Chapman & Hall/CRC, Boca

Raton, Fla, USA, 2000.[9] S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear

Differential Equations, vol. 181 of Monographs and Textbooks in Pure and Applied Mathematics, MarcelDekker, New York, NY, USA, 1994.

[10] R. Manka, “On generalized methods of successive approximations,” to appear in Nonlinear Analysis.[11] S. Abian and A. B. Brown, “A theorem on partially ordered sets, with applications to fixed point

theorems,” Canadian Journal of Mathematics, vol. 13, pp. 78–82, 1961.[12] T. Buber and W. A. Kirk, “A constructive proof of a fixed point theorem of Soardi,” Mathematica

Japonica, vol. 41, no. 2, pp. 233–237, 1995.[13] T. Buber and W. A. Kirk, “Constructive aspects of fixed point theory for nonexpansive mappings,”

in Proceedings of the 1st World Congress of Nonlinear Analysts, vol. 1, pp. 2115–2125, Walter de Gruyter,Tampa, Fla, USA, 1996.

[14] S. Heikkila, “On chain methods used in fixed point theory,” Nonlinear Studies, vol. 6, no. 2, pp. 171–180, 1999.

[15] B. Fuchssteiner, “Iterations and fixpoints,” Pacific Journal of Mathematics, vol. 68, no. 1, pp. 73–80, 1977.[16] K. Baclawski and A. Bjorner, “Fixed points in partially ordered sets,” Advances in Mathematics, vol. 31,

no. 3, pp. 263–287, 1979.


Research ArticleSome Weak Convergence Theorems for a Family ofAsymptotically Nonexpansive Nonself Mappings

Yan Hao,1 Sun Young Cho,2 and Xiaolong Qin3

1 School of Mathematics, Physics and Information Science, Zhejiang Ocean University,Zhoushan 316004, China

2 Department of Mathematics, Gyeongsang National University, Jinju 660-701, South Korea3 Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Correspondence should be addressed to Yan Hao, [email protected]

Received 31 August 2009; Accepted 16 November 2009


Copyright q 2010 Yan Hao et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A one-step iteration with errors is considered for a family of asymptotically nonexpansive nonselfmappings. Weak convergence of the purposed iteration is obtained in a Banach space.


Let E be a real Banach space and E∗ the dual space of E. Let 〈·, ·〉 denote the pairing betweenE and E∗. The normalized duality mapping J : E → 2E

∗is defined by

J(x) ={f ∈ E∗ :

⟨x, f

⟩= ‖x‖2 =

∥∥f∥∥2}, ∀x ∈ E. (1.1)

Let UE = {x ∈ E : ‖x‖ = 1}, where E is said to be smooth or said to have a Gateauxdifferentiable norm if the limit

limt→ 0

∥∥x + ty∥∥ − ‖x‖t

(1.2)

exists for each x, y ∈ UE, where E is said to have a uniformly Gateaux differentiable normif for each y ∈ UE, the limit is attained uniformly for all x ∈ UE, where E is said to beuniformly smooth or said to have a uniformly Frechet differentiable norm if the limit is


attained uniformly for all x, y ∈ UE, whereE is said to be uniformly convex if for any ε ∈ (0, 2]there exists δ > 0 such that for any x, y ∈ UE:

∥∥x − y∥∥ ≥ ε implies∥∥∥∥x + y

2

∥∥∥∥ ≤ 1 − δ. (1.3)

It is known that a uniformly convex Banach space is reflexive and strictly convex.In this paper, we use → and ⇀ to denote the strong convergence and weak

convergence, respectively. Recall that a Banach space E is said to have the Kadec-Kleeproperty if for any sequence {xn} ⊂ E and x ∈ E with xn ⇀ x and ‖xn‖ → ‖x‖, then‖xn − x‖ → 0 as n → ∞ for more details on Kadec-Klee property, the reader is referred to[1, 2] and the references therein. It is well known that if E is a uniformly convex Banach space,then E enjoys the Kadec-Klee property.

Recall that a Banach space E is said to satisfy the Opial condition [3] if, for eachsequence {xn} in E, the convergence xn ⇀ x implies that

lim infn→∞

‖xn − x‖ < lim infn→∞

∥∥xn − y∥∥, ∀y ∈ E (y /=x

). (1.4)

Let C be a nonempty closed and convex subset of E and T a mapping. In this paper,we use F(T) to denote the fixed point set of T. Recall that the mapping T is said to benonexpansive if

∥∥Tx − Ty∥∥ ≤ ∥∥x − y∥∥, ∀x, y ∈ C. (1.5)

T is said to be asymptotically nonexpansive if there exists a sequence {kn} ⊂ [1,∞) withkn → 1 as n → ∞ such that

∥∥Tnx − Tny∥∥ ≤ kn∥∥x − y∥∥, ∀x, y ∈ C, ∀n ≥ 1. (1.6)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk[4] as a generalization of the class of nonexpansive mappings. They proved that if C isa nonempty closed convex and bounded subset of a real uniformly convex Banach space,then every asymptotically nonexpansive self-mapping has a fixed point; see [4] for moredetails. Some classical results on asymptotically nonexpansive mappings and other importantnonlinear mappings have been established by Kirk et al.; see [5–13].

However, T is said to be uniformly L-lipschitz if there exists a positive constant L suchthat

∥∥Tnx − Tny∥∥ ≤ L∥∥x − y∥∥, ∀x, y ∈ C, ∀n ≥ 1. (1.7)

Recall that the Mann iteration was introduced by Mann [14] in 1953. The Manniteration sequence {xn} is defined in the following manner:

∀x1 ∈ C, xn+1 = (1 − αn)xn + αnTxn, ∀n ≥ 1, (1.8)

where {αn} is a sequence in the interval (0, 1) and T : C → C is a mapping.


In 1979, Reich [15] obtained the following celebrated weak convergence theorem.

Theorem R-1. Let C be a closed convex subset of a uniformly convex Banach space E with a Frechetdifferential norm, T : C → C a nonexpansive mapping with a fixed point, and {αn} a real sequencesuch that 0 ≤ αn ≤ 1 and

∑n=1 αn(1 − αn) = ∞. Let {xn} be a sequence generated in (1.8). Then the

sequence {xn} converges weakly to a fixed point of T.

Note that the dual of reflexive Banach spaces with a Frechet differentiable norm havethe Kadec-Klee property. In 2001, Garcıa Falset et al. [16] obtained a new weak convergencetheorem without the restriction E enjoys the Frechet differential norm. To be more precise,they obtained the following results.

Theorem FKKR. Let C be a closed convex subset of a uniformly convex Banach space E such that E∗

has the Kadec-Klee property, T : C → C a nonexpansive mapping with a fixed point, and {αn} a realsequence such that 0 ≤ αn ≤ 1 and

∑∞n=1 αn(1 − αn) = ∞. Let {xn} be a sequence generated in (1.8).

Then the sequence {xn} converges weakly to a fixed point of T.

Recall that the modified Mann iteration which was introduced by Schu [17] generatesa sequence {xn} in the following manner:

x1 ∈ C, xn+1 = (1 − αn)xn + αnTnxn, ∀n ≥ 1, (1.9)

where {αn} is a sequence in the interval (0, 1) and T : C → C is an asymptoticallynonexpansvie mapping.

In 1991, Schu [17] obtained the following weak convergence results for asymptoticallynonexpansive mappings in a uniformly convex Banach space. To be more precise, theyobtained the following results.

Theorem S. Let E be a uniformly convex Banach space satisfying the Opial condition, ∅/=C ⊂ Eclosed bounded and convex and S : C → C asymptotically nonexpansive with sequence {kn} ⊂ [1,∞)for which

∑∞n=1(kn − 1) <∞ and {αn} ∈ [0, 1] is bounded away. Let {xn} be a sequence generated in

(1.9). Then the sequence {xn} converges weakly to some fixed point of T .

Note that each lp (1 ≤ p <∞) satisfies the Opial condition, while all Lp do not have theproperty unless p = 2. In 1994, Tan and Xu [18] obtained the following results.

Theorem TX. Let E be a uniformly convex Banach space whose norm is Frechet differentiable, C anonempty closed and convex subset of E, and T : K → K an asymptotically nonexpansive mappingwith a sequence {kn} ⊂ [1,∞) such that

∑∞n=1(kn − 1) < ∞ such that F(T) is nonempty. Let {xn}

be sequence generated in (1.9), where {αn} is a real sequence bounded away from 0 and 1. Then thesequence {xn} converges weakly to some point in F(T).

Let E be a Banach space, K a nonempty subset of E, and T : K → E a mapping. Forall x ∈ K, define a set IK(x) by

IK(x) ={x + λ

(y − x) : λ > 0, y ∈ K}

, (1.10)

where T is said to be inward if Tx ∈ IK(x) for all x ∈ K and T is said to be weakly inwardif Tx ∈ IK(x) for all x ∈ K. Recall that the subset K of E is said to be retract if there exists


a continuous mapping P : E → K such that Px = x for all x ∈ K. It is well known that everyclosed convex subset of a uniformly convex Banach space is a retract. A mapping P : E → Eis said to be a retraction if P 2 = P. Let C and D be subsets of E. Then a mapping P : C → Dis said to be sunny if P(Px + t(x − Px)) = Px, whenever Px + t(x − Px) ∈ C for all x ∈ C andt ≥ 0.

The following result describes a characterization of sunny nonexpansive retractionson a smooth Banach space. See Reich [19].

Theorem R-2. Let E be a smooth Banach space and let C be a nonempty subset of E. Let Q : E → Cbe a retraction and let J be the normalized duality mapping on E. Then the following are equivalent:

(1) P is sunny and nonexpansive;(2) ‖Px − Py‖2 ≤ 〈x − y, J(Px − Py)〉, ∀x, y ∈ E;(3) 〈x − Px, J(y − Px)〉 ≤ 0, ∀x ∈ E, y ∈ C.Recently, fixed point problems of nonself mappings have been studied by a number

of authors; see, for example, [20–30]. Next, we draw our attention to nonself mappings. LetK be a nonempty subset of a Banach space E, T : K → E be a mapping and P a sunnynonexpansive retraction from E onto K.

The mapping T is said to be asymptotically nonexpansive with respect to P if thereexists a sequence {kn} ⊂ [1,∞) with kn → 1 as n → ∞ such that

∥∥(PT)nx − (PT)ny∥∥ ≤ kn∥∥x − y∥∥, ∀x, y ∈ K, ∀n ≥ 1. (1.11)

The mapping T is said to be uniformly L-lipschitz with respect to P if there exists apositive constant L such that

∥∥(PT)nx − (PT)ny∥∥ ≤ L∥∥x − y∥∥, ∀x, y ∈ K, ∀n ≥ 1. (1.12)

We remark that if T is a self mapping, then P is reduced to the identity mapping. Itfollows that (1.11) is reduced to (1.6).

In this paper, we consider a one-step iteration for a finite family of asymptoticallynonexpansive nonself mappings. Weak convergence theorems are established in a realsmooth and uniformly convex Banach space.

In order to prove our main results, we need the following lemmas.

Lemma 1.1 (see [16, 31]). LetE be a uniformly convex Banach space such that its dual has the Kadec-Klee property. Suppose that {xn} is a bounded sequence such that limn→∞‖axn+(1−a)f1−f2‖ existsfor all a ∈ [0, 1] and f1, f2 ∈ ωw(xn). Then ωw(xn) is a singleton.

Lemma 1.2 (see [2, 25]). Let E be a real smooth Banach space, K a nonempty closed convex subsetof E with P as a sunny nonexpansive retraction, and T : K → E a mapping which enjoys the weaklyinward condition. Then F(PT) = F(T).

Lemma 1.3 (see [32]). Let {an} and {bn} be two nonnegative sequences satisfying the followingcondition:

an+1 ≤ an + bn, ∀n ≥ 1. (1.13)

If∑∞

n=1 bn <∞, then limn→∞an exists.


Lemma 1.4 (see [33]). Let p > 1 and s > 0 be two fixed real numbers. Then a Banach space Eis uniformly convex if and only if there exists a continuous strictly increasing convex function g :[0,∞) → [0,∞) with g(0) = 0 such that

∥∥λx + (1 − λ)y∥∥p ≤ λ‖x‖p + (1 − λ)∥∥y∥∥p −wp(λ)g(∥∥x − y∥∥) (1.14)

for all x, y ∈ Bs(0) = {x ∈ E : ‖x‖ ≤ s} and λ ∈ [0, 1], where wp(λ) = λp(1 − λ) + λ(1 − λ)p.

The following lemma is an immediate result of Lemma 1.4. See also Zhang [34].

Lemma 1.5. Let E be a uniformly convex Banach space, s > 0 a positive number, and Bs(0) a closedball of E. There exits a continuous, strictly increasing, and convex function g : [0,∞) → [0,∞) withg(0) = 0 such that

∥∥∥∥∥N∑i=1

(αixi)

∥∥∥∥∥2

≤N∑i=1

(αi‖xi‖2

)− α1α2g(‖x1 − x2‖) (1.15)

for all x1, x2, . . . , xN ∈ Bs(0) = {x ∈ E : ‖x‖ ≤ s} and α1, α2, . . . , αN ∈ [0, 1] such that∑N

i=1 αi = 1.

Proof. We prove it by inductions. For N = 2, we from Lemma 1.4 see that (1.15) holds. ForN = j, where j ≥ 3 is some positive integer, suppose that (1.15) holds. We see that (1.15) stillholds for N = j + 1. Indeed, from Lemma 1.4, we see that

∥∥α1x1 + α2x2 + · · · + αjxj + αj+1xj+1∥∥2

=

∥∥∥∥∥(1 − αj+1

)( α1

1 − αj+1x1 +

α2

1 − αj+1x2 + · · · +

αj

1 − αj+1xj

)+ αj+1xj+1

∥∥∥∥∥2

≤ (1 − αj+1

)∥∥∥∥∥α1

1 − αj+1x1 +

α2

1 − αj+1x2 + · · · +

αj

1 − αj+1xj

∥∥∥∥∥2

+ αj+1∥∥xj+1

∥∥2

− αj(1 − αj+1

)g

(∥∥∥∥∥(

α1

1 − αj+1x1 +

α2

1 − αj+1x2 + · · · +

αj

1 − αj+1xj

)− xj+1

∥∥∥∥∥)

≤ (1 − αj+1

)( α1

1 − αj+1‖x1‖2 +

α2

1 − αj+1‖x2‖2 + · · · + αj

1 − αj+1

∥∥xj∥∥2

− α1α2(1 − αj+1

)(1 − αj+1

)g(‖x1 − x2‖))

+ αj+1∥∥xj+1

∥∥2

= α1‖x1‖2 + α2‖x2‖2 + · · · + αj∥∥xj∥∥2 + αj+1

∥∥xj+1∥∥2 − α1α2

1 − αj+1g(‖x1 − x2‖)

≤ α1‖x1‖2 + α2‖x2‖2 + · · · + αj∥∥xj∥∥2 + αj+1

∥∥xj+1∥∥2 − α1α2g(‖x1 − x2‖).

(1.16)

This completes the proof.


Lemma 1.6 (see [35]). Let E be a real uniformly convex Banach space, K a nonempty closed, andconvex subset ofE and T : K → K an asymptotically nonexpansive mapping. Then I−T is demiclosedat zero, that is, xn ⇀ x and xn − Txn → 0 imply that x = Tx.

2. Main Results

Lemma 2.1. Let E be a real uniformly convex Banach space,K a nonempty closed and convex subsetof E, and P a sunny nonexpansive retraction from E onto K. Let Ti : K → E be an asymptoticallynonexpansive mapping with respect to P with a sequence {kn,i} ⊂ [1,∞) such that

∑∞n=1(kn,i−1) <∞

for each i ∈ {1, 2, . . . ,N}. Assume that F =⋂Ni=1 F(Ti) is nonempty. Let {xn} be sequence generated

in the following manner: x1 ∈ K and

xn+1 = αn,0xn +N∑i=1

αn,i(PTi)nxn + αn,N+1un, ∀n ≥ 1, (HCQ)

where {αn,i} is a real sequence in (0, 1) and {un} is a bounded sequence in K. Assume that

(a)∑N+1

i=0 αn,i = 1;

(b) lim infn→∞αn,0αn,i > 0 for each i ∈ {1, 2, . . . ,N};

(c)∑∞

n=1 αn,N+1 <∞.

Then limn→∞‖xn − (PTi)xn‖ = 0 for each i ∈ {1, 2, . . . ,N}.

Proof. Fix q ∈ F and kn = max{kn,1, kn,2, . . . , kn,N}. It follows that∑∞

n=1(kn −1) <∞. Since {un}is a bounded sequence in K, we set M = sup{‖un − q‖ : n ≥ 1}. It follows that

∥∥xn+1 − q∥∥ =

∥∥∥∥∥αn,0xn +N∑i=1

αn,i(PTi)nxn + αn,N+1un − q∥∥∥∥∥

≤ αn,0∥∥xn − q∥∥ +

N∑i=1

αn,i∥∥(PTi)nxn − q∥∥ + αn,N+1

∥∥un − q∥∥

≤ αn,0∥∥xn − q∥∥ +

N∑i=1

αn,ikn,i∥∥xn − q∥∥ + αn,N+1

∥∥un − q∥∥

≤ [1 + (kn − 1)]∥∥xn − q∥∥ + αn,N+1M.

(2.1)

In view of the condition (c), we obtain from Lemma 1.3 that limn→∞‖xn − q‖ exists for anyq ∈ F(T). This in turn shows that the sequence {xn} is bounded.


On the other hand, we conclude from Lemma 1.4 that

∥∥xn+1 − q∥∥2 =

∥∥∥∥∥αn,0xn +N∑i=1

αn,i(PTi)nxn + αn,N+1un − q∥∥∥∥∥

2

≤ αn,0∥∥xn − q∥∥2 +

N∑i=1

αn,i∥∥(PTi)nxn − q∥∥2 + αn,N+1

∥∥un − q∥∥2

− αn,0αn,1g(∥∥xn − (PT1)nxn

∥∥)

≤ αn,0∥∥xn − q∥∥2 +

N∑i=1

αn,ik2n,i

∥∥xn − q∥∥2 + αn,N+1∥∥un − q∥∥2

− αn,0αn,1g(∥∥xn − (PT1)nxn

∥∥)

≤[1 +

(k2n − 1

)]∥∥xn − q∥∥2 + αn,N+1∥∥un − q∥∥2 − αn,0αn,1g

(∥∥xn − (PT1)nxn∥∥).

(2.2)

This shows that

αn,0αn,1g(∥∥xn − (PT1)nxn

∥∥)

≤ ∥∥xn − q∥∥2 − ∥∥xn+1 − q∥∥2 +

(k2n − 1

)∥∥xn − q∥∥2 + αn,N+1∥∥un − q∥∥2

≤ (∥∥xn − q∥∥ − ∥∥xn+1 − q∥∥)R1 +

(k2n − 1

)R2 + αn,N+1

∥∥un − q∥∥2.

(2.3)

where R1 = sup{‖xn − q‖ + ‖xn+1 − q‖ : n ≥ 1} and R2 = sup{‖xn − q‖2 : n ≥ 1}. In view of theconditions (b) and (c), we arrive at limn→∞g(‖xn − (PT1)

nxn‖) = 0. In view of the property ofthe function g, we conclude that

limn→∞

∥∥xn − (PT1)nxn∥∥ = 0. (2.4)

By repeating (2.2) and (2.3), we can conclude that

limn→∞

∥∥xn − (PTi)nxn∥∥ = 0, ∀i ∈ {1, 2, . . . ,N}. (2.5)

Note that

‖xn+1 − xn‖ ≤N∑i=1

αn,i∥∥(PTi)nxn − xn∥∥ + αn,N+1‖un − xn‖. (2.6)

From (2.5) and condition (c), we see that

limn→∞

‖xn+1 − xn‖ = 0. (2.7)


On the other hand, we have

‖xn − (PTi)xn‖ ≤ ‖xn − xn+1‖ +∥∥∥xn+1 − (PTi)n+1xn+1

∥∥∥+∥∥∥(PTi)n+1xn+1 − (PTi)n+1xn

∥∥∥ +∥∥∥(PTi)n+1xn − (PTi)xn

∥∥∥.(2.8)

Since Ti is Lipschitz with respective to P for each i ∈ {1, 2, . . . ,N}, we obtain that

limn→∞

‖xn − (PTi)xn‖ = 0, ∀i ∈ {1, 2, . . . ,N}. (2.9)


Next, we give some weak convergence theorems.

Theorem 2.2. Let E be a real smooth and uniformly convex Banach space which enjoys the Opialcondition, K a nonempty closed and convex subset of E, and P a sunny nonexpansive retraction fromE on K. Let Ti : K → E be a weakly inward and asymptotically nonexpansive mapping with respectto P with a sequence {kn,i} ⊂ [1,∞) such that

∑∞n=1(kn,i −1) <∞ for each i ∈ {1, 2, . . . ,N}. Assume

that F =⋂Ni=1 F(Ti) is nonempty. Let {xn} be sequence generated in (HCQ), where {αn,i} is a real

sequence in (0, 1) and {un} is a bounded sequence in K. Assume that

(a)∑N+1

i=0 αn,i = 1;

(b) lim infn→∞αn,0αn,i > 0 for each i ∈ {1, 2, . . . ,N};(c)

∑∞n=1 αn,N+1 <∞.

Then the sequence {xn} converges weakly to some point in F.

Proof. Since E is reflexive and {xn} is bounded, we from Lemmas 1.2 and 1.6 conclude thatωw(xn) ⊂ F(PTi) = F(Ti) for each i ∈ {1, 2, . . . ,N}. On the other hand, since the space Eenjoys the Opial condition, we see that ωw(xn) is singleton. This completes the proof.

If T = Ti for each i ∈ {1, 2, . . . ,N} and αn,N+1 = 0 for each n ≥ 1, then we have fromTheorem 2.2 the following results.

Corollary 2.3. Let E be a real smooth and uniformly convex Banach space which enjoys the Opialcondition, K a nonempty closed and convex subset of E, and P a sunny nonexpansive retraction fromE ontoK. Let T : K → E be a weakly inward and asymptotically nonexpansive mapping with respectto P with a sequence {kn} ⊂ [1,∞) such that

∑∞n=1(kn − 1) <∞. Assume that F(T) is nonempty. Let

{xn} be sequence generated in the following manner: x1 ∈ K and

xn+1 = (1 − αn)xn + αn(PT)nxn, ∀n ≥ 1, (2.10)

where {αn} is a real sequence in (0, 1) such that lim infn→∞αn(1 − αn) > 0. Then the sequence {xn}converges weakly to some point in F(T).


Theorem 2.4. Let E be a real smooth and uniformly convex Banach space whose norm is Frechetdifferentiable, K a nonempty closed and convex subset of E, and P a sunny nonexpansive retractionfrom E onto K. Let Ti : K → E be a weakly inward and asymptotically nonexpansive mapping withrespect to P with a sequence {kn,i} ⊂ [1,∞) such that

∑∞n=1(kn,i − 1) <∞ for each i ∈ {1, 2, . . . ,N}.

Assume that F =⋂Ni=1 F(Ti) is nonempty. Let {xn} be sequence generated in (HCQ), where {αn,i} is

a real sequence in (0, 1) and {un} is a bounded sequence in K. Assume that

(a)∑N+1

i=0 αn,i = 1;

(b) lim infn→∞αn,0 αn,i > 0 for each i ∈ {1, 2, . . . ,N};(c)

∑∞n=1 αn,N+1 <∞.


Proof. Since E is reflexive and {xn} is bounded, we from Lemma 1.2 and 1.6 conclude thatωw(xn) ⊂ F(PTi) = F(Ti) for each i ∈ {1, 2, . . . ,N}. From the proof of Tan and Xu [18, Lemma2.2] (see also Cho et al. [35, Lemma 1.8]), we can show that, for every f1, f2 ∈ F,

⟨p − q, J(f1 − f2

)⟩= 0, ∀p, q ∈ ωw(xn). (2.11)

Let p, q ∈ ωw(xn). It follows that p, q ∈ F; that is,

∥∥p − q∥∥ =⟨p − q, J(p − q)⟩ = 0. (2.12)

Therefore, p = q. This completes the proof.

If T = Ti for each i ∈ {1, 2, . . . ,N} and αn,N+1 = 0 for each n ≥ 1, then we fromTheorem 2.4 have the following results.

Corollary 2.5. Let E be a real smooth and uniformly convex Banach space whose norm is Frechetdifferentiable, K a nonempty closed and convex subset of E, and P a sunny nonexpansive retractionfrom E onto K. Let T : K → E be a weakly inward and asymptotically nonexpansive mapping withrespect to P with a sequence {kn} ⊂ [1,∞) such that

∑∞n=1(kn − 1) < ∞. Assume that F(T) is

nonempty. Let {xn} be sequence generated in (2.10), where {αn} is a real sequence in (0, 1) such thatlim infn→∞αn(1 − αn) > 0. Then the sequence {xn} converges weakly to some point in F(T).

Theorem 2.6. Let E be a real smooth and uniformly convex Banach space such that its dual E∗ hasthe Kadec-Klee property, K a nonempty closed and convex subset of E, and P a sunny nonexpansiveretraction from E onto K. Let Ti : K → E be a weakly inward and asymptotically nonexpansivemapping with respect to P with a sequence {kn,i} ⊂ [1,∞) such that

∑∞n=1(kn,i − 1) < ∞ for each

i ∈ {1, 2, . . . ,N}. Assume that F =⋂Ni=1 F(Ti) is nonempty. Let {xn} be sequence generated in

(HCQ), where {αn,i} is a real sequence in (0, 1) and {un} is a bounded sequence in K. Assume that

(a)∑N+1

i=0 αn,i = 1;

(b) lim infn→∞αn,0αn,i > 0 for each i ∈ {1, 2, . . . ,N};(c)

∑∞n=1 αn,N+1 <∞.



Proof. Since E is reflexive and {xn} is bounded, we from Lemma 1.2 and Lemma 1.6 concludethat ωw(xn) ⊂ F(PTi) = F(Ti) for each i ∈ {1, 2, . . . ,N}. From the proof of Lemma 2.2 of Tanand Xu [18] (see also of Cho et al. [35, Lemma 1.8]), we can show that limn→∞‖axn + (1 −a)f1 − f2‖ exists for all a ∈ [0, 1] and f1, f2 ∈ ωw(xn). In view of Lemma 1.1, we see thatωw(xn) is singleton. This completes the proof.

If T = Ti for each i ∈ {1, 2, . . . ,N} and αn,N+1 = 0 for each n ≥ 1, then we fromTheorem 2.6 have the following results.

Corollary 2.7. Let E be a real smooth and uniformly convex Banach space such that its dual E∗ hasthe Kadec-Klee property, K a nonempty closed and convex subset of E and P a sunny nonexpansiveretraction from E onto K. Let T : K → E be a weakly inward and asymptotically nonexpansivemapping with respect to P with a sequence {kn} ⊂ [1,∞) such that

∑∞n=1(kn − 1) < ∞. Assume that

F(T) is nonempty. Let {xn} be sequence generated in (2.10), where {αn} is a real sequence in (0, 1)such that lim infn→∞αn(1−αn) > 0. Then the sequence {xn} converges weakly to some point in F(T).

Acknowledgments

This project is supported by the National Natural Science Foundation of China (no.10901140). The authors are extremely grateful to the referees for useful suggestions thatimproved the contents of the paper.

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Research ArticleProperties WORTH and WORTHH∗,(1 + δ) Embeddings in Banach Spaces with1-Unconditional Basis and wFPP

Helga Fetter and Berta Gamboa de Buen

Centro de Investigacion en Matematicas (CIMAT), Apdo. Postal 402, 36000 Guanajuato, GTO, Mexico

Correspondence should be addressed to Helga Fetter, [email protected]



Copyright q 2010 H. Fetter and B. Gamboa de Buen. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We will use Garcıa-Falset and Llorens Fuster’s paper on the AMC-property to prove that a Banachspace X that (1+ δ) embeds in a subspace Xδ of a Banach space Y with a 1-unconditional basis hasthe property AMC and thus the weak fixed point property. We will apply this to some results byCowell and Kalton to prove that every reflexive real Banach space with the property WORTH andits dual have the FPP and that a real Banach space X such that BX∗ is w∗ sequentially compact andX∗ has WORTH∗ has the wFPP.

1. Introduction

In 1988 Sims [1] introduced the notion of weak orthogonality (WORTH) and asked whetherspaces with WORTH have the weak fixed point property (wFPP). Since then several partialanswers have been given. For instance, in 1993 Garcıa-Falset [2] proved that if X is uniformlynonsquare and has WORTH then it has the wFPP, although Mazcunan Navarro in herdoctoral dissertation [3] showed that uniform nonsquareness is enough. In this work shealso showed that WORTH plus 2-UNC implies the wFPP. In both of these cases the space Xturns out to be reflexive. In 1994 Sims [4] himself proved that WORTH plus ε0-inquadrate inevery direction for some ε0 < 2 implies the wFPP and in 2003 Dalby [5] showed that if X∗ hasWORTH∗ and is ε0-inquadrate in every direction for some ε0 < 2, then X has the wFPP.

Recently in 2008 Cowell and Kalton [6] studied properties au and au∗ in a Banachspace X, where au coincides with WORTH if X is separable and au∗ in X coincides withWORTH∗ in X∗ if X is a separable Banach space. Among other things they proved thata real Banach space with au∗ embeds almost isometrically in a space with a shrinking 1-unconditional basis and observed that au and au∗ are equivalent if X is reflexive.

We proved, using property AMC shown by Garcıa-Falset and Llorens Fuster [7] toimply the wFPP, that spaces that (1 + δ) embed in a space with a 1-unconditional basis have


the wFPP. Combining this with Cowell and Kalton’s results we were able to show that areflexive real Banach space with WORTH and its dual both have FPP, giving a partial answerto Sims’ question. We also showed that a separable space X such that X∗ has WORTH∗ andBX∗ is w∗ sequentially compact has the wFPP.

2. Notations and Definitions

Let (X, ‖ · ‖X) be a real Banach space and K a closed nonempty bounded convex subset of X.

Definition 2.1. If x ∈ X, we define

R(x,K) = sup{‖x − z‖X : z ∈ K}. (2.1)

If x, y ∈ K the set of quasi-midpoints of x and y in K is given by

M(x, y

)={z ∈ K : max

{‖z − x‖X,

∥∥z − y∥∥X ≤ 12∥∥x − y∥∥X

}}. (2.2)

Definition 2.2. [X] is the quotient space l∞(X)/c0(X) endowed with the norm ‖z‖ =lim supn‖zn‖X where z is the equivalence class of (zn) in l∞(X), which we also will denote by[zn]. For x ∈ X we will also denote by x the equivalence class [(x, x, x, . . .)] in [X]. If K is asabove, let K = {[zn] ∈ [X] : zn ∈ K, n = 1, 2, . . .}. If Y is a Banach space and Tn : X → Y forn = 1, 2, . . . , we define [Tn] = T , T : [X] → [Y ] by

T([zn]) = [Tnzn]. (2.3)

If Tn = T for n = 1, 2, . . . we denote T by T .

It is known that K is also closed bounded and convex in [X] and that ‖T‖ =lim supn‖Tn‖.

Definition 2.3. Let S(N) be the set of strictly increasing sequences of natural numbers andK a nonempty bounded convex subset of a Banach space X. A sequence (xn) in K is calledequilateral in K, if for every β, γ ∈ S(N) such that β(n)/= γ(n) for every n ∈ N, the followingequality holds in [X]. If xβ = [xβ(n)] and xγ = [xγ(n)], then

∥∥xβ∥∥ =∥∥xγ∥∥ =

∥∥xβ − xγ∥∥ = D(xn) = diam(K), (2.4)

where D(xn) = lim supn(lim supm‖xn − xm‖X).

It is easy to see that if (xn) is equilateral in K, and β, γ ∈ S(N) are as above, then

∥∥xβ − xγ∥∥ = limn

∥∥xβ(n) − xγ(n)∥∥X = limn

∥∥xβ(n)∥∥X = limn

∥∥xγ(n)∥∥X. (2.5)

Now we define the property which interests us in this paper, it was given by Garcıa-Falset and Llorens Fuster in 1990 [7].


Definition 2.4. A bounded closed convex subset K of a Banach space (X, ‖ · ‖X) with 0 ∈ Khas the AMC property, if for every weakly null sequence (xn) which is equilateral in K, thereexist ρ ∈ (0, 1), x ∈ K, β, γ ∈ S(N) with β(n)/= γ(n) for every n ∈ N, such that the set

Mρ

((xn), β, γ

)=M

(xβ, xγ

) ∩ {[zn] : d([zn], K) ≤ ρdiam(K)}

(2.6)

is nonempty and R(x,Mρ((xn), β, γ)) < diam(K). X is said to have AMC if every weaklycompact nonempty subset K of X with 0 ∈ K has the AMC property.

3. Embeddings into Spaces with 1-Unconditional Basis and the wFPP

Lin in [8] showed that if X has an unconditional basis (en) with unconditional constant λ <(331/2−3)/2, thenX has the wFPP. Garcıa-Falset and Llorens Fuster proved that in fact underthese conditions X has the AMC property which in turn implies the wFPP. We will follow theproof of this closely to establish the next theorem.

Theorem 3.1. Let (X, ‖ · ‖X) be a Banach space and suppose that there exists a Banach space(Y, ‖ · ‖Y ) with a 1-unconditional basis (en) and a subspace Xδ of Y such that d(X,Xδ) < (1 + δ)where δ < (

√13 − 3)/2. Then X has AMC and thus the wFPP.

Proof. Let S : X → Xδ be an isomorphism with ‖S‖ ≤ 1 + δ and ‖S−1‖ ≤ 1. Let K ⊂ X be anonempty weakly compact convex subset of X with 0 ∈ K and diam(K) = 1. We will showthat K has the AMC property.

Let (xn) be a weakly null equilateral sequence in K and let Kδ = S(K). Then Kδ isweakly compact and (Sxn) is weakly null in Y . Hence there exists a sequence β ∈ S(N) andprojections with respect to the basis (en) in Y with

(a) Pn : Y → sp(emn, emn+1, . . . , ern) where mn ≤ rn < mn+1,

(b) limn‖Pny‖Y = 0 for all y ∈ Y ,

(c) limn‖Sxβ(n) − PnSxβ(n)‖Y = 0.

Let γ ∈ S(N) be given by γ(n) = β(n + 1). Then clearly β(n)/= γ(n) for every n ∈ N.Let P , Q : [Y ] → [Y ] be given by P = [Pn] and Q = [Pn+1] and let S : [X] → [Y ] beS = [(S, S, . . .)]. Recall that we will write S instead of S. By (a), (b), and (c) and since (xn) isequilateral we have that

(1) ‖Sxβ‖ ≤ 1 + δ, ‖Sxγ‖ ≤ 1 + δ and ‖Sxβ − Sxγ‖ ≤ 1 + δ,

(2) PSxβ = Sxβ, QSxγ = Sxγ ,

(3) QSxβ = 0, PSxγ = 0,

(4) for all y ∈ [Y ], P y = Qy = 0.

Therefore, since (en) is 1-unconditional

∥∥Sxβ + Sxγ∥∥ =∥∥∥QSxβ + PSxγ

∥∥∥ =∥∥∥QSxβ − PSxγ

∥∥∥ =∥∥Sxβ − Sxγ∥∥ ≤ 1 + δ. (3.1)


Thus ‖xβ+xγ‖ = ‖S−1S(xβ+xγ)‖ ≤ 1+δ. Since by hypothesis ‖xβ−xγ‖ ≤ 1, we obtain that (xβ+xγ)/2 ∈Mρ((xn), β, γ) if δ < 1 and ρ = (1+δ)/2. Next we will show thatR(0,Mρ((xn), β, γ)) <1.

To this effect let w = [wn] ∈ Mρ((xn), β, γ). Define u = (P + Q)Sw and r = (I − (P +Q))Sw. Then for every x ∈ K

2u = (u + r − Sx) + (u − r + Sx). (3.2)

By the unconditionality of (en) and since by (4) we have that PSx = QSx = 0,

‖u − r + Sx‖ =∥∥∥(P + Q

)(Sw + Sx) −

(I −

(P + Q

))(Sw − Sx)

∥∥∥=∥∥∥(P + Q

)(Sw + Sx) +

(I −

(P + Q

))(Sw − Sx)

∥∥∥ = ‖u + r − Sx‖.(3.3)

Therefore

2‖u‖ ≤ 2‖u + r − Sx‖ = 2‖Sw − Sx‖. (3.4)

Now let v ∈ K be such that ‖w − v‖ ≤ ρ. Such an element exists since w ∈ Mρ((xn, β, γ)).Recalling that ρ = (1 + δ)/2, we obtain that

2‖u‖ ≤ 2‖Sw − Sv‖ ≤ (1 + δ)2. (3.5)

Using again that w ∈Mρ((xn, β, γ)), we get

2‖w‖ ≤∥∥∥∥∥w +

xβ + xγ2

∥∥∥∥∥ +

∥∥∥∥∥w − xβ

2

∥∥∥∥∥ +

∥∥∥∥∥w − xγ

2

∥∥∥∥∥

≤∥∥∥∥∥w +

xβ + xγ2

∥∥∥∥∥ +12

(3.6)

or equivalently

∥∥∥∥∥w +xβ + xγ

2

∥∥∥∥∥ ≥ 2‖w‖ − 12. (3.7)

On the other hand, since by (2) PSxβ = Sxβ, we have

∥∥∥Sw + Sxβ − 2PSw∥∥∥ =

∥∥∥Sw + PSxβ − 2PSw∥∥∥ =

∥∥∥(I − P)Sw + P(Sxβ − Sw

)∥∥∥=∥∥∥(I − P)Sw − P(Sxβ − Sw)∥∥∥ =

∥∥∥Sw − PSxβ∥∥∥ =

∥∥Sw − Sxβ∥∥

≤ (1 + δ)∥∥w − xβ∥∥ ≤ 1 + δ

2.

(3.8)


Similarly

∥∥∥Sw + Sxγ − 2QSw∥∥∥ ≤ 1 + δ

2. (3.9)

By (3.8) and (3.9), and (3.5) we obtain

2

∥∥∥∥∥Sw +Sxβ + Sxγ

2

∥∥∥∥∥ ≤ (1 + δ) + 2∥∥∥(P + Q

)Sw

∥∥∥ = (1 + δ) + 2‖u‖

≤ (1 + δ) + (1 + δ)2.

(3.10)

Hence

∥∥∥∥∥w +xβ + xδ

2

∥∥∥∥∥ ≤(1 + δ)(2 + δ)

2. (3.11)

Finally, from (3.7) and (3.11) we have 2‖w‖ − 1/2 ≤ (1 + δ)(2 + δ)/2 and

‖w‖ ≤ δ2 + 3δ + 3

4. (3.12)

Therefore, if δ < (√

13 − 3)/2 we conclude that R(0,Mρ((xn), β, γ)) < 1 and thus X has theAMC property.

Remark 3.2. It is evident that if the space Y has a (1 + λ) unconditional basis, if λ is smallenough, the above result remains true for some δ.

4. Some Consequences

There has always been the conjecture that a space with property WORTH has the wFPP. Weshow here that this is correct as long as X is reflexive. We also show that property WORTH∗

in X∗ implies the wFPP in Banach spaces X so that BX∗ is w∗ sequentially compact and thatWORTH together with WABS implies the wFPP as well. All these results are consequences ofsome theorems by Cowell and Kalton [6]. First we need to recall some definitions.

Definition 4.1. A Banach space X has the WORTH property if for every weakly null sequence(xn) ⊂ X and every x ∈ X, the following equality holds:

limn(‖xn − x‖ − ‖xn + x‖) = 0. (4.1)

This definition was given by Sims in [1]. The next definition was stated by Dalby [5].


Definition 4.2. A Banach space X∗ has the WORTH∗ property if for every weak∗ null sequence(x∗n) ⊂ X∗ and every x∗ ∈ X∗, the following equality holds:

limn(‖x∗n − x∗‖ − ‖x∗n + x∗‖) = 0. (4.2)

If X is separable and X∗ has WORTH∗, this coincides with the property au∗ defined in[6].

Definition 4.3. A Banach space X has the Weak Alternating Banach-Saks (WABS) property ifevery bounded sequence (xn) in X has a convex block sequence (yn) such that

limn

supr1<r2<···<rn

∥∥∥∥∥∥1n

n∑j=1

(−1)jyrj

∥∥∥∥∥∥ = 0. (4.3)

Cowell and Kalton in [6] proved the following three results.

Theorem 4.4. If X is a separable real Banach space, then X∗ has the property WORTH∗ if and only iffor any δ > 0 there is a Banach space Y with a shrinking 1-unconditional basis and a subspace Xδ ofY such that d(X,Xδ) < 1 + δ.

Dalby [5] observed that property WORTH∗ in a space X∗ implies property WORTHin X and it follows that if X is reflexive, then both properties are equivalent. From this andanother theorem we are not going to mention here, Cowell and Kalton obtained the nexttheorem.

Theorem 4.5. If X is a separable real reflexive space, then X has property WORTH if and only if forany δ > 0 there is a reflexive Banach space Y with a 1-unconditional basis and a subspace Xδ of Ysuch that d(X,Xδ) < 1 + δ.

The third result we are going to use is as follows.

Theorem 4.6. If X is a separable real Banach space, then X has both the properties WORTH andWABS if and only if for any δ > 0 there is a Banach space Y with a shrinking 1-unconditional basisand a subspace Xδ of Y such that d(X,Xδ) < 1 + δ.

From this and our previous work it follows directly the following:

Theorem 4.7. If X is a real separable space such that either

(I) X∗ has property WORTH∗,

(II) X is reflexive and has property WORTH, or

(III) X has both the properties WORTH and WABS,

then X has the property AMC and thus the wFPP.

It is known that reflexivity implies WABS, and thus (II) implies (III), but we want toinclude (II) in order to deduce the next corollary. Properties WORTH and WABS are inheritedby subspaces, and if X∗ has property WORTH∗ and BX∗ is w∗ sequentially compact, then thedual of any subspace of X also has this property. Hence we have the following result.


Corollary 4.8. Let X be a real Banach space.

(1) If X is reflexive and has property WORTH, then X and X∗ both have the FPP.

(2) If X has properties WORTH and WABS, then X has the wFPP.

(3) If X∗ has WORTH∗ and BX∗ is w∗ sequentially compact, then X has the wFPP.

Proof. If X is a Banach space that satisfies (1), (2), or (3), every separable subspace has thewFPP and hence, since the wFPP is separably determined, X has the wFPP. If X is separableand reflexive and has property WORTH, then it has property WORTH∗ as well and thisimplies by definition that X∗ also has property WORTH. Therefore both have the FPP andhence the result follows for nonseparable reflexive spaces.

Acknowledgment

This work is partially supported by SEP-CONACYT Grant 102380. It is dedicated to W. A.Kirk.

References

[1] B. Sims, “Orthogonality and fixed points of nonexpansive maps,” in Workshop/Miniconference onFunctional Analysis and Optimization (Canberra, 1988), vol. 20 of Proceedings of the Centre for MathematicalAnalysis, Australian National University, pp. 178–186, The Australian National University, Canberra,Australia, 1988.

[2] J. Garcıa-Falset, “The fixed point property in Banach spaces whose characteristic of uniform convexityis less than 2,” Journal of the Australian Mathematical Society. Series A, vol. 54, no. 2, pp. 169–173, 1993.

[3] E. M. Mazcunan Navarro, Geometrıa de los espacios de Banach en teorıa metrica del punto fijo, Tesis doctoral,Universitat de Valencia, Valencia, Spain, 2003.

[4] B. Sims, “A class of spaces with weak normal structure,” Bulletin of the Australian Mathematical Society,vol. 49, no. 3, pp. 523–528, 1994.

[5] T. Dalby, “The effect of the dual on a Banach space and the weak fixed point property,” Bulletin of theAustralian Mathematical Society, vol. 67, no. 2, pp. 177–185, 2003.

[6] S. R. Cowell and N. J. Kalton, “Asymptotic unconditionality,” http://arxiv.org/abs/0809.2294.[7] J. Garcıa-Falset and E. Llorens Fuster, “A geometric property of Banach spaces related to the fixed point

property,” Journal of Mathematical Analysis and Applications, vol. 172, no. 1, pp. 39–52, 1993.[8] P.-K. Lin, “Unconditional bases and fixed points of nonexpansive mappings,” Pacific Journal of

Mathematics, vol. 116, no. 1, pp. 69–76, 1985.


Research ArticleNonexpansive Matrices with Applicationsto Solutions of Linear Systems by FixedPoint Iterations

Teck-Cheong Lim

Department of Mathematical Sciences, George Mason University, 4400, University Drive,Fairfax, VA 22030, USA

Correspondence should be addressed to Teck-Cheong Lim, [email protected]

Received 28 August 2009; Accepted 19 October 2009


Copyright q 2010 Teck-Cheong Lim. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We characterize (i) matrices which are nonexpansive with respect to some matrix norms, and (ii)matrices whose average iterates approach zero or are bounded. Then we apply these results toiterative solutions of a system of linear equations.

Throughout this paper, R will denote the set of real numbers, C the set of complex numbers,and Mn the complex vector space of complex n × n matrices. A function ‖ · ‖ : Mn → R is amatrix norm if for all A,B ∈Mn, it satisfies the following five axioms:

(1) ‖A‖ ≥ 0;

(2) ‖A‖ = 0 if and only if A = 0;

(3) ‖cA‖ = |c|‖A‖ for all complex scalars c;

(4) ‖A + B‖ ≤ ‖A‖ + ‖B‖;(5) ‖AB‖ ≤ ‖A‖ ‖B‖.

Let | · | be a norm on Cn. Define ‖ · ‖ on Mn by

‖A‖ = max|x|=1|Ax|. (1)

This norm on Mn is a matrix norm, called the matrix norm induced by | · |. A matrix norm onMn is called an induced matrix norm if it is induced by some norm on C

n. If ‖ · ‖1 is a matrixnorm on Mn, there exists an induced matrix norm ‖ · ‖2 on Mn such that ‖A‖2 ≤ ‖A‖1 for all


A ∈ Mn (cf. [1, page 297]). Indeed one can take ‖ · ‖2 to be the matrix norm induced by thenorm | · | on C

n defined by

|x| = ‖C(x)‖1, (2)

where C(x) is the matrix in Mn whose columns are all equal to x. For A ∈Mn, ρ(A) denotesthe spectral radius of A.

Let | · | be a norm in Cn. A matrix A ∈ Mn is a contraction relative to | · | if it is a

contraction as a transformation from Cn into C

n; that is, there exists 0 ≤ λ < 1 such that

∣∣Ax −Ay∣∣ ≤ λ∣∣x − y∣∣, x, y ∈ Cn. (3)

Evidently this means that for the matrix norm ‖ · ‖ induced by | · |, ‖A‖ < 1. The followingtheorem is well known (cf. [1, Sections 5.6.9–5.6.12]).

Theorem 1. For a matrix A ∈Mn, the following are equivalent:

(a) A is a contraction relative to a norm in Cn;

(b) ‖A‖ < 1 for some induced matrix norm ‖ · ‖;

(c) ‖A‖ < 1 for some matrix norm ‖ · ‖;

(d) limk→∞Ak = 0;

(e) ρ(A) < 1.

That (b) follows from (c) is a consequence of the previous remark about an induced matrixnorm being less than a matrix norm. Since all norms on Mn are equivalent, the limit in (d)can be relative to any norm on Mn, so that (d) is equivalent to all the entries of Ak convergeto zero as k → ∞, which in turn is equivalent to limk→∞Akx = 0 for all x ∈ C

n.In this paper, we first characterize matrices inMn that are nonexpansive relative to some

norm | · | on Cn, that is,

∣∣Ax −Ay∣∣ ≤ ∣∣x − y∣∣, x, y ∈ Cn. (4)

Then we characterize those A ∈Mn such that

Ak =1k

(I +A +A2 + · · · +Ak−1

)(5)

converges to zero as k → ∞, and those that {Ak : k = 0, 1, 2, . . .} is bounded.Finally we apply our theory to approximation of solution of Ax = b using iterative

methods (fixed point iteration methods).


Theorem 2. For a matrix A ∈Mn, the following are equivalent:

(a) A is nonexpansive relative to some norm on Cn;

(b) ‖A‖ ≤ 1 for some induced matrix norm ‖ · ‖;(c) ‖A‖ ≤ 1 for some matrix norm ‖ · ‖;(d) {Ak : k = 0, 1, 2, . . .} is bounded;(e) ρ(A) ≤ 1, and for any eigenvalue λ of A with |λ| = 1, the geometric multiplicity is equal to

the algebraic multiplicity.

Proof. As in the previous theorem, (a), (b), and (c) are equivalent. Assume that (b) holds. Letthe norm ‖ · ‖ be induced by a vector norm | · | of C

n. Then

∣∣∣Ak(x)∣∣∣ ≤∥∥∥Ak

∥∥∥ |x| ≤ ‖A‖k|x| ≤ |x|, k = 0, 1, 2, . . . , (6)

proving that Ak(x) is bounded in norm | · | for every x ∈ Cn. Taking x = ei, we see that the set

of all columns ofAk, k = 0, 1, 2, . . . , is bounded. This proves thatAk, k = 0, 1, 2, . . . , is boundedin maximum column sum matrix norm ([1, page 294]), and hence in any norm in Mn. Notethat the last part of the proof also follows from the Uniform Boundedness Principle (see, e.g.,[2, Corollary 21, page 66])

Now we prove that (d) implies (e). Suppose that A has an eigenvalue λ with λ > 1.Let x be an eigenvector corresponding to λ. Then

∥∥∥Akx∥∥∥ = |λ|k‖x‖ −→ ∞ (7)

as k → ∞, where ‖ · ‖ is any vector norm of Cn. This contradicts (d). Hence |λ| ≤ 1. Now

suppose that λ is an eigenvalue with |λ| = 1 and the Jordan block corresponding to λ is notdiagonal. Then there exist nonzero vectors v1, v2 such that Av1 = λv1, A(v2) = v1 + λv2. Letu = v1 + v2. Then

Aku = λk−1(λ + k)v1 + λkv2, (8)

and ‖Ak(u)‖ ≥ k‖v1‖ − ‖v1‖ − ‖v2‖. It follows that Aku, k = 0, 1, 2, . . . , is unbounded,contradicting (d). Hence (d) implies (e).

Lastly we prove that (e) implies (c). Assume that (e) holds. A is similar to its Jordancanonical form J whose nonzero off-diagonal entries can be made arbitrarily small bysimilarity ([1, page 128]). Since the Jordan block for each eigenvalue with modulus 1 isdiagonal, we see that there is an invertible matrix S such that the l1-sum of each row ofSAS−1 is less than or equal to 1, that is, ‖SAS−1‖∞ ≤ 1, where ‖ · ‖∞ is the maximum row summatrix norm ([1, page 295]). Define a matrix norm ‖ · ‖ by ‖M‖ = ‖SMS−1‖∞. Then we have‖A‖ ≤ 1.

Let λ be an eigenvalue of a matrix A ∈Mn. The index of λ, denoted by index(λ) is thesmallest value of k for which rank(A − λI)k = rank(A − λI)k+1 ([1, pages 148 and 131]). Thuscondition (e) above can be restated as ρ(A) ≤ 1, and for any eigenvalue λ of A with |λ| = 1,index(λ) = 1.


Let A ∈Mn. Consider

Ak =1k

(I +A + · · · +Ak−1

). (9)

We call Ak the k-average of A. As with Ak, we have Akx → 0 for every x if and only ifAk → 0 in Mn, and that Akx is bounded for every x if and only if Ak is bounded in Mn. Wehave the following theorem.

Theorem 3. Let A ∈Mn. Then

(a) Ak, k = 1, 2, . . . , converges to 0 if and only if ‖A‖ ≤ 1 for some matrix norm ‖ · ‖ and that1 is not an eigenvalue of A,

(b) Ak, k = 1, 2, . . . , is bounded if and only if ρ(A) ≤ 1, index(λ) ≤ 2 for every eigenvalue λwith |λ| = 1 and that index(1) = 1 if 1 is an eigenvalue of A.

Proof. First we prove the sufficiency part of (a). Let x be a vector in Cn. Let

yk =1k

(I +A + · · · +Ak−1

)(x). (10)

By Theorem 2 for any eigenvalues λ of A either |λ| < 1 or |λ| = 1 and index(λ) = 1.If A is written in its Jordan canonical form A = SJS−1, then the k-average of A is

SJ ′S−1, where J ′ is the k-average of J . J ′ is in turn composed of the k-average of each of itsJordan blocks. For a Jordan block of J of the form

⎛⎜⎜⎜⎜⎜⎜⎝

λ 1λ 1· ·· 1λ

⎞⎟⎟⎟⎟⎟⎟⎠, (11)

|λ| must be less than 1. Its k-average has constant diagonal and upper diagonals. Let Dj bethe constat value of its jth upper diagonal (D0 being the diagonal) and let Sj = kDj . Then(C(m,n) = 0 for n > m)

S0 =1 − λk1 − λ ,

Sj = C(j, j)+ C(j + 1, j

)λ + · · · + C(k − 1, j

)λk−1−j , j = 1, 2, . . . , n − 1.

(12)

Using the relation C(m + 1, j) − C(m, j) = C(m, j − 1), we obtain

Sj − λSj = Sj−1 − λk−jC(k, j). (13)

Thus, we have S0 → 1/(1 − λ) as k → ∞. By induction, using (13) above and the fact thatλk−jC(k, j) → 0 as k → ∞, we obtain Sj → 1/(1 − λ)j+1 as k → ∞. Therefore Dj = Sj/k =O(1/k) as k → ∞.


If the Jordan block is diagonal of constant value λ, then λ/= 1, |λ| ≤ 1 and the k-averageof the block is diagonal of constant value (1 − λk)/k(1 − λ) = O(1/k).

We conclude that ‖Ak‖ = O(1/k) and hence ‖yk‖ ≤ ‖Ak‖‖x‖ = O(1/k) as k → ∞.Now we prove the necessity part of (a). If 1 is an eigenvalue of A and x is a

corresponding eigenvector, then Akx = x /= 0 for every k and of course Bkx fails to convergeto 0. If λ is an eigenvalue of A with |λ| > 1 and x is a corresponding eigenvector, then

‖Akx‖ =∣∣∣∣∣λk − 1k(λ − 1)

∣∣∣∣∣ ‖x‖ ≥|λ|k − 1k|λ − 1| ‖x‖. (14)

which approaches to ∞ as k → ∞. If λ is an eigenvalue of A with |λ| = 1, λ /= 1, andindex(λ) ≥ 2, then there exist nonzero vectors v1, v2 such that A(v1) = λv1, A(v2) = v1 + λv2.Then by using the identity

1 + 2λ + 3λ2 + · · · + (k − 1)λk−2 =1 − λk−1

(1 − λ)2− (k − 1)

λk−1

1 − λ (15)

we get

Ak(v2) =

(1 − λk−1

k(1 − λ)2−(

1 − 1k

)λk−1

1 − λ

)v1 +

1 − λkk(1 − λ)v2. (16)

It follows that limk→∞Ak(v2) does not exist. This completes the proof of part (a).Suppose that A satisfies the conditions in (b) and that A = SJS−1 is the Jordan

canonical form of A. Let λ be an eigenvalue of A and let v be a column vector of Scorresponding to λ. If |λ| < 1, then the restriction B of A to the subspace spanned byv,Av,A2v, . . . is a contraction, and we have ‖Akv‖ = ‖Bkv‖ ≤ ‖v‖. If |λ| = 1, and λ/= 1,then by conditions in (b) either Av = λv, or there exist v1, v2 with v = v2 such thatA(v1) = λv1, A(v2) = v1 + λv2. In the former case, we have ‖Ak‖ ≤ ‖v‖ and in the latter case,we see from (16) that Ak(v) = Ak(v2) is bounded. Finally if λ = 1 then since index(1) = 1, wehave Av = v and hence Akv = v. In all cases, we proved that Akv, k = 0, 1, 2, . . . , is bounded.Since column vectors of S form a basis for C

n, the sufficiency part of (b) follows.Now we prove the necessity part of (b). If A has an eigenvalue λ with |λ| > 1 and

eigenvector v, then as shown above Ak(v) → ∞ as k → ∞. If A has 1 as an eigenvalue andindex(1) ≥ 2, then there exist nonzero vectors v1, v2 such that Av1 = v1 and Av2 = v1 + v2.ThenAk(v2) = ((k−1)/2)+v2 which is unbounded. If λ is an eigenvalue ofAwith |λ| = 1, λ /= 1and index(λ) ≥ 3, then there exist nonzero vectors v1, v2 and v3 such that Av1 = λv1, A(v2) =v1 +λv2 and A(v3) = v2 +λv3. By expanding Aj(v3), j = 0, 1, 2, . . . , k − 1 and using the identity

k−1∑j=2

C(j, 2)λj−2 =

1

(1 − λ)2

(1 − λk−2

1 − λ +12(k − 2)λk−2((k − 1)λ − (k + 1))

), (17)


we obtain

Ak(v3) =1

(1 − λ)2

(1 − λk−2

k(1 − λ) +12(k − 2)λk−2

(k − 1k

λ − k + 1k

))v1

+

(1 − λk−1

k(1 − λ)2−(

1 − 1k

)λk−1

1 − λ

)v2 +

1 − λkk(1 − λ)v3

(18)

which approaches to∞ as k → ∞. This completes the proof.

We now consider applications of preceding theorems to approximation of solution ofa linear system Ax = b, where A ∈Mn and b a given vector in C

n. Let Q be a given invertiblematrix in Mn. x is a solution of Ax = b if and only if x is a fixed point of the mapping Tdefined by

Tx =(I −Q−1A

)x +Q−1b. (19)

T is a contraction if and only if I − Q−1A is. In this case, by the well known ContractionMapping Theorem, given any initial vector x0, the sequence of iterates xk = Tkx0, k =0, 1, 2, . . . , converges to the unique solution of Ax = b. In practice, given x0, each successivexk is obtained from xk−1 by solving the equation

Q(xk) = (Q −A)xk−1 + b. (20)

The classical methods of Richardson, Jacobi, and Gauss-Seidel (see, e.g., [3]) have Q = I,D,and L respectively, where I is the identity matrix, D the diagonal matrix containing thediagonal of A, and L the lower triangular matrix containing the lower triangular portionof A. Thus by Theorem 1 we have the following known theorem.

Theorem 4. Let A,Q ∈ Mn, with Q invertible. Let b, x0 ∈ Cn. If ρ(I − Q−1A) < 1, then A is

invertible and the sequence xk, k = 1, 2, . . . , defined recursively by

Q(xk) = (Q −A)xk−1 + b (21)

converges to the unique solution of Ax = b.

Theorem 4 fails if ρ(I −Q−1A) = 1, For a simple 2 × 2 example, let Q = I, b = 0, A = 2Iand x0 any nonzero vector.

We need the following lemma in the proof of the next two theorems. For a matrixA ∈Mn, we will denote R(A) and N(A) the range and the null space of A respectively.

Lemma 5. Let A be a singular matrix in Mn such that the geometric multiplicity and the algebraicmultiplicity of the eigenvalue 0 are equal, that is, index(0) = 1. Then there is a unique projectionPA whose range is the range of A and whose null space is the null space of A, or equivalently, C

n =R(A)⊕N(A). Moreover,A restricted to R(A) is an invertible transformation from R(A) onto R(A).


Proof. If A = SJS−1 is a Jordan canonical form of A where the eigenvalues 0 appear at the endportion of the diagonal of J , then the matrix

PA = S

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1

··

1

0

··

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

S−1 (22)

is the required projection. Obviously A maps R(A) into R(A). If z ∈ R(A) and Az = 0, thenz ∈N(A) ∩ R(A) = {0} and so z = 0. This proves that A is invertible on R(A).

Remark 6. Under the assumptions of Lemma 5, we will call the component of a vector c inN(A) the projection of c on N(A) along R(A). Note that by definition of index, the conditionin the lemma is equivalent to N(A2) =N(A).

Theorem 7. Let A be a matrix inMn and b a vector in Cn. LetQ be an invertible matrix inMn and

let B = I − Q−1A. Assume that ρ(B) ≤ 1 and that index(λ) = 1 for every eigenvalue λ of B withmodulus 1, that is, B is nonexpansive relative to a matrix norm. Starting with an initial vector x0 inCn define xk recursively by

Q(xk) = (Q −A)xk−1 + b (23)

for k = 1, 2, . . . . Let

yk =x0 + x1 + · · · + xk−1

k. (24)

If Ax = b is consistent, that is, has a solution, then yk, k = 1, 2, . . . , converge to a solution vector zwith rate of convergence ‖yk − z‖ = O(1/k). IfAx = b is inconsistent, then limk‖xk‖ = limk‖yk‖ =∞. More precisely, limkxk/k = c′ and limkyk/k = c′/2, where c = Q−1b and c′ is the projection of conN(A) =N(Q−1A) along R(Q−1A).

Proof. First we assume that A is invertible so that I − B = Q−1A is also invertible. Let T be themapping defined by Tx = Bx+c. Then Tkx = Bkx+c+Bc+· · ·+Bk−1c. Let s = c+Bc+· · ·+Bk−1c.Then s − Bs = c − Bkc and hence s = (I − B)−1c − (I − B)−1Bkc = (I − B)−1c − Bk(I − B)−1c. Letz = (I − B)−1c = A−1b. z is the unique solution of Ax = b and

Tkx = Bkx + z − Bkz = Bk(x − z) + z. (25)


Since the sequence xk in the theorem is Tkx0, we have

yk =1k

(I + B + · · · + Bk−1

)(x0 − z) + z = Bk(x0 − z) + z. (26)

Since I − B is invertible, 1 is not an eigenvalue of B, and by Theorem 3 part (a) ‖yk − z‖ =‖Bk(x0−z‖ → 0 as k → ∞. Moreover, from the proof of the same theorem, ‖yk−z‖ = O(1/k).

Next we consider the case when A is not invertible. Since Q is invertible, we haveR(Q−1A) = Q−1(R(A)) and N(Q−1A) = N(A). The index of the eigenvalue 0 of Q−1A is theindex of eigenvalue 1 of B = I −Q−1A. Thus by Lemma 5, C

n = Q−1(R(A)) ⊕N(A). For everyvector v ∈ C

n, let v(r) and v(n) denote the component of v in the subspace Q−1(R(A)) andN(A), respectively.

Assume that Ax = b is consistent, that is, b ∈ R(A). Then c ∈ R(Q−1A). By Lemma 5,the restriction of Q−1A on its range is invertible, so there exists a unique z′ in R(Q−1A) suchthat Q−1Az′ = c, or equivalently, (I − B)z′ = c. For any vector x, we have

Tkx = Bkx + c + Bc + · · · + Bk−1c

= BK(x(r) + x(n)

)+(I + B + · · · + Bk−1

)(I − B)z′

= Bk(x(r))+ x(n) + z′ − Bk(z′)

= Bk(x(r) − z′

)+ x(n) + z′.

(27)

Since B maps R(Q−1A) into R(Q−1A) and I −B = Q−1A restricted to R(Q−1A) is invertible, wecan apply the preceding proof and conclude that the sequence yk as defined before convergesto z = x(n)

0 + z′ and ‖yk − z‖ = O(1/k). Now Az = A(x(n)0 ) +A(z′) = A(z′) = Qc = b, showing

that z is a solution of Ax = b.Assume now that b /∈R(A), that is, Ax = b is inconsistent. Then c /∈R(Q−1A) and c =

c(r) + c(n) with c(n) /= 0. As in the preceding case there exists a unique z′ ∈ R(Q−1A) such that(I − B)z′ = c(r). Note that for all y ∈ N(A), B(y) = (I −Q−1A)(y) = y. Thus for any vector xand any positive integer j

xj = Tjx

= Bjx + c + Bc + · · · + Bj−1c

= Bj(x(r) + x(n)

)+(I + B + · · · + Bj−1

)(I − B)z′ + jc(n)

= Bj(x(r))+ x(n) + z′ − Bj(z′) + jc(n)

= Bj(

x(r) − z′)+ x(n) + z′ + jc(n),

yk =1k

(x + Tx + · · · + Tk−1x

)

= Bk(x(r) − z′

)+ x(n) + z′ +

k − 12

c(n),

(28)


where Bk = (I +B+ · · ·+Bk−1). As in the preceding case, Bk(x(r) −z′), k = 0, 1, 2, . . . is boundedand Bk(x(r)−z′), k = 1, 2, . . . , converges to 0. Thus limk→∞(xk/k) = c(n) and limk→∞(yk/k) =c(n)/2, and hence limk→∞‖xk‖ = limk→∞‖yk‖ =∞. This completes the proof.

Next we consider another kind of iteration in which the nonlinear case was consideredin Ishikawa [4]. Note that the type of mappings in this case is slightly weaker thannonexpansivity (see condition (c) in the next lemma).

Lemma 8. Let B be an n × n matrix. The following are equivalent:

(a) for every 0 < μ < 1, there exists a matrix norm ‖ · ‖μ such that ‖μI + (1 − μ)B‖μ ≤ 1,

(b) for every 0 < μ < 1, there exists an induced matrix norm ‖·‖μ such that ‖μI+(1−μ)B‖μ ≤ 1,

(c) ρ(B) ≤ 1 and index(1) = 1 if 1 is an eigenvalue of B.

Proof. As in the proof of Theorem 2, (a) and (b) are equivalent. For 0 < μ < 1, denote μI +(1 − μ)B by B(μ). Suppose now that (a) holds. Let λ be an eigenvalue of B. Then μ + (1 −μ)λ is an eigenvalue of B(μ). By Theorem 2 |μ + (1 − μ)λ| ≤ 1 for every 0 < μ < 1 andhence |λ| ≤ 1. If 1 is an eigenvalue of B, then it is also an eigenvalue of B(μ). By Theorem 2,the index of 1, as an eigenvalue of B(μ), is 1. Since obviously B and B(μ) have the sameeigenvectors corresponding to the eigenvalue 1, the index of 1, as an eigenvalue of B, is also1. This proves (c).

Now assume (c) holds. Since |μ + (1 − μ)λ| < 1 for |λ| = 1, λ /= 1, every eigenvalue ofB(μ), except possibly for 1, has modulus less than 1. Reasoning as above, if 1 is an eigenvalueof B(μ), then its index is 1. Therefore by Theorem 2, (a) holds. This completes the proof.

Theorem 9. Let A ∈ Mn and b ∈ Cn. Let Q be an invertible matrix in Mn, and B = I − Q−1A.

Suppose ρ(B) ≤ 1 and that index(1) = 1 if 1 is an eigenvalue of B. Let 0 < μ < 1 be fixed. Startingwith an initial vector x0, define xk, yk, k = 0, 1, 2, . . . , recursively by

y0 = x0,

Q(xk) = (Q −A)(yk−1

)+ b,

yk = μyk−1 +(1 − μ)xk.

(29)

If Ax = b is consistent, then yk, k = 0, 1, 2, . . . , converges to a solution vector z of Ax = b with rateof convergence given by

∥∥yk − z∥∥ = o(ζk), (30)

where ζ is any number satisfying

max{∣∣μ +

(1 − μ)λ∣∣ : λ an eigenvalue of B, λ/= 1

}< ζ < 1. (31)


If Ax = b is inconsistent, then limk→∞‖yk‖ =∞; more precisely,

limk→∞

ykk

=(1 − μ)c(n), (32)

where c(n) is the projection of c onN(A) along R(Q−1A).

Proof. Let c = Q−1b, B1 = μI + (1 − μ)B = I − (1 − μ)Q−1A, and Tx = B1x + (1 − μ)c. Thenyk = Tk(x0).

First we assume that A is invertible. Then I − B1 = (1 − μ)Q−1A is invertible and 1 isnot an eigenvalue of B1; thus ρ(B1) < 1. Let z = (1 − μ)(I − B1)

−1c = A−1b. We have

yk = Tkx0

= Bk1x0 +(1 − μ)(c + B1c + · · · + Bk−1

1 c)

= Bk1x0 +(1 − μ) 1

1 − μ(I + B1 + · · · + Bk−1

1

)(I − B1)z

= Bk1 (x0 − z) + z.

(33)

By a well known theorem (see, e.g. [1]), ‖yk − z‖ = o(ζk) for every ζ > ρ(B1).Assume now that A is not invertible and b ∈ R(A). Then c is in the range of Q−1A.

Since B = I − Q−1A satisfies the condition in Lemma 8, Q−1A satisfies the condition inLemma 5. Thus the restriction ofQ−1A on its range is invertible and there exists z′ in R(Q−1A)such that Q−1Az′ = c, or equivalently, (I − B1)z′ = (1 − μ)c. For any vector x = x0, wehave

yk = Tk(x)

= Bk1x +(1 − μ)(c + B1c + · · · + Bk−1

1 c)

= Bk1(x(r) + x(n)

)+(I + B1 + · · · + Bk−1

1

)(I − B1)z′

= Bk1(x(r))+ x(n) + z′ − Bk1

(z′)

= Bk1(x(r) − z′

)+ x(n) + z′.

(34)

Since B1 maps R(Q−1A) into R(Q−1A) and I − B = Q−1A restricted to R(Q−1A) is invertible,we can apply the preceding proof and conclude that the sequence yk, k = 0, 1, 2, . . . convergesto z = x(n) + z′ and ‖yk − z‖ = o(ζk). z solves Ax = b since Az = A(x(n)) + A(z′) = A(z′) =Qc = b.


Assume lastly that b /∈R(A), that is, Ax = b is inconsistent. Then c /∈R(Q−1A) andc = c(r) + c(n) with c(n) /= 0. As before there exists z′ ∈ R(Q−1A) such that (I −b1)z′ = (1−μ)c(r).Note that B1(p) = p for p ∈N(A). Then

yk = Tk(x)

= Bk1x +(1 − μ)(c + B1c + · · · + Bk−1

1 c)

= Bk1(x(r) + x(n)

)+(I + B1 + · · · + Bk−1

1

)(I − B1)z′ + k

(1 − μ)c(n)

= Bk1(x(r) − z′

)+ x(n) + z′ + k

(1 − μ)c(n).

(35)

Since Bk1 (x(r) − z′), k = 0, 1, 2, . . . , converges to 0, we have

limk→∞

ykk

=(1 − μ)c(n), (36)

and hence limk→∞‖yk‖ =∞. This completes the proof.

By taking Q = I and considering only nonexpansive matrices in Theorems 7 and 9, weobtain the following corollary.

Corollary 10. Let A be an n × n matrix such that ‖I −A‖ ≤ 1 for some matrix norm ‖ · ‖. Let b be avector in C

n. Then:(a) starting with an initial vector x0 in C

n define xk recursively as follows:

xk = (I −A)(xk−1) + b (37)

for k = 1, 2, . . . . Let

yk =x0 + x1 + · · · + xk−1

k(38)

for k = 1, 2, . . . . If Ax = b is consistent, then yk, k = 1, 2, . . . , converges to a solution vector z withrate of convergence given by

∥∥yk − z∥∥ = O(

1k

). (39)

If Ax = b is inconsistent, then limk→∞‖xk‖ = limk→∞‖yk‖ =∞.(b) let 0 < μ < 1 be a fixed number. Starting with an initial vector x0, let

y0 = x0,

xk = (I −A)(yk−1

)+ b,

yk = μyk−1 +(1 − μ)xk.

(40)



∥∥yk − z∥∥ = o(ζk)

(41)


max{∣∣μ +


}< ζ < 1. (42)

If Ax = b is inconsistent, then limk→∞‖yk‖ =∞.

Remark 11. If in the previous corollary, ‖I −A‖ < 1, and μ = 0 in part (b), the sequence yk = xkconverges to a solution. This is the Richardson method, see for example, [3]. Even in this case,our method in part (b) may yield a better approximation. For example if

A =

(1 0.9

−0.9 1

), (43)

b = 0, and x0 = e1, then the nth iterate in the Richardson method is 0.9n away from thesolution 0, while the nth iterate using the method in the corollary part (b) with μ = 1/2 is lessthan (0.5)n/2.

An n × n matrix A = (aij) is called diagonally dominant if

|aii| ≥n∑

j=1,j /= i

∣∣aij∣∣ (44)

for all i = 1, . . . , n. If A is diagonally dominant with aii /= 0 for every i and if Q = D or L, whereD is the diagonal matrix containing the diagonal of A, and L the lower triangular matrixcontaining the lower triangular entries of A, then it is easy to prove that ‖I − Q−1A‖∞ ≤1 where ‖ · ‖∞ denotes the maximum row sum matrix norm; see, for example, [1, 3]. Thefollowing follows from Theorems 7 and 9.

Corollary 12. LetA be a diagonally dominant n×nmatrix with aii /= 0 for all i = 1, . . . , n. LetQ = Dor L, where D is the diagonal matrix containing the diagonal of A, and L the lower triangular matrixcontaining the lower triangular entries of A. Let b be a vector in C

n. Then:(a) starting with an initial vector x0 in C

n define xk recursively as follows:

Q(xk) = (Q −A)(xk−1) + b (45)

for k = 1, 2, . . . . Let

yk =x0 + x1 + · · · + xk−1

k(46)


for k = 1, 2, . . . . If Ax = b is consistent, then yk, k = 1, 2, . . . converges to a solution vector z withrate of convergence given by

∥∥yk − z∥∥ = O(

1k

). (47)

If Ax = b is inconsistent, then limk→∞‖xk‖ = limk→∞‖yk‖ =∞.(b) Let 0 < μ < 1 be a fixed number. Starting with an initial vector x0, let

y0 = x0,

Q(xk) = (Q −A)(yk−1

)+ b,

yk = μyk−1 +(1 − μ)xk.

(48)


∥∥yk − z∥∥ = o(ζk), (49)


max{∣∣μ +


}< ζ < 1. (50)

If Ax = b is inconsistent, then limk→∞‖yk‖ =∞.

References

[1] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.[2] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience Publishers, New York, NY, USA,

1957.[3] D. Kincaid and W. Cheney, Numerical Analysis, Brooks/Cole, Pacific Grove, Calif, USA, 1991.[4] S. Ishikawa, “Fixed points and iteration of a nonexpansive mapping in a Banach space,” Proceedings of

the American Mathematical Society, vol. 59, no. 1, pp. 65–71, 1976.


Research ArticleFixed Points for Pseudocontractive Mappings onUnbounded Domains

Jesus Garcıa-Falset and E. Llorens-Fuster

Departamento de Analisis Matematico, Facultad de Matematicas, Universitat de Valencia,46100 Burjassot, Spain

Correspondence should be addressed to Jesus Garcıa-Falset, [email protected]

Received 4 September 2009; Accepted 14 October 2009


Copyright q 2010 J. Garcıa-Falset and E. Llorens-Fuster. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We give some fixed point results for pseudocontractive mappings on nonbounded domains whichallow us to obtain generalizations of recent fixed point theorems of Penot, Isac, and Nemeth. Anapplication to integral equations is given.

1. Introduction

Let C be a nonempty subset of a Banach space X with norm ‖ · ‖. Recall that a mappingT : C → X is said to be nonexpansive whenever ‖T(x)−T(y)‖ ≤ ‖x−y‖ for every x, y ∈ C. Xis said to have the fixed point property ((FPP) for short) if every nonexpansive selfmapping ofeach nonempty bounded closed and convex subset of X has a fixed point. It has been knownfrom the outset of the study of this property (around the early sixties of the last century) thatit depends strongly on “nice” geometrical properties of the space. For instance, a celebratedresult due to Kirk [1] establishes that those reflexive Banach spaces with normal structure(NS) have the (FPP). In particular, uniformly convex Banach spaces have normal structure(see [2, 3] for more information).

If C is a closed convex of a Banach space enjoying the (FPP), in general it is not truethat T : C → C has a fixed point due to the possible unboundedness of C (it is enough toconsider any translation map, with nonnull vector, in the Banach space X). In 2003 Penot [4]showed that if C is a closed convex subset of a uniformly convex Banach space X, T : C → Cis a nonexpansive mapping, and for some x0 ∈ C,

lim supx∈C,‖x‖→∞

‖T(x) − T(x0)‖‖x − x0‖ < 1 (1.1)

(in other words if T is asymptotically contractive), then T has a fixed point.


A celebrated fixed point result due to Altman [5] is the following.Let H be a separable Hilbert space, with inner product [·, ·] and induced norm ‖x‖ =√

[x, x]. Let F : Br → H be a weakly closed mapping where Br is the closed ball with center0 and radius r. Suppose that T maps the sphere Sr into a bounded set in H. If the followingcondition is satisfied:

[T(x), x] ≤ [x, x] (1.2)

for all x ∈ Sr , then T has a fixed point in Br .In 2006, Isac and Nemeth [6] gave some fixed point results for nonexpansive

nonlinear mappings in Banach spaces inspired by Penot’s results where the asymptoticallycontractiveness was stated in similar terms to condition (1.2).

In this paper we generalize some Penot, Isac, and Nemeth’s fixed point results inseveral ways. First, we will be concerned with pseudocontractive mappings, a more generalclass of mappings than the nonexpansive ones. Second, we use an inwardness conditionweaker than T(C) ⊂ C, and finally our Altmann type assumptions are more general thanthose required in [4, 6].

We prove our fixed point results as a consequence of some results on the existenceof zeroes for accretive operators. Among the problems treated by accretive operators theory,one of the most studied is just this one (see, e.g., Kirk and Schoneberg’s paper [7] as wellas [3, 8, 9] and the references therein). We obtain here several results of this type, and inparticular we give a characterization in the setting of the Banach spaces with (FPP) of thosem-accretive operators which have zeroes.

2. Preliminaries

Throughout this paper we suppose that X is a real Banach space and that X∗ is its topologicaldual. We use Br to denote the closed ball centered at 0X ∈ X with radius r > 0. We also usethe notation |B| := inf{‖y‖ : y ∈ B}, B ⊂ X.

If x ∈ X, we will denote by J(x) the normalized duality mapping at x defined byJ(x) := {j ∈ X∗ : j(x) = ‖x‖2, ‖j‖ = ‖x‖}. We will often use the mapping 〈·, ·〉+ : X × X → R

defined by 〈y, x〉+ := max{j(y) : j ∈ J(x)}.A mapping A : D(A) → 2X will be called an operator on X. The domain of A is

denoted by D(A) and its range by R(A). It is well known that an operator A : D(A) ⊂ X →2X is accretive if and only if 〈u − v, x − y〉+ ≥ 0 for all (x, u), (y, v) ∈ A.

If, in addition, R(I + λA) is for one, hence for all, λ > 0, precisely X, then A is calledm-accretive. We say that A satisfies the range condition if D(A) ⊂ R(I + λA) for all λ > 0.

We now recall some important facts regarding accretive operators which will be usedin our paper (see, e.g., [10]).

Proposition 2.1. Let A : D(A) → 2X be an operator on X. The following conditions are equivalent:

(i) A is an accretive operator,

(ii) the inequality ‖x − y‖ ≤ |x − y + λ(A(x) − A(y))| holds for all λ ≥ 0, and for everyx, y ∈ D(A),

(iii) for each λ > 0 the resolvent Jλ := (I + λA)−1 : R(I + λA) → D(A) is a single-valuednonexpansive mapping.


Let C be a nonempty subset of X and let T : C → X be a mapping. Recall that asequence (xn) of elements of C is said to be an a.f.p sequence for T whenever limn→∞‖xn −T(xn)‖ = 0. It is well known that if T is a nonexpansive mapping which maps a closed convexbounded subset C of X into itself, then such a mapping always has a.f.p. sequences in C.

When the Banach space X has the (FPP), Morales [9] gave a characterization of thosem-accretive operators A such that 0X ∈ R(A). Let us recall such result.

Theorem 2.2. Let X be a Banach space with the FPP, and let A : D(A) → 2X be an m-accretiveoperator. Then 0X ∈ R(A) if and only if the set E = {x ∈ D(A) : tx ∈ A(x), t < 0} is bounded.

A mapping T : C → X is said to be pseudocontractive if for every x, y ∈ C, and for allpositive r, ‖x − y‖ ≤ ‖(1 + r)(x − y) − r(T(x)− T(y))‖. Pseudocontractive mappings are easilyseen to be more general than nonexpansive mappings ones. The interest in these mappingsalso stems from the fact that they are firmly connected to the well-known class of accretivemappings. Specifically T is pseudocontractive if and only if I − T is accretive where I is theidentity mapping.

We say that a mapping T : C → X is demiclosed at 0X if for any sequence (xn) in Cweakly convergent to x0 ∈ C with (T(xn)) norm convergent to 0X one has that T(x0) = 0X .It is well known that if C is weakly compact and convex, T : C → C is nonexpansive, andI − T : C → X is demiclosed at 0X , then T has a fixed point in C.

We say that the mapping T : C → X is weakly inward on C if limλ→ 0+d((1 − λ)x +λT(x), C) = 0 for all x ∈ C. Such condition is always weaker than the assumption of Tmapping the boundary of C into C. Recall that if A : D(A) → X is a continuous accretivemapping, D(A) is convex and closed, and I − A is weakly inward on D(A), then A has therange condition (see [11]).

We say that a semi-inner-product is defined on X, if to any x, y ∈ X there correspondsa real number denoted by [x, y] satisfying the following properties:

(s1) [x + y, z] = [x, z] + [y, z] for x, y, z ∈ X,

(s2) [λx, y] = λ[x, y] for x, y ∈ X, and λ ∈ R,

(s3) [x, x] > 0 for x /= 0X ,

(s4) |[x, y]|2 ≤ [x, x][y, y] .

It is known (see [12, 13]) that a semi-inner-product space is a normed linear space withthe norm ‖x‖s = [x, x]1/2 and that every Banach space can be endowed with a semi-inner-product (and in general in infinitely many different ways, but a Hilbert space in a uniqueway).

In [6] the authors considered several fixed point results for nonexpansive mappingswith unbounded domains satisfying additional asymptotic contractive-type conditions interms of a function G : X ×X → R under the following assumptions:

(G1) G(λx, y) = λG(x, y) for any x, y ∈ X and λ > 0,

(G2) ‖x‖2 ≤ G(x, x) for any x ∈ X,

(G3) G(x + y, z) = G(x, z) +G(y, z) for any x, y, z ∈ X,

(G4) there exists an M > 0 such that |G(x, y)| ≤M‖x‖‖y‖ for every x, y ∈ X.


3. Zeroes for Accretive Operators

We begin with the definition of a certain kind of functions on which we will be concerned.This class is more general than the corresponding one considered in [6]. Let (X, ‖ · ‖) be a realBanach space and G : X ×X → R a mapping which satisfies the following conditions:

(g1) G(λx, y) ≤ λG(x, y) for any x, y ∈ X and λ > 0,

(g2) there exists S > 0 such that 0 < G(x, x) for any x ∈ X with ‖x‖ ≥ S,

(g3) G(x + y, z) ≤ G(x, z) +G(y, z) for any x, y, z ∈ X,

(g4) for each y ∈ X, there exists t > 0 (depending on y), such that if ‖x‖ ≥ t, then|G(y, x)| < G(x, x).

Notice that if we consider either G(x, y) = [x, y] or G(x, y) = 〈x, y〉+, then G satisfies(g1)–(g4).

Let X be a Banach space with the (FPP). If A : D(A) → 2X is an m-accretive operatorsuch that its domain D(A) is a bounded set, then it is well known that 0X ∈ R(A) (see, e.g.,[7, 9]). If D(A) is not bounded, then we give the following result.

Theorem 3.1. Let X be a Banach space with the (FPP). Let G : X ×X → R be a mapping satisfying(g1) and (g2). If A : D(A) → 2X is anm-accretive operator such that there exists R > 0 with

supy∈Ax

G(x − y, x) ≤ G(x, x) (3.1)

whenever ‖x‖ ≥ R, then, 0X ∈ R(A).

Proof. Since A is m-accretive and X has the (FPP), by Theorem 2.2 we know that 0X ∈ R(A) ifand only if the set E = {x ∈ D(A) : tx ∈ A(x); t < 0} is bounded.

In order to get a contradiction we assume that E is an unbounded set. This fact meansthat for each n ∈ N there exists xn ∈ E such that ‖xn‖ ≥ n.

Since xn ∈ E, then there exist tn < 0 and yn ∈ A(xn) such that tnxn = yn. This meansthat (1 − tn)xn = xn − yn.

Consequently, for every n ≥ max{R, S}, we have

0 < G(xn, xn) = G(

11 − tn

(xn − yn

), xn

)≤ 1

1 − tn G(xn − yn, xn

) ≤ 11 − tnG(xn, xn), (3.2)

which is a contradiction.

In the following theorem we are going to give a characterization in terms of a particularfunction G, (in the framework of the Banach spaces with the (FPP)), of those m-accretiveoperators which have zeroes.

Theorem 3.2. Let X be a Banach space with the (FPP). Let G : X ×X → R be the mapping

G(x, y)=

⎧⎨⎩λ, if x = λy, λ > 0, x /= 0,

0, otherwise.(3.3)


If A : D(A) → 2X is anm-accretive operator, then the following conditions are equivalent:

(1) there exists R > 0 such that supy∈AxG(x − y, x) ≤ G(x, x) whenever x ∈ D(A) and‖x‖ ≥ R;

(2) 0X ∈ R(A).

Proof. (1)⇒(2) It is clear that G satisfies conditions (g1) and (g2), thus by Theorem 3.1 weobtain that 0 ∈ R(A).

(2)⇒(1) In order to get a contradiction, assume that for each n ∈ N there exists xn ∈D(A) with ‖xn‖ ≥ n such that

supyn∈A(xn)

G(xn − yn, xn

)> G(xn, xn) = 1. (3.4)

The above inequality implies that for each n ∈ N, there exist yn ∈ A(xn) and λn > 1such that G(xn − yn, xn) = λn.

By definition of G, we have that xn − yn = λnxn, and thus (1 − λn)xn = yn ∈ A(xn).From the above fact, we derive that for each n ∈ N,

xn ∈ E = {x ∈ D(A) : tx ∈ A(x), t < 0}, (3.5)

that is, E is unbounded. By Theorem 2.2, it follows that if E is unbounded, then 0X /∈R(A);therefore, we have a contradiction.

As a consequence of the above characterization it is easy to capture the following resultwhich is related to [7, Theorems 2 and 3].

Corollary 3.3. Let X be a real Banach space with the (FPP). Suppose that A : D(A) ⊆ X → 2X isanm-accretive operator for which there exist x0 ∈ D(A) and R > 0 such that

|A(x0)| < |A(x)| (3.6)

for all x ∈ D(A) with ‖x‖ ≥ R. Then 0X ∈ R(A).

Proof. Without loss of generality we may assume that x0 = 0X . Otherwise, we work with theoperator A : D(A) \ {x0} → 2X defined by A(x − x0) = A(x).

If we take G(·, ·) as in Theorem 3.2, to obtain the conclusion it is enough to see that

supy∈A(x)

G(x − y, x) ≤ G(x, x) = 1, (3.7)

whenever x ∈ D(A) \ BR.In order to get a contradiction, assume that for each n ∈ N there exists xn ∈ D(A) with

‖xn‖ ≥ n such that

supyn∈A(xn)

G(xn − yn, xn

)> G(xn, xn) = 1. (3.8)


The above inequality implies that for each n ∈ N, there exist yn ∈ A(xn) and λn > 1such that G(xn − yn, xn) = λn.

By definition of G, we have that xn − yn = λnxn, and thus (1 − λn)xn = yn ∈ A(xn).By hypothesis, we know that the inequality |A(0X)| < ‖yn‖ = (λn − 1)‖xn‖ holds for

every n ≥ R.This means that there exists zn ∈ A(0X) such that ‖zn‖ < ‖yn‖. Therefore

(1 − λn)‖xn‖ < −‖zn‖. (3.9)

On the other hand, since A is an accretive operator, it is clear that

0 ≤ ⟨yn − zn, xn − 0X⟩+ = 〈(1 − λn)xn − zn, xn〉+≤ (1 − λn)‖xn‖2 + ‖zn‖‖xn‖≤ ((1 − λn)‖xn‖ + ‖zn‖)‖xn‖ < 0,

(3.10)


The above corollary allows us to recapture the following well-known result.

Corollary 3.4. Let X be a real Banach space with the (FPP). Suppose that A : D(A) ⊆ X → 2X isanm-accretive operator; if

lim||x||→∞,x∈D(A)

|A(x)| =∞,(3.11)

then 0X ∈ R(A).

Corollary 3.5. Let X be a Banach space with the (FPP). Let G : X ×X → R be a mapping satisfying(g1) and (g2). If A : D(A) → 2X is anm-accretive operator such that

limx∈D(A),||x||→∞

supy∈Ax

G(x − y, x)G(x, x)

< 1, (3.12)

then 0X ∈ R(A).

Proof. It is clear that condition (3.12) implies assumption (3.1).

Corollary 3.6. Let H be a real Hilbert space. Let ϕ : H → (−∞,∞] be a convex proper lowersemicontinuous mapping with effective domain D(ϕ). Suppose that for some z0 ∈ D(ϕ) there existsr > 0 such that ϕ(z0) < ϕ(x) for all x ∈ H with ‖x‖ ≥ r. Then ϕ has an absolute minimum onH.

Proof. Consider ∂ϕ : H → 2H the subdifferential associated to ϕ, that is

∂ϕ(x) ={y ∈ H :

[y, z − x] ≤ ϕ(z) − ϕ(x), ∀z ∈ H}. (3.13)


It is well known that ∂ϕ is an m-accretive operator on H (see [14]). Now, we consider G :H ×H → R defined as in Theorem 3.2.

In order to get a contradiction, suppose that given n ∈ N there is xn ∈ D(∂ϕ) with‖xn‖ ≥ n such that

supyn∈∂ϕ(xn)

G(xn − yn, xn

)> G(xn, xn). (3.14)

By definition of G we have that there exists yn ∈ ∂ϕ(xn) such that

G(xn − yn, xn

)= λn > 1. (3.15)

This means that xn − yn = λnxn, hence (1 − λn)xn = yn ∈ ∂ϕ(xn). Consequently

[(1 − λn)xn, z0 − xn] ≤ ϕ(z0) − ϕ(xn). (3.16)

By hypothesis, when ‖xn‖ ≥ max{r, ‖z0‖}, we obtain the following contradiction:

0 ≤ (λn − 1)(‖xn‖ − ‖z0‖)‖xn‖ ≤ ϕ(z0) − ϕ(xn) < 0. (3.17)

This contradiction allows us to conclude that there exists R > 0 such that if x ∈ D(∂ϕ)with ‖x‖ ≥ R then

supy∈∂ϕ(x)

G(x − y, x) ≤ G(x, x). (3.18)

Since H has the (FPP), from Theorem 3.1 we conclude that 0H ∈ R(∂ϕ); that is, there existsx ∈ H such that

0H ∈ ∂ϕ(x), (3.19)

and therefore x is an absolute minimum of ϕ.

If X has the (FPP), A : D(A) → 2X is an accretive operator with the range condition,and D(A) is convex and bounded, then, 0X ∈ R(A); see [8]. For the case that D(A) is notbounded we have the following result.

Theorem 3.7. Let X be a Banach space. Suppose that G : X × X → R is a mapping satisfyingconditions (g1)–(g4).

If X has the (FPP), A : D(A) → 2X is an accretive operator with the range condition, D(A)is convex, and A satisfies condition (3.1), then 0X ∈ R(A).

Proof. Since A is accretive with the range condition, then the following two conditions hold:

(i) D(A) =⋂λ>0R(I + λA),

(ii) g := (I +A)−1 : R(I +A) → D(A) is a nonexpansive mapping.


Fix x0 ∈ D(A). For each positive integer n, from (i) there exist xn ∈ D(A) and yn ∈ A(xn)such that

x0 = xn + nyn. (3.20)

Hence, x0 = (1 + n)xn + n(yn − xn). It follows that

xn − yn =n + 1n

xn − 1nx0. (3.21)

We claim that (xn) is a bounded sequence. Indeed, otherwise we can assume that there existsa subsequence (xnk) of (xn) such that ‖xnk‖ → ∞. Without loss of generality we may assumethat ‖xnk‖ ≥ max{R, S, t}, k ∈ N, where the constants S and t are given in the definitions ofconditions (g2) and (g4), respectively.

Therefore, we have

0 < G(xnk , xnk) = G(

nknk + 1

(xnk − ynk

)+

1nk + 1

x0, xnk

)

≤ nknk + 1

G(xnk − ynk , xnk

)+

1nk + 1

G(x0, xnk).

(3.22)

Consequently,

G(xnk , xnk) <(

nknk + 1

G(xnk , xnk) +1

nk + 1G(xnk , xnk)

)= G(xnk , xnk). (3.23)

This is a contradiction which proves our claim.Since (xn) is a bounded sequence, it is clear that (yn) goes to 0X as n goes to infinity.Now we claim that g has a bounded a.f.p. sequence. Indeed, consider for each positive

integer n, wn = xn + yn. It is not difficult to see that g(wn) = xn because yn ∈ A(xn). In thiscase, we obtain

wn − g(wn) = yn −→ 0X. (3.24)

Finally, if we call r0 = lim sup ‖wn − x0‖, we obtain that the following set

K ={x ∈ D(A) : lim sup‖wn − x‖ ≤ r0

}(3.25)

is bounded closed convex and g-invariant. Thus, since X enjoys the (FPP), there exists z ∈ Ksuch that z = g(z) and then 0X ∈ A(z).

Remark 3.8. If we check the proof of Theorem 3.7, we may notice that such theorem still holdsif we omit conditions (g3) and (g4) but we add 0X ∈ D(A).


Corollary 3.9. Let X be a Banach space. Suppose that G : X × X → R is a mapping satisfyingconditions (g1)–(g4).

If X has the (FPP), A : D(A) → 2X is an accretive operator with the range condition, D(A)is convex, and A satisfies condition (3.12), then 0X ∈ R(A).

4. Fixed Point Results

Theorem 4.1. Let X be a Banach space with the (FPP). Suppose that G : X × X → R is a mappingsatisfying conditions (g1) and (g2). Let C be a closed convex and unbounded subset of X with 0X ∈ C.Let T : C → X be a continuous pseudocontractive mapping. Assume that the following conditionsare satisfied.

(a) T is weakly inward on C.

(b) There exists R > 0 such that for every x ∈ C with ‖x‖ ≥ R the inequality

G(T(x), x) ≤ G(x, x) (4.1)

holds.

Then T has a fixed point in C.

Proof. Since T : C → X is a continuous, pseudocontractive mappings weakly inward on C,then A = I − T : C → X is an accretive operator with the range condition (see [11, 15]).

Let us see that condition (3.1) is satisfied. Indeed, if x ∈ C with ‖x‖ ≥ R,

supy∈Ax

G(x − y, x) = G(x − (x − T(x)), x) = G(T(x), x). (4.2)

The above equality along with (4.1) allows us conclude that condition (3.1) holds.On the other hand, since 0X ∈ C = D(A), by Remark 3.8 and following the same

argument developed in the proof of Theorem 3.7, it is not difficult to see that

g := (I +A)−1 : R(I +A) −→ C (4.3)

has a bounded a.f.p. sequence (wn), and thus, if we call r0 = lim sup ‖wn‖, we obtain that theset

K ={x ∈ C : lim sup‖wn − x‖ ≤ r0

}(4.4)

is bounded closed convex and g-invariant. Thus, since X enjoys the (FPP), there exists z ∈ Ksuch that z = g(z) and then 0X = A(z) = z − T(z).


Corollary 4.2. Let X be a Banach space with the (FPP). Suppose that G : X ×X → R is a mappingsatisfying conditions (g1) and (g2). Let C be a closed convex and unbounded subset of X with 0X ∈ C.If T : C → X is a continuous pseudocontractive mapping weakly inward on C and

limx∈C,||x||→∞

G(T(x), x)G(x, x)

< 1, (4.5)

then T has a fixed point in C.

Proof. Clearly inequality (4.5) implies condition (4.1).

Corollary 4.3. Let X be a Banach space with the (FPP). Suppose that G : X ×X → R is a mappingsatisfying conditions (g1)–(g4). Let C be a closed convex and unbounded subset of X. If T : C → Xis a continuous pseudocontractive mapping weakly inward on C and satisfies condition (4.1), then Thas a fixed point in C.

Proof. From the above theorem, we know that A = I − T : C → X is an accretive operatorwith the range condition and with condition (3.1). Therefore by Theorem 3.7 we obtain theresult.

Remark 4.4. In order to give an alternative proof of Corollary 4.3, it is enough to see thatcondition (4.5) implies that T has an a.f.p. sequence (xn), and thus, using [16, Theorem 4.3],we obtain the same conclusion. In this case, if we assume that X is a reflexive Banach spaceand I − T is demiclosed at zero, then we can remove the assumption on the (FPP) for thespace X. Nevertheless, it is well known that there exist nonreflexive Banach spaces withthe FPP (see [13]). On the other hand, if X is a reflexive Banach space such that for everynonexpansive mapping, say T , the mapping I − T is demiclosed at 0X , then the Banach spacehas the FPP.

Remark 4.5 (Theorem 3.2 in [6] reads). LetX be a reflexive Banach space. Suppose thatG : X×X → R satisfies conditions (G1), (G2), (G3), and (G4). Let C ⊆ X be a nonempty unboundedclosed convex set. If T : C → X is a nonexpansive mapping such that T(C) ⊂ C, I − T isdemiclosed and


G(T(x) − x0, x)

‖x‖2< 1 (4.6)

for some x0 ∈ C, then T has a fixed point in C.

Notice that Corollary 4.3 generalizes this theorem in several senses.

(i) Our assumptions (g1)–(g4) on mapping G are weaker than the corresponding inthat theorem.

(ii) Every nonexpansive mapping is in fact continuous and pseudocontractive.

(iii) The inwardness condition is more general than the assumption T(C) ⊂ C.


(iv) Condition (4.6) implies that

G(T(x), x)

‖x‖2≤ G(T(x) − x0 + x0, x)

‖x‖2

≤ G(T(x) − x0, x) +G(x0, x)

‖x‖2

≤ G(T(x) − x0, x)

‖x‖2+M‖x0‖‖x‖‖x‖2

.

(4.7)

Interchanging the roles of x0 and 0X we can conclude that


G(T(x), x)

‖x‖2= lim

x∈C,||x||→∞

G(T(x) − x0, x)

‖x‖2 (4.8)

for every x0 ∈ X. Therefore, there exists R > 0 such that if ‖x‖ ≥ R, then

G(T(x), x) ≤ ‖x‖2 ≤ G(x, x), (4.9)

which is just condition (4.1) of Theorem 4.1.In the same sense, Theorem 4.1 is a generalization of Theorem 3.1 of [6].

Corollary 4.6. Let X be a Banach space with the (FPP). Let C be a closed convex and unboundedsubset of X such that 0X ∈ C. Let T : C → X be a continuous pseudocontractive mapping. Assumethat the following conditions are satisfied.

(a) T is weakly inward on C.(b) There exists R > 0 such that for every x ∈ C \ BR and for every λ > 1, T(x)/=λx.

Then T has a fixed point in C.

Proof. It is enough to apply Theorem 4.1, where G : X ×X → R is defined by

G(x, y)=

⎧⎨⎩λ, if x = λy, λ > 0, x /= 0,

0, otherwise,(4.10)

and if x ∈ C \ BR, then G(Tx, x) ≤ 1 = G(x, x).

Remark 4.7. Notice that the above condition (b) is similar to the well-known Leray-Schauderboundary condition. Some results of this type can be found in [17–19].

Corollary 4.8. Let X be a Banach space with the (FPP). Let C be a closed convex and unboundedsubset of X. If T : C → X is a continuous pseudocontractive mapping weakly inward on C and forevery x ∈ C and ‖x‖ large enough

‖T(x) − x0‖ ≤ ‖x − x0‖ (4.11)

for some x0 ∈ X, then T has a fixed point in C.


Proof. Let G : X × X → R be the function defined by G(x, y) = 〈x, y − x0〉+. It is clear that Gsatisfies conditions (g1) and (g3) . Moreover,

G(x, x) = 〈x, x − x0〉+ = j(x) (4.12)

for some j ∈ J(x − x0). Therefore,

G(x, x) = 〈x, x − x0〉+ = j(x) = j(x − x0) + j(x0) ≥ ‖x − x0‖2 − ‖x − x0‖‖x0‖. (4.13)

Since x0 is a fix element ofX, clearly there exists S > 0 such thatG(x, x) > 0 whenever ‖x‖ ≥ S.This means that G satisfies (g2).

To see that G satisfies condition (g4) we argue as follows.Given a fix y ∈ X, we know that |G(y, x)| ≤ ‖y‖‖x − x0‖.Since ‖x −x0‖ − ‖x0‖ → ∞ as ‖x‖ → ∞, we can find t > 0 such that (‖x −x0‖ − ‖x0‖) >

‖y‖ for every ‖x‖ ≥ t.Then,

∣∣G(y, x)∣∣ ≤ ∥∥y∥∥‖x − x0‖ < (‖x − x0‖ − ‖x0‖)‖x − x0‖ ≤ G(x, x). (4.14)

Now, we will see that G satisfies inequality (4.1) in Corollary 4.3. Indeed, if ‖x‖ ≥ R, we have,for some j ∈ J(x − x0), that

G(Tx, x) = 〈Tx, x − x0〉+ = j(Tx) = j(Tx − x0) + j(x0)

≤ ‖Tx − x0‖‖x − x0‖ + j(x0)

≤ ‖x − x0‖2 + j(x0) = j(x)

≤ 〈x, x − x0〉+= G(x, x).

(4.15)

Thus the conclusion follows from Corollary 4.3.

Remark 4.9. In the case that for all x ∈ C, ‖T(x) − x0‖ ≤ ‖x − x0‖, then the mapping T is saidto have x0 as a center; see [20], where some fixed point theorems are given for this class ofmappings.

On the other hand, in [21, Corollary 1.6, page 54] one can read a similar condition,where the domain of the mapping is required to be bounded.

If T : C → X is asymptotically contractive in the sense due to Penot, then it is easy tosee that

lim supx∈C,||x||→∞

‖T(x)‖‖x‖ < 1, (4.16)

which implies condition (4.11) of Corollary 4.8 for x0 = 0X , and therefore Penot’s fixed pointtheorem is a consequence of Corollary 4.8.


Example 4.10. Next, we are concerned with the solvability of the following Hammerstein’sintegral equation:

u(t) = w(t) +∫Ωζ(t, s)f(s, u(s))ds (4.17)

in Lp(Ω). Here 1 < p < ∞, Ω is a bounded domain of Rn, such that its Lebesgue’s measure

μ(Ω) = 1, and w ∈ Lp(Ω). Suppose that ζ and f satisfy the following conditions:

(1) f : Ω × R → R is a Caratheodory function,

(2) |f(s, x)| ≤ a(s) + b|x|, where a ∈ Lp(Ω) and b ≥ 0,

(3) |f(s, x) − f(s, y)| ≤ k|x − y|,(4) the function ζ : Ω × Ω → R is strongly measurable and

∫Ωζ(·, s)u(s)ds ∈ Lp(Ω)

whenever u ∈ Lp(Ω),

(5) there exists a function τ : Ω → R, belonging to Lp(Ω) such that |ζ(t, s)| ≤ τ(t) forall (t, s) ∈ Ω ×Ω,

(6) k‖τ‖p ≤ 1 and b‖τ‖p < 1.

Proposition 4.11. Assume that conditions (1)–(6) are satisfied, then problem (4.17) has at least onesolution in Lp(Ω).

Proof. First notice that (4.17) may be written in the form u = T(u) where T is given by

T : Lp(Ω) −→ Lp(Ω) : u −→ T(u)(t) := w(t) +∫Ωζ(t, s)f(s, u(s))ds. (4.18)

Let us see that T satisfies the conditions of Corollary 4.8. In this sense, we are going toprove that T is a nonexpansive mapping. Indeed,

‖T(u) − T(v)‖pp =∫Ω

∣∣∣∣∫Ωζ(t, s)

(f(s, u(s)) − f(s, v(s)))ds

∣∣∣∣p

dt

≤∫Ω

(∫Ω

∣∣ζ(t, s)(f(s, u(s)) − f(s, v(s)))∣∣ds)pdt

≤∫Ω|τ(t)|p

(∫Ω

∣∣f(s, u(s)) − f(s, v(s))∣∣ds)pdt

≤∫Ω|τ(t)|p

(∫Ωk|u(s) − v(s)|ds

)pdt.

(4.19)

Since μ(Ω) = 1, by Holder’s inequality, we obtain that

‖T(u) − T(v)‖p ≤ k‖τ‖p‖u − v‖p ≤ ‖u − v‖p. (4.20)


Finally, we are going to show that there exists R > 0 such that if ‖u‖p ≥ R, then‖T(u)‖p ≤ ‖u‖p. Indeed, we know that

‖T(u)‖p ≤ ‖w‖p +∥∥∥∥∫Ωζ(·, s)f(s, u(s))ds

∥∥∥∥p

, (4.21)

hence,

∥∥∥∥∫Ωζ(·, s)f(s, u(s))ds

∥∥∥∥p

p

=∫Ω

∣∣∣∣∫Ωζ(t, s)f(s, u(s))ds

∣∣∣∣p

dt

≤∫Ω

(∫Ω|τ(t)|(a(s) + b|u(s)|)ds

)pdt.

(4.22)

Applying again Holder’s inequality, we derive that

∥∥∥∥∫Ωζ(·, s)f(s, u(s))ds

∥∥∥∥p

≤(‖a‖p + b‖u‖p

)‖τ‖p. (4.23)

Moreover, it is clear that

lim||u||p→∞

‖w‖p +(‖a‖p + b‖u‖p

)‖τ‖p

‖u‖p= b‖τ‖p < 1, (4.24)

therefore there exists R > 0 such that if ‖u‖p ≥ R, then ‖T(u)‖p ≤ ‖u‖p as we claimed.

Notice that if k‖τ‖p = 1 then, Corollary 3 in [4] does not apply because under thiscondition we cannot guarantee that T is asymptotically contractive on Lp(Ω).

Let C be a closed convex subset of a Banach space X. A family of mappings {T(t) :C → C : t ≥ 0} is called a one-parametric strongly continuous semigroup of nonexpansive mappings(nonexpansive semigroup, for short) on C if the following assumptions are satisfied:

(1) T(s + t) = T(s) ◦ T(t) for all s, t ≥ 0,

(2) for each x ∈ C, the mapping t �→ T(t)x from [0,∞[ into C is continuous,

(3) for each t ≥ 0, T(t) : C → C is a nonexpansive mapping.

In the next result we study when a nonexpansive semigroup has a common fixed point.

Theorem 4.12. Let X be a Banach space with the (FPP). Suppose that G : X ×X → R is a mappingsatisfying conditions (g1)–(g4). Let C be a closed convex and unbounded subset of X. If {T(t) : C →C : t ≥ 0} is a nonexpansive semigroup such that there exist α, β ∈ ]0,∞[ with (α/β) ∈ R \ Q

satisfying that

max{G(T(α)x, x);G

(T(β)x, x)} ≤ G(x, x) (4.25)

whenever x ∈ C large enough, then the semigroup has a least one common fixed point.


Proof. By Theorem 1 of [22] in order to get the conclusion it is enough to show that, givenλ ∈ (0, 1), the mapping Tλ : C → C defined by

Tλ(x) = λT(α)(x) + (1 − λ)T(β)(x) (4.26)

has a fixed point.By hypotheses we know that there exists R > 0 such that for every x ∈ C with ‖x‖ ≥ R

the inequality

max{G(T(α)x, x);G

(T(β)x, x)} ≤ G(x, x) (4.27)

holds. Since G satisfies conditions (g1)–(g4), we have

G(Tλ(x), x) = G(λT(α)(x) + (1 − λ)T(β)(x), x)

≤ G(λT(α)(x), x) +G((1 − λ)T(β)(x), x)≤ λG(T(α)(x), x) + (1 − λ)G(T(β)(x), x)≤ G(x, x).

(4.28)

The above inequality means that Tλ satisfies the conditions of Corollary 4.3 andtherefore Tλ has a fixed point, which implies by Theorem 1 of [22] that the semigroup hasa common fixed point.

Corollary 4.13. Let X be a Banach space with the (FPP). Let C be a closed convex and unboundedsubset of X. If {T(t) : C → C : t ≥ 0} is a nonexpansive semigroup such that there exist x0 ∈ X,α, β ∈ ]0,∞[ with α/β ∈ R \Q satisfying that

max{‖T(α)x − x0‖;

∥∥T(β)x − x0∥∥} ≤ ‖x − x0‖ (4.29)

whenever x ∈ C large enough, then the semigroup has a least one common fixed point.

Proof. It is enough to apply the above theorem with G(x, y) = 〈x, y − x0〉+ (see the proof ofCorollary 4.8).

We conclude this section by presenting a corollary of Theorem 4.1 which guaranteesthe existence of positive eigenvalues.

Corollary 4.14. Let X be a Banach space with the (FPP). Suppose that G : X ×X → R is a mappingsatisfying conditions (g1) and (g2). Let C be a closed convex and unbounded subset of X with 0X ∈ C.Let T : C → C be a continuous pseudocontractive mapping. Assume that the following conditionsare satisfied.

(a) T(0X)/= 0X .(b) There exists R > 0 such that for every x ∈ C with ‖x‖ ≥ R the inequality

G(T(x), x) ≤ lG(x, x) (4.30)

holds for some l ≥ 0.

Then any λ ≥ max{l, 1} is an eigenvalue of T associated to an eigenvector in C.


Proof. Consider a fixed λ ≥ max{l, 1}. Let us see that (1/λ)T is a continuous pseudocon-tractive mapping such that (1/λ)T(C) ⊆ C. Indeed, since 0X ∈ C, C is convex, λ ≥ 1, andT(C) ⊆ C, then (1/λ)T(C) ⊆ C.

To see that (1/λ)T is a pseudocontractive mapping, it is enough to prove that I−(1/λ)Tis an accretive mapping:

⟨x − 1

λT(x) −

(y − 1

λT(y)), x − y

⟩+

=⟨

1λ

(x − T(x) − (y − T(y))) +

(1 − 1

λ

)(x − y), x − y

⟩+

≥ 1λ

⟨x − T(x) − (y − T(y)), x − y⟩+ +

(1 − 1

λ

)‖x − y‖2 ≥ 0.

(4.31)

The above inequality holds since T is a pseudocontractive mapping and therefore〈x − T(x) − (y − T(y)), x − y〉+ ≥ 0.

Finally, if x ∈ C with ‖x‖ ≥ R, we have

G

(1λT(x), x

)≤ 1λG(Tx, x) ≤ l

λG(x, x) ≤ G(x, x). (4.32)

The above facts show that (1/λ)T is under the assumption of Theorem 4.1 and hencethere exists xλ ∈ C \ {0} such that T(xλ) = λxλ.

Acknowledgments

Both authors were partially supported by MTM 2009-10696-C02-02. This work is dedicatedto Professor W. A. Kirk.

References


[2] W. A. Kirk and B. Sims, Eds., Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers,Dordrecht, The Netherlands, 2001.


[4] J.-P. Penot, “A fixed-point theorem for asymptotically contractive mappings,” Proceedings of theAmerican Mathematical Society, vol. 131, no. 8, pp. 2371–2377, 2003.

[5] M. Altman, “A fixed point theorem in Hilbert space,” Bulletin de l’Academie Polonaise des Sciences, vol.5, pp. 19–22, 1957.

[6] G. Isac and S. Z. Nemeth, “Fixed points and positive eigenvalues for nonlinear operators,” Journal ofMathematical Analysis and Applications, vol. 314, no. 2, pp. 500–512, 2006.

[7] W. A. Kirk and R. Schoneberg, “Zeros of m-accretive operators in Banach spaces,” Israel Journal ofMathematics, vol. 35, no. 1-2, pp. 1–8, 1980.

[8] J. Garcıa-Falset and S. Reich, “Zeroes of accretive operators and the asymptotic behavior of nonlinearsemigroups,” Houston Journal of Mathematics, vol. 32, no. 4, pp. 1197–1225, 2006.

[9] C. Morales, “Nonlinear equations involving m-accretive operators,” Journal of Mathematical Analysisand Applications, vol. 97, no. 2, pp. 329–336, 1983.


[10] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.

[11] R. H. Martin Jr., “Differential equations on closed subsets of a Banach space,” Transactions of theAmerican Mathematical Society, vol. 179, pp. 399–414, 1973.

[12] J. R. Giles, “Classes of semi-inner-product spaces,” Transactions of the American Mathematical Society,vol. 129, pp. 436–446, 1967.

[13] G. Lumer, “Semi-inner-product spaces,” Transactions of the American Mathematical Society, vol. 100, pp.29–43, 1961.

[14] H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,North-Holland Mathematics Studies, no. 5. Notas de Matematica (50), North-Holland, Amsterdam,The Netherlands, 1973.

[15] F. E. Browder, “Nonlinear mappings of nonexpansive and accretive type in Banach spaces,” Bulletinof the American Mathematical Society, vol. 73, pp. 875–882, 1967.

[16] J. Garcıa-Falset, “Fixed points for mappings with the range type condition,” Houston Journal ofMathematics, vol. 28, no. 1, pp. 143–158, 2002.

[17] C. Gonzalez, A. Jimenez-Melado, and E. Llorens-Fuster, “A Monch type fixed point theorem underthe interior condition,” Journal of Mathematical Analysis and Applications, vol. 352, no. 2, pp. 816–821,2009.

[18] W. A. Kirk, “Fixed point theorems for nonexpansive mappings satisfying certain boundaryconditions,” Proceedings of the American Mathematical Society, vol. 50, pp. 143–149, 1975.

[19] C. H. Morales, “The Leray-Schauder condition for continuous pseudo-contractive mappings,”Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 1013–1020, 2009.

[20] J. Garcıa-Falset, E. Llorens-Fuster, and S. Prus, “The fixed point property for mappings admitting acenter,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 6, pp. 1257–1274, 2007.

[21] A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, NewYork, NY, USA, 2003.

[22] T. Suzuki, “Common fixed points of one-parameter nonexpansive semigroups,” The Bulletin of theLondon Mathematical Society, vol. 38, no. 6, pp. 1009–1018, 2006.


Research ArticleBrowder’s Convergence for UniformlyAsymptotically Regular Nonexpansive Semigroupsin Hilbert Spaces

Genaro Lopez Acedo1 and Tomonari Suzuki2

1 Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Sevilla,41080 Sevilla, Spain

2 Department of Mathematics, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan

Correspondence should be addressed to Genaro Lopez Acedo, [email protected]

Received 6 October 2009; Accepted 14 October 2009


Copyright q 2010 G. Lopez Acedo and T. Suzuki. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We give a sufficient and necessary condition concerning a Browder’s convergence type theoremfor uniformly asymptotically regular one-parameter nonexpansive semigroups in Hilbert spaces.

1. Introduction

Let C be a closed convex subset of a Hilbert space E. A mapping T on C is called anonexpansive mapping if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ C. We denote by F(T) the setof fixed points of T . Browder, see [1], proved that F(T) is nonempty provided that C is, inaddition, bounded. Kirk in a very celebrated paper, see [2], extended this result to the settingof reflexive Banach spaces with normal structure.

Browder [3] initiated the investigation of an implicit method for approximating fixedpoints of nonexpansive self-mappings defined on a Hilbert space. Fix u ∈ C, he studied theimplicit iterative algorithm

zt = tu + (1 − t)Tzt. (1.1)

Namely, zt, t ∈ (0, 1), is the unique fixed point of the contraction x �→ tu + (1 − t)Tx, x ∈ C.Browder proved that limt→+0zt = Pu, where Pu is the element of F(T) nearest to u. Extensionsto the framework of Banach spaces of Browder’s convergence results have been done bymany authors, including Reich [4], Takahashi and Ueda [5], and O’Hara et al. [6].


A family of mappings {T(t) : t ≥ 0} is called a one-parameter strongly continuoussemigroup of nonexpansive mappings (nonexpansive semigroup, for short) on C if the followingare satisfied.

(NS1) For each t ≥ 0, T(t) is a nonexpansive mapping on C.

(NS2) T(s + t) = T(s) ◦ T(t) for all s, t ≥ 0.

(NS3) For each x ∈ C, the mapping t �→ T(t)x from [0,∞) into C is strongly continuous.

There are many papers concerning the existence of common fixed points of {T(t) : t ≥ 0};see, for instance, [7–13]. As a matter of fact, Browder [8] proved that if C is bounded, then⋂t≥0F(T(t)) is nonempty.

Browder’s type convergence theorem for nonexpansive semigroups is proved in [11,14–18] and others. For example, the following theorem is proved in [17].

Theorem 1.1 (see [17]). Let C be a closed convex subset of a Hilbert space E. Let {T(t) : t ≥ 0}be a nonexpansive semigroup on C such that

⋂t≥0F(T(t))/= ∅. Let {αn} and {tn} be sequences in R

satisfying

(C1) 0 < αn < 1 and 0 ≤ tn;(C2) limntn = limnαn/tn = 0, where 1/0 =∞.

Fix u ∈ C and define a sequence {xn} in C by

xn = αnu + (1 − αn)T(tn)xn. (1.2)

Then {xn} converges strongly to the element of⋂t≥0F(T(t)) nearest to u.

We note that (C1) is needed to define {xn}.A nonexpansive semigroup {T(t) : t ≥ 0} on C is said to be uniformly asymptotically

regular (u.a.r.) if for every t ≥ 0 and for every bounded subset K of C,

lims→∞

supx∈K‖T(s + t)x − T(s)x‖ = 0 (1.3)

holds. The following is proved by Domınguez Benavides et al. [16]; see also [15].

Theorem 1.2 (see [16]). Let E, C, and {T(t) : t ≥ 0} be as in Theorem 1.1. Assume that {T(t) : t ≥0} is u.a.r. Let {αn} and {tn} be sequences in R satisfying (C1) and

(D2) limnαn = 0 and limntn =∞.

Fix u ∈ C and define a sequence {xn} in C by (1.2). Then {xn} converges strongly to the element of⋂t≥0F(T(t)) nearest to u.

There is an interesting difference between Theorems 1.1 and 1.2, that is, {tn} inTheorem 1.1 converges to 0 and {tn} in Theorem 1.2 diverges to∞. By the way, very recently,Akiyama and Suzuki [14] generalized Theorem 1.1. They replaced (C2) of Theorem 1.1 by


the following:

(C2′) {tn} is bounded;

(C3′) limnαn/(tn − τ) = 0 for all τ ∈ [0,∞).

They also showed that the conjunction of (C2′) and (C3′) is best possible; see also [18].In this paper, motivated by the previous considerations, we generalize Theorem 1.2

concerning {αn} and {tn}. Also, we will show that our new condition is best possible.

2. Main Results

We denote by N the set of all positive integers and by R the set of all real numbers. For t ∈ R,we denote by [t] the maximum integer not exceeding t.

The following proposition plays an important role in this paper.

Proposition 2.1. LetC be a set of a separated topological vector space E. Let {T(t) : t ≥ 0} be a familyof mappings on C such that T(s) ◦ T(t) = T(s + t) for all s, t ∈ [0,∞). Assume that {T(t) : t ≥ 0} isasymptotic regular, that is,

lims→∞

(T(t + s)x − T(s)x) = 0 (2.1)

for all t ∈ [0,∞) and x ∈ C. Then

F(T(t)) =⋂s≥0

F(T(s)) (2.2)

holds for all t ∈ (0,∞).

Proof. Fix t ∈ (0,∞). It is obvious that F(T(t)) ⊃ ⋂sF(T(s)) holds. Let z ∈ C be a fixed pointof T(t). For every h ∈ [0,∞), we have

T(h)z − z = limn→∞

(T(h) ◦ T(t)nz − T(t)nz)

= limn→∞

(T(h + nt)z − T(nt)z)

= lims→∞

(T(h + s)z − T(s)z)

= 0,

(2.3)

and hence z is a common fixed point of {T(t) : t ≥ 0}.

It is well known that every Hilbert space has the Opial property.

Proposition 2.2 (Opial [19]). Let E be a Hilbert space. Let {xn} be a sequence in E convergingweakly to z0 ∈ H. Then the inequality lim infn‖xn − z‖ ≤ lim infn‖xn − z0‖ implies z = z0.

We generalize Theorem 1.2.


Theorem 2.3. Let C be a closed convex subset of a Hilbert space E. Let {T(t) : t ≥ 0} be au.a.r. nonexpansive semigroup on C such that

⋂t≥0F(T(t))/= ∅. Let {αn} and {tn} be sequences in

R satisfying (C1) and

(D2′) limnαn = limnαn/tn = 0.

Fix u ∈ C and define a sequence {xn} in C by (1.2). Then {xn} converges strongly to the element of⋂t≥0F(T(t)) nearest to u.

Proof. Put F(T) = ⋂t≥0F(T(t)). Let v be the element of F(T) nearest to u. Since

‖xn − v‖ = ‖(1 − αn)T(tn)xn + αnu − v‖≤ (1 − αn)‖T(tn)xn − v‖ + αn‖u − v‖≤ (1 − αn)‖xn − v‖ + αn‖u − v‖,

(2.4)

we have ‖xn − v‖ ≤ ‖u − v‖. Therefore {xn} is bounded. Hence {T(t)xn : n ∈ N, t ≥ 0} is alsobounded.

We put

M := sup{‖T(t)xn − u‖ : n ∈ N, t ≥ 0} <∞. (2.5)

Let {f(n)} be an arbitrary subsequence of {n}. Then there exists a subsequence {g(n)} of {n}such that {xf◦g(n)} converges weakly to x. We choose a subsequence {h(n)} of {n} such that

τ := limn→∞

tf◦g◦h(n) = lim supn→∞

tf◦g(n). (2.6)

Put yj = xf◦g◦h(j), βj = αf◦g◦h(j), and sj = tf◦g◦h(j). We will show x ∈ F(T), dividing thefollowing three cases:

(i) τ =∞,

(ii) 0 < τ <∞,

(iii) τ = 0.

In the first case, we fix t ≥ 0. For sufficiently large j ∈ N, we have

‖T(t)x − yj‖ ≤ ‖T(t)x − T(t)yj‖ + ‖T(t)yj − yj‖≤ ‖x − yj‖ + βj‖T(t)yj − u‖ +

(1 − βj

)‖T(t)yj − T(sj)yj‖≤ ‖x − yj‖ + βjM +

(1 − βj

)‖T(sj − t)yj − yj‖≤ ‖x − yj‖ + βjM +

(1 − βj

)βj‖T

(sj − t

)yj − u‖ +

(1 − βj

)2‖T(sj − t)yj − T(sj)yj‖≤ ‖x − yj‖ + βj

(2 − βj

)M +

(1 − βj

)2‖T(sj − t + t)yj − T(sj − t)yj‖,(2.7)


and hence

lim infj→∞

∥∥T(t)x − yj∥∥ ≤ lim infj→∞

∥∥x − yj∥∥. (2.8)

By the Opial property, we obtain T(t)x = x. Thus x ∈ F(T).In the second case, we have

‖T(τ)x − yj‖ ≤ ‖T(τ)x − T(sj)x‖ + ‖T(sj)x − T(sj)yj‖ + ‖T(sj)yj − yj‖

≤ ‖T(τ)x − T(sj)x‖ + ‖x − yj‖ + βj‖T(sj)yj − u‖≤ ‖T(∣∣τ − sj∣∣)x − T(0)x‖ + ‖x − yj‖ + βjM,

(2.9)

and hence

lim infj→∞

∥∥T(τ)x − yj∥∥ ≤ lim infj→∞

∥∥x − yj∥∥. (2.10)

By the Opial property, we obtain T(τ)x = x. By Proposition 2.1, we obtain x ∈ F(T).In the third case, we fix t ≥ 0. For sufficiently large j ∈ N, we have

‖T(t)x − yj‖ ≤ ‖T(t)x − T([t/sj]sj)x‖ + ‖T([t/sj]sj)x − T([t/sj]sj)yj‖

+[t/sj ]−1∑k=0

‖T(ksj)yj − T((k + 1)sj)yj‖ + ‖T(0)yj − yj‖

≤ ‖T(t − [t/sj]sj)x − T(0)x‖ + ‖x − yj‖+[t/sj]‖T(sj)yj − yj‖ + ‖T(0)yj − T(sj)yj‖ + ‖T(sj)yj − yj‖

≤ ‖T(t − [t/sj]sj)x − T(0)x‖ + ‖x − yj‖+[t/sj]‖T(sj)yj − yj‖ + ‖yj − T(sj)yj‖ + ‖T(sj)yj − yj‖

= ‖T(t − [t/sj]sj)x − T(0)x‖ + ‖x − yj‖ + ([t/sj] + 2)‖T(sj)yj − yj‖

= ‖T(t − [t/sj]sj)x − T(0)x‖ + ‖x − yj‖ + ([t/sj] + 2)βj‖T

(sj)yj − u‖

≤ max{‖T(s)x − T(0)x‖ : 0 ≤ s ≤ sj

}+ ‖x − yj‖ +

(tβj/sj + 2βj

)M.

(2.11)

Hence (2.8) holds. Thus we obtain x ∈ F(T).We next prove that {yj} converges strongly to v. Since

βj∥∥yj − v∥∥2 +

(1 − βj

)〈(yj − T(sj)yj) − (v − T(sj)v), yj − v〉= βj〈u − v, yj − v〉,

〈(yj − T(sj)yj) − (v − T(sj)v), yj − v〉≥ ‖yj − v‖2 − ‖T(sj)yj − T(sj)v‖‖yj − v‖ ≥ 0,

(2.12)


we obtain ‖yj − v‖2 ≤ 〈u − v, yj − v〉. Since 〈u − v, x − v〉 ≤ 0, we have

∥∥yj − v∥∥2 ≤ 〈u − v, yj − v〉= 〈u − v, yj − x〉 + 〈u − v, x − v〉≤ ⟨u − v, yj − x⟩,

(2.13)

and hence {yj} converges strongly to v. Since {xf(n)} is arbitrary, we obtain that {xn}converges strongly to v.

Using [20, Theorem 7], we obtain the following Moudafi’s type convergence theorem;see [21].

Corollary 2.4. Let E, C, {T(t) : t ≥ 0}, {αn}, and {tn} be as in Theorem 2.3. Let Φ be a contractionon C; that is, there exists r ∈ [0, 1) such that ‖Φx −Φy‖ ≤ r‖x − y‖ for x, y ∈ C. Define a sequence{xn} in C by

xn = αnΦxn + (1 − αn)T(tn)xn. (2.14)

Then {xn} converges strongly to the unique point z ∈ C satisfying P ◦Φz = z, where P is the metricprojection from C onto

⋂t≥0F(T(t)).

We will show that (D2′) is best possible.

Example 2.5. Put E = �2(N), that is, E is a Hilbert space consisting of all the functions x fromN into R satisfying

∑k∈N |x(k)|2 < ∞ with inner product 〈x, y〉 =

∑k∈N x(k)y(k). Define a

bounded closed convex subset C of E by

C ={x ∈ E : 0 ≤ x(k) ≤ pk

}, (2.15)

where pk = 2−k/2. Define a u.a.r. nonexpansive semigroup {T(t) : t ≥ 0} on C by

(T(t)x)(k) = max{x(k) − tpk2, 0

}. (2.16)

Let {ek} be the canonical basis of E and put u =∑∞

k=1 pkek. Let {αn} and {tn} be sequences inR satisfying (C1) and define {xn} in C by (1.2). Then {xn} converges to a common fixed pointof {T(t) : t ≥ 0} only if limnαn = limnαn/tn = 0.

Proof. For α ∈ (0, 1) and t ≥ 0, we define x(α, t) by

x(α, t) = αu + (1 − α)T(t)x(α, t). (2.17)


We note

x(α, t)(k) =

⎧⎪⎨⎪⎩αpk, if α ≤ tpk,(

1 + tpk −tpkα

)pk, if α ≥ tpk.

(2.18)

So, x(α, t)(k) ≥ αpk. It is obvious that⋂t≥0F(T(t)) = {0}. We assume limnxn = limnx(αn, tn) =

Pu = 0. Then

0 = limn→∞

xn(1)p1

≥ limn→∞

αn. (2.19)

Arguing by contradiction, we assume lim supnαn/tn > 0. Then there exist κ ∈ N and asubsequence {f(n)} of {n} such that

αf(n)

tf(n)≥ 2pκ. (2.20)

Since limnxf(n)(κ) = 0, we have

0 = limn→∞

xf(n)(κ)pκ

= limn→∞

(1 + tf(n)pκ −

tf(n)pκ

αf(n)

)

≥ lim supn→∞

(1 − tf(n)pκ

αf(n)

)≥ 1

2> 0,

(2.21)

which is a contradiction. Therefore we obtain limnαn/tn = 0.

By Theorem 2.3 and Example 2.5, we obtain the following.

Theorem 2.6. Let E be an infinite-dimensional Hilbert space. Let {αn} and {tn} be sequences in R

satisfying (C1). Then the following are equivalent:

(i) limnαn = limnαn/tn = 0,

(ii) if C is a bounded closed convex subset C of E, {T(t) : t ≥ 0} is a u.a.r. nonexpansivesemigroup on C, u ∈ C, and {xn} is a sequence in C defined by (1.2), then {xn} convergesstrongly to the element of

⋂t≥0F(T(t)) nearest to u.

Compare (D2′) with the conjunction of (C2′) and (C3′). We can tell that the differencebetween both conditions is u.a.r.

Acknowledgments

The first author was partially supported by DGES, Grant MTM2006-13997-C02-01 and Juntade Andalucıa, Grant FQM-127. The second author is supported in part by Grants-in-Aid forScientific Research from the Japanese Ministry of Education, Culture, Sports, Science andTechnology.


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[14] S. Akiyama and T. Suzuki, “Browder’s convergence for one-parameter nonexpansive semigroups,”to appear in Canadian Mathematical Bulletin.

[15] A. Aleyner and Y. Censor, “Best approximation to common fixed points of a semigroup ofnonexpansive operators,” Journal of Nonlinear and Convex Analysis., vol. 6, no. 1, pp. 137–151, 2005.

[16] T. Domınguez Benavides, G. L. Acedo, and H.-K. Xu, “Construction of sunny nonexpansiveretractions in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 66, no. 1, pp. 9–16,2002.

[17] T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbertspaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003.

[18] T. Suzuki, “Browder’s type convergence theorems for one-parameter semigroups of nonexpansivemappings in Banach spaces,” Israel Journal of Mathematics, vol. 157, no. 1, pp. 239–257, 2007.


[20] T. Suzuki, “Moudafi’s viscosity approximations with Meir-Keeler contractions,” Journal of Mathemati-cal Analysis and Applications, vol. 325, no. 1, pp. 342–352, 2007.



Research ArticleCoincidence Theorems for Certain Classes ofHybrid Contractions

S. L. Singh and S. N. Mishra

Department of Mathematics, School of Mathematical & Computational Sciences, Walter Sisulu University,Nelson Mandela Drive Mthatha 5117, South Africa

Correspondence should be addressed to S. N. Mishra, [email protected]

Received 27 August 2009; Accepted 9 October 2009


Copyright q 2010 S. L. Singh and S. N. Mishra. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Coincidence and fixed point theorems for a new class of hybrid contractions consisting of a pairof single-valued and multivalued maps on an arbitrary nonempty set with values in a metricspace are proved. In addition, the existence of a common solution for certain class of functionalequations arising in dynamic programming, under much weaker conditions are discussed. Theresults obtained here in generalize many well known results.

1. Introduction

Nadler’s multivalued contraction theorem [1] (see also Covitz and Nadler, Jr. [2]) wassubsequently generalized among others by Reich [3] and Ciric [4]. For a fundamentaldevelopment of fixed point theory for multivalued maps, one may refer to Rus [5].Hybrid contractive conditions, that is, contractive conditions involving single-valued andmultivalued maps are the further addition to metric fixed point theory and its applications.For a comprehensive survey of fundamental development of hybrid contractions andhistorical remarks, refer to Singh and Mishra [6] (see also Naimpally et al. [7] and Singhand Mishra [8]).

Recently Suzuki [9, Theorem 2] obtained a forceful generalization of the classicalBanach contraction theorem in a remarkable way. Its further outcomes by Kikkawa andSuzuki [10, 11], Mot and Petrusel [12] and Dhompongsa and Yingtaweesittikul [13],are important contributions to metric fixed point theory. Indeed, [10, Theorem 2] (seeTheorem 2.1 below) presents an extension of [9, Theorem 2] and a generalization of themultivalued contraction theorem due to Nadler, Jr. [1]. In this paper we obtain a coincidencetheorem (Theorem 3.1) for a pair of single-valued and multivalued maps on an arbitrary


nonempty set with values in a metric space and derive fixed point theorems which generalizeTheorem 2.1 and certain results of Reich [3], Zamfirescu [14], Mot and Petrusel [12], andothers. Further, using a corollary of Theorem 3.1, we obtain another fixed point theorem formultivalued maps. We also deduce the existence of a common solution for Suzuki-Zamfirescutype class of functional equations under much weaker contractive conditions than those inBellman [15], Bellman and Lee [16], Bhakta and Mitra [17], Baskaran and Subrahmanyam[18], and Pathak et al. [19].

2. Suzuki-Zamfirescu Hybrid Contraction

For the sake of brevity, we follow the following notations, wherein P and T are maps to bedefined specifically in a particular context while x, and y are the elements of specific domains:

M(P ;x, y

)=

{d(x, y),d(x, Px) + d

(y, Py

)2

,d(x, Py

)+ d(y, Px

)2

},

M(P ; Tx, Ty

)=

{d(Tx, Ty

),d(Tx, Px) + d

(Ty, Py

)2

,d(Tx, Py

)+ d(Ty, Px

)2

},

m(P ;x, y

)=

{d(x, y), d(x, Px), d

(y, Py

),d(x, Py

)+ d(y, Px

)2

}.

(2.1)

Consistent with Nadler, Jr. [20, page 620], Y will denote an arbitrary nonempty set,(X, d) a metric space, and CL(X) (resp. CB(X)) the collection of nonempty closed (resp.,closed and bounded) subsets of X. For A,B ∈ CL(X) and ε > 0,

N(ε,A) = {x ∈ X : d(x, a) < ε for some a ∈ A},EA,B = {ε > 0 : A ⊆N(ε, B), B ⊆N(ε,A)},

H(A,B) =

⎧⎨⎩

infEA,B, if EA,B /=φ

+∞, if EA,B = φ.

(2.2)

The hyperspace (CL(X), H) is called the generalized Hausdorff metric space inducedby the metric d on X.

For any subsets A,B of X, d(A,B) denotes the ordinary distance between the subsetsA and B, while

ρ(A,B) = sup{d(a, b) : a ∈ A, b ∈ B},BN(X) =

{A : φ/=A ⊆ X and the diameter of A is finite

}.

(2.3)

As usual, we write d(x, B) (resp., ρ(x, B)) for d(A,B) (resp., ρ(A,B)) when A = {x}.


In all that follows η is a strictly decreasing function from [0, 1) onto (1/2, 1] defined by

η(r) =1

1 + r. (2.4)

Recently Kikkawa and Suzuki [10] obtained the following generalization of Nadler, Jr.[1].

Theorem 2.1. Let (X, d) be a complete metric space and P : X → CB(X). Assume that there existsr ∈ [0, 1) such that

(KSC) η(r)d(x, Px) ≤ d(x, y) implies H(Px, Py) ≤ rd(x, y)

for all x, y ∈ X. Then P has a fixed point.For the sake of brevity and proper reference, the assumption (KSC) will be called Kikkawa-

Suzuki multivalued contraction.

Definition 2.2. Maps P : Y → CL(X) and T : Y → X are said to be Suzuki-Zamfirescu hybridcontraction if and only if there exists r ∈ [0, 1) such that

(S-Z) η(r)d(Tx, Px) ≤ d(Tx, Ty) implies H(Px, Py) ≤ r ·maxM(P ; Tx, Ty)

for all x, y ∈ Y.A map P : X → CL(X) satisfying

(CG) H(Px, Py) ≤ r ·maxm(P ;x, y)

for all x, y ∈ X, where 0 ≤ r < 1, is called Ciric-generalized contraction. Indeed, Ciric [4]showed that a Ciric generalized contraction has a fixed point in a P -orbitally complete metricspace X.

It may be mentioned that in a comprehensive comparison of 25 contractive conditionsfor a single-valued map in a metric space, Rhoades [21] has shown that the conditions (CG)and (Z) are, respectively, the conditions (21′) and (19′′) when P is a single-valued map, where

(Z) H(Px, Py) ≤ r ·maxM(P ;x, y) for all x, y ∈ X.

Obiviously, (Z) implies (CG). Further, Zamfirescu’s condition [14] is equivalent to (Z)when P is single-valued (see Rhoades [21, pages 259 and 266]).

The following example indicates the importance of the condition (S-Z).

Example 2.3. Let X = {1, 2, 3} be endowed with the usual metric and let P and T be definedby

Px =

⎧⎨⎩

2, 3 if x /= 3,

3 if x = 3,

Tx =

⎧⎨⎩

1 if x /= 1,

3 if x = 1.

(2.5)


Then P does not satisfy the condition (KSC). Indeed, for x = 2, y = 3,

η(r)d(2, P2) = 0 ≤ d(2, 3), (2.6)

and this does not imply

1 = H(P2, P3) ≤ d(2, 3) = r. (2.7)

Further, as easily seen, P does not satisfy (CG) for x = 2, y = 3. However, it can beverified that the pair P and T satisfies the assumption (S-Z). Notice that P does not satisfythe condition (S-Z) when Y = X and T is the identity map.

We will need the following definitions as well.

Definition 2.4 (see [4]). An orbit for P : X → CL(X) at x0 ∈ X is a sequence {xn : xn ∈Pxn−1}, n = 1, 2, . . . . A space X is called P -orbitally complete if and only if every Cauchysequence of the form {xni : xni ∈ Pxni−1}, i = 1, 2, . . . converges in X.

Definition 2.5. Let P : Y → CL(X) and T : Y → X. If for a point x0 ∈ Y, there exists asequence {xn} in Y such that Txn+1 ∈ Pxn, n = 0, 1, 2, . . . , then

OT (x0) = {Txn : n = 1, 2, . . .} (2.8)

is the orbit for (P, T) at x0. We will use OT (x0) as a set and a sequence as the situationdemands. Further, a space X is (P, T)-orbitally complete if and only if every Cauchy sequenceof the form {Txni : Txni ∈ Pxni−1} converges in X.

As regards the existence of a sequence {Txn} in the metric space X, the sufficientcondition is that P(Y ) ⊆ T(Y ). However, in the absence of this requirement, for somex0 ∈ Y, a sequence {Txn}may be constructed some times. For instance, in the above example,the range of P is not contained in the range of T, but we have the sequence {Txn} forx0 = 2, x1 = x2 = · · · = 1. So we have the following definition.

Definition 2.6. If for a point x0 ∈ Y, there exists a sequence {xn} in Y such that the sequenceOT (x0) converges in X, then X is called (P, T)-orbitally complete with respect to x0 or simply(P, T, x0)-orbitally complete.

We remark that Definitions 2.5 and 2.6 are essentially due to Rhoades et al. [22] whenY = X. In Definition 2.6, if Y = X and T is the identity map on X, the (P, T, x0)-orbitalcompleteness will be denoted simply by (P, x0)-orbitally complete.

Definition 2.7 ([23], see also [8]). Maps P : X → CL(X) and T : X → X are IT-commuting atz ∈ X if TPz ⊆ PTz.

We remark that IT-commuting maps are more general than commuting maps, weaklycommuting maps and weakly compatible maps at a point. Notice that if P is also single-valued, then their IT-commutativity and commutativity are the same.


3. Coincidence and Fixed Point Theorems

Theorem 3.1. Assume that the pair of maps P : Y → CL(X) and T : Y → X is a Suzuki-Zamfirescu hybrid contraction such that P(Y ) ⊆ T(Y ). If there exists an u0 ∈ Y such that T(Y ) is(P, T, u0)-orbitally complete, then P and T have a coincidence point; that is, there exists z ∈ Y suchthat Tz ∈ Pz.

Further, if Y = X, then P and T have a common fixed point provided that P and T are IT-commuting at z and Tz is a fixed point of T .

Proof. Without any loss of generality, we may take r > 0 and T a nonconstant map. Let q =r−1/2. Pick u0 ∈ Y. We construct two sequences {un} ⊆ Y and {yn = Tun} ⊆ T(Y ) in thefollowing manner. Since P(Y ) ⊆ T(Y ), we take an element u1 ∈ Y such that Tu1 ∈ Pu0.Similarly, we choose Tu2 ∈ Pu1 such that

d(Tu1, Tu2) ≤ qH(Pu0, Pu1). (3.1)

If Tu1 = Tu2, then Tu1 ∈ Pu1 and we are done as u1 is a coincidence point of T and P.So we take Tu1 /= Tu2. In an analogous manner, choose Tu3 ∈ Pu2 such that

d(Tu2, Tu3) ≤ qH(Pu1,Pu2). (3.2)

If Tu2 = Tu3, then Tu2 ∈ Pu2 and we are done. So we take Tu2 /= Tu3, andcontinue the process. Inductively, we construct sequences {un} and {Tun} such that Tun+2 ∈Pun+1, Tun+1 /= Tun+2 and

d(Tun+1, Tun+2) ≤ qH(Pun, Pun+1). (3.3)

Now we see that

η(r)d(Tun, Pun) ≤ η(r)d(Tun, Tun+1) ≤ d(Tun, Tun+1). (3.4)

Therefore by the condition (S-Z),

d(yn+1, yn+2

) ≤ qH(Pun, Pun+1)

≤ qr ·max{d(Tun, Tun+1),

d(Tun, Pun) + d(Tun+1, Pun+1)2

,

d(Tun, Pun+1) + d(Tun+1, Pun)2

}

≤ qr ·max

⎧⎪⎪⎨⎪⎪⎩d(yn, yn+1

),d(yn, yn+1

)+ d(yn+1, yn+2

)2

,

12d(yn, yn+2

)

⎫⎪⎪⎬⎪⎪⎭.

(3.5)


This yields

d(yn+1, yn+2

) ≤ r1d(yn, yn+1

), (3.6)

where r1 = qr < 1.Therefore the sequence {yn} is Cauchy in T(Y ). Since T(Y ) is (P, T, u0)-orbitally

complete, it has a limit in T(Y ). Call it u. Let z ∈ T−1u. Then z ∈ Y and u = Tz.Now as in [10], we show that

d(Tz, Px) ≤ rd(Tz, Tx) (3.7)

for any Tx ∈ T(Y ) − {Tz}. Since yn → Tz, there exists a positive integer n0 such that

d(Tz, Tun) ≤ 13d(Tz, Tx) ∀n ≥ n0. (3.8)

Therefore for n ≥ n0,

η(r)d(Tun, Pun) ≤ d(Tun, Pun) ≤ d(Tun, Tun+1)

≤ d(Tun, Tz) + d(Tun+1,Tz)

≤ 23d(Tz, Tx) = d(Tz, Tx) − 1

3d(Tz, Tx)

≤ d(Tz, Tx) − d(Tz, Tun) ≤ d(Tun, Tx).

(3.9)

Therefore by the condition (S-Z),

d(yn+1, Px

) ≤ H(Pun, Px)

≤ r ·max

{d(yn, Tx

),d(yn, Pun

)+ d(Tx, Px)2

,d(yn, Px

)+ d(Tx, Pun)2

}

≤ r ·max

{d(yn, Tx

),d(yn, yn+1

)+ d(Tx, Px)2

,d(yn, Px

)+ d(Tx, yn+1

)2

}.

(3.10)

Making n → ∞,

d(Tz, Px) ≤ r ·max{d(Tz, Tx),

12d(Tx, Px),

d(Tz, Px) + d(Tx, Tz)2

}. (3.11)

This yields (3.7); Tx/= Tz.Next we show that

H(Px, Pz) ≤ r ·max{d(Tx, Tz),

d(Tx, Px) + d(Tz, Pz)2

,d(Tx, Pz) + d(Tz, Px)

2

}(3.12)


for any x ∈ Y. If x = z, then it holds trivially. So we suppose x /= z such that Tx /= Tz. Such achoice is permissible as T is not a constant map.

Therefore using (3.7),

d(Tx, Px) ≤ d(Tx, Tz) + d(Tz, Px)≤ d(Tx, Tz) + rd(Tx, Tz).

(3.13)

Hence

1(1 + r)

d(Tx, Px) ≤ d(Tx, Tz). (3.14)

This implies (3.12), and so

d(yn+1, Pz

) ≤ H(Pun, Pz)

≤ r ·max{d(Tun, Tz),

d(Tun, Pun) + d(Tz, Pz)2

,d(Tun, Pz) + d(Tz, Pun)

2

}

≤ r ·max

{d(yn, Tz

),d(yn, yn+1

)+ d(Tz, Pz)

2,d(yn, Pz

)+ d(Tz, yn+1

)2

}.

(3.15)

Making n → ∞,

d(Tz, Pz) ≤ rd(Tz, Pz). (3.16)

So Tz ∈ Pz, since Pz is closed.Further, if Y = X, TTz = Tz, and P, T are IT-commuting at z, that is, TPz ⊆ PTz, then

Tz ∈ Pz⇒ TTz ∈ TPz ⊆ PTz, and this proves that Tz is a fixed point of P.

We remark that, in general, a pair of continuous commuting maps at their coincidencesneed not have a common fixed point unless T has a fixed point (see, e.g., [6–8]).

Corollary 3.2. Let P : X → CL(X). Assume that there exists r ∈ [0, 1) such that

η(r)d(x, Px) ≤ d(x, y) implies H(Px, Py

) ≤ r ·maxM(P ;x, y

)(3.17)

for all x, y ∈ X. If there exists a u0 ∈ X such that X is (P, u0)-orbitally complete, then P has a fixedpoint.

Proof. It comes from Theorem 3.1 when Y = X and T is the identity map on X.

The following two results are the extensions of Suzuki [9, Theorem 2]. Corollary 3.3also generalizes the results of Kikkawa and Suzuki [10, Theorem 3] and Jungck [24].


Corollary 3.3. Let f, T : Y → X be such that f(Y ) ⊆ T(Y ) and T(Y ) is an (f, T)-orbitally completesubspace of X. Assume that there exists r ∈ [0, 1) such that

η(r)d(Tx, fx

) ≤ d(Tx, Ty) (3.18)

implies

d(fx, fy

) ≤ r ·maxM(f ; Tx, Ty

)(3.19)

for all x, y ∈ Y. Then f and T have a coincidence point; that is, there exists z ∈ Y such that fz = Tz.

Further, if Y = X and f and T commute at z, then f and T have a unique commonfixed point.

Proof. Set Px = {fx} for every x ∈ Y. Then it comes from Theorem 3.1 that there exists z ∈ Ysuch that fz = Tz. Further, if Y = X and f, and T commute at z, then ffz = fTz = Tfz. Also,η(r)d(Tz, fz) = 0 ≤ d(Tz, Tfz), and this implies

d(fz, ffz

) ≤ r ·maxM(f ; Tz, Tfz

)= rd

(fz, ffz

).

(3.20)

This yields that fz is a common fixed point of f and T. The uniqueness of the commonfixed point follows easily.

Corollary 3.4. Let f : X → X be such that X is f-orbitally complete. Assume that there existsr ∈ [0, 1) such that

η(r)d(x, fx

) ≤ d(x, y) implies d(fx, fy

) ≤ r ·maxM(f ;x, y

)(3.21)

for all x, y ∈ X. Then f has a unique fixed point.

Proof. It comes from Corollary 3.2 that f has a fixed point. The uniqueness of the fixed pointfollows easily.

Theorem 3.5. Let P : Y → BN(X) and T : Y → X be such that P(Y ) ⊆ T(Y ) and let T(Y ) be(P, T)-orbitally complete. Assume that there exists r ∈ [0, 1) such that

η(r)ρ(Tx, Px) ≤ d(Tx, Ty) (3.22)

implies

ρ(Px, Py

) ≤ r ·max

{d(Tx, Ty

),ρ(Tx, Px) + ρ

(Ty, Py

)2

,d(Tx, Py

)+ d(Ty, Px

)2

}(3.23)

for all x, y ∈ Y. Then there exists z ∈ Y such that Tz ∈ Pz.


Proof. Choose λ ∈ (0, 1). Define a single-valued map f : Y → X as follows. For each x ∈ Y,let fx be a point of Px, which satisfies

d(Tx, fx

) ≥ rλρ(Tx, Px). (3.24)

Since fx ∈ Px, d(Tx, fx) ≤ ρ(Tx, Px). So (3.22) gives

η(r)d(Tx, fx

) ≤ η(r)ρ(Tx, Px) ≤ d(Tx, Ty), (3.25)

and this implies (3.23). Therefore

d(fx, fy

) ≤ ρ(Px, Py)

≤ r · r−λ ·max

{rλd(Tx, Ty

),rλρ(Tx, Px) + rλρ

(Ty, Py

)2

,

rλd(Tx, Py

)+ rλd

(Ty, Px

)2

}

≤ r1−λ ·max

{d(Tx, Ty

),d(Tx, fx

)+ d(Ty, fy

)2

,d(Tx, fy

)+ d(Ty, fx

)2

}.

(3.26)

This means that Corollary 3.3 applies as

f(Y ) = ∪{fx ∈ Px} ⊆ P(Y ) ⊆ T(Y ). (3.27)

Hence f and T have a coincidence at z ∈ Y. Clearly fz = Tz implies Tz ∈ Pz.

Now we have the following.

Theorem 3.6. Let P : X → BN(X) and let X be P -orbitally complete. Assume that there existsr ∈ [0, 1) such that η(r)ρ(x, Px) ≤ d(x, y) implies

ρ(Px, Py

) ≤ r ·max

{d(x, y),ρ(x, Px) + ρ

(y, Py

)2

,d(x, Py

)+ d(y, Px

)2

}(3.28)

for all x, y ∈ X. Then P has a unique fixed point.

Proof. For λ ∈ (0, 1), define a single-valued map f : X → X as follows. For each x ∈ X, let fxbe a point of Px such that

d(x, fx

) ≥ rλρ(x, Px). (3.29)

Now following the proof technique of Theorem 3.5 and using Corollary 3.4, weconclude that f has a unique fixed point z ∈ X. Clearly z = fz implies that z ∈ Pz.


Now we close this section with the following.

Question 1. Can we replace Assumption (3.17) in Corollary 3.2 by the following:

η(r)d(x, Px) ≤ d(x, y) (3.30)

implies

H(Px, Py

) ≤ r ·max{d(x, y), d(x, Px), d

(y, Py

),

12[d(x, Py

)+ d(y, Px

)]}(3.31)

for all x, y ∈ X?

4. Applications

Throughout this section, we assume that U and V are Banach spaces, W ⊆ U, and D ⊆ V. LetR denote the field of reals, τ : W ×D → W, g, g ′ : W ×D → R, and G,F : W ×D × R → R.Viewing W and D as the state and decision spaces respectively, the problem of dynamicprogramming reduces to the problem of solving the functional equations:

p := supy∈D

{g(x, y)+G(x, y, p

(τ(x, y)))}

, x ∈W, (4.1)

q := supy∈D

{g ′(x, y)+ F(x, y, q

(τ(x, y)))}

, x ∈W. (4.2)

In the multistage process, some functional equations arise in a natural way (cf. Bellman[15] and Bellman and Lee [16]); see also [17–19, 25]). In this section, we study the existence ofthe common solution of the functional equations (4.1), (4.2) arising in dynamic programming.

Let B(W) denote the set of all bounded real-valued functions on W. For an arbitraryh ∈ B(W), define ‖h‖ = supx∈W |h(x)|. Then (B(W), ‖ · ‖) is a Banach space. Suppose that thefollowing conditions hold:

(DP-1) G,F, g and g ′ are bounded.

(DP-2) Let η be defined as in the previous section. There exists r ∈ [0, 1) such that for every(x, y) ∈W ×D, h, k ∈ B(W) and t ∈W,

η(r)|Kh(t) − Jh(t)| ≤ |Jh(t) − Jk(t)| (4.3)

implies

∣∣G(x, y, h(t)) −G(x, y, k(t))∣∣

≤ r ·max{|Jh(t) − Jk(t)|, |Jh(t) −Kh(t)| + |Jk(t) −Kk(t)|

2,

|Jh(t) −Kk(t)| + |Jk(t) −Kh(t)|2

},

(4.4)


where K and J are defined as follows:

Kh(x) = supy∈D

{g(x, y)+G(x, y, h

(τ(x, y)))}

, x ∈W, h ∈ B(W), (∗)

Jh(x) = supy∈D

{g ′(x, y)+ F(x, y, h

(τ(x, y)))}

, x ∈W, h ∈ B(W). (4.5)

(DP-3) For any h ∈ B(W), there exists k ∈ B(W) such that

Kh(x) = Jk(x), x ∈W. (4.6)

(DP-4) There exists h ∈ B(W) such that

Jh(x) = Kh(x) implies JKh(x) = KJh(x). (4.7)

Theorem 4.1. Assume that the conditions (DP-1)–(DP-4) are satisfied. If J(B(W)) is a closed convexsubspace of B(W), then the functional equations (4.1) and (4.2) have a unique common boundedsolution.

Proof. Notice that (B(W), d) is a complete metric space, where d is the metric induced by thesupremum norm on B(W). By (DP-1), J and K are self-maps of B(W). The condition (DP-3) implies that K(B(W)) ⊆ J(B(W)). It follows from (DP-4) that J and K commute at theircoincidence points.

Let λ be an arbitrary positive number and h1, h2 ∈ B(W). Pick x ∈ W and choosey1, y2 ∈ D such that

Khj < g(x, yj

)+G(x, yj , hj

(xj))

+ λ, (4.8)

where xj = τ(x, yj), j = 1, 2.Further,

Kh1(x) ≥ g(x, y2

)+G(x, y2, h1(x2)

), (4.9)

Kh2(x) ≥ g(x, y1

)+G(x, y1, h2(x1)

). (4.10)

Therefore, the first inequality in (DP-2) becomes

η(r)|Kh1(x) − Jh1(x)| ≤ |Jh1(x) − Jh2(x)|, (4.11)


and this together with (4.8) and (4.10) implies

Kh1(x) −Kh2(x) < G(x, y1, h1(x1)

) −G(x, y1, h2(x1))+ λ

≤ ∣∣G(x, y1, h1(x1)) −G(x, y1, h2(x1)

)∣∣ + λ≤ r ·maxM(K; Jh1, Jh2) + λ.

(4.12)

Similarly, (4.8), (4.9), and (4.11) imply

Kh2(x) −Kh1(x) ≤ r ·maxM(K; Jh1, Jh2) + λ. (4.13)

So, from (4.12) and (4.13), we have

|Kh1(x) −Kh2(x)| ≤ r ·maxM(K; Jh1, Jh2) + λ. (4.14)

Since the above inequality is true for any x ∈ W, and λ > 0 is arbitrary, we find from(4.17) that

η(r)d(Kh1, Jh1) ≤ d(Jh1, Jh2) (4.15)

implies

d(Kh1, Kh2) ≤ r ·maxM(K; Jh1, Jh2). (4.16)

Therefore Corollary 3.3 applies, whereinK and J correspond, respectively, to the mapsf and T, Therefore, K and J have a unique common fixed point h∗, that is, h∗(x) is the uniquebounded common solution of the functional equations (4.1) and (4.2).

Corollary 4.2. Suppose that the following conditions hold.

(i) G and g are bounded.

(ii) For η defined earlier (cf. (DP-2) above), there exists r ∈ [0, 1) such that for every (x, y) ∈W ×D, h, k ∈ B(W) and t ∈W,

η(r)|h(t) −Kh(t)| ≤ |h(t) − k(t)| (4.17)

implies

∣∣G(x, y, h(t)) −G(x, y, k(t))∣∣ ≤ r ·maxM(K;h(t), k(t)), (4.18)

where K is defined by (∗). Then the functional equation (4.1) possesses a unique boundedsolution inW.

Proof. It comes from Theorem 4.1 when q = p, F = G, and g = g ′ as the conditions (DP-3) and(DP-4) become redundant in the present context.


Acknowledgments

The authors thank the referees and Professor M. A. Khamsi for their appreciation andsuggestions regarding this work. This research is supported by the Directorate of ResearchDevelopment, Walter Sisulu University.

References


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Research ArticleA Hybrid Projection Algorithm for FindingSolutions of Mixed Equilibrium Problem andVariational Inequality Problem

Filomena Cianciaruso,1 Giuseppe Marino,1Luigi Muglia,1 and Yonghong Yao2

1 Dipartimento di Matematica, Universita della Calabria, 87036 Arcavacata di Rende (CS), Italy2 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Correspondence should be addressed to Giuseppe Marino, [email protected]

Received 3 June 2009; Accepted 16 September 2009


Copyright q 2010 Filomena Cianciaruso et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We propose a modified hybrid projection algorithm to approximate a common fixed point of ak-strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strongconvergence theorem of the proposed method and we obtain, as a particular case, approximationof solutions of systems of two equilibrium problems.

1. Introduction

In this paper, we define an iterative method to approximate a common fixed point of a k-strict pseudocontraction and of two sequences of nonexpansive mappings generated by twosequences of firmly nonexpansive mappings and two nonlinear mappings. Let us recall from[1] that the k-strict pseudocontractions in Hilbert spaces were introduced by Browder andPetryshyn in [2].

Definition 1.1. S : C → C is said to be k-strict pseudocontractive if there exists k ∈ [0, 1[ suchthat

∥∥Sx − Sy∥∥2 ≤ ∥∥x − y∥∥2 + k∥∥(I − S)x − (I − S)y∥∥2

, ∀x, y ∈ C. (1.1)

The iterative approximation problems for nonexpansive mappings, asymptoticallynonexpansive mappings, and asymptotically pseudocontractive mappings were studiedextensively by Browder [3], Goebel and Kirk [4], Kirk [5], Liu [6], Schu [7], and Xu [8, 9]


in the setting of Hilbert spaces or uniformly convex Banach spaces. Although nonexpansivemappings are 0-strict pseudocontractions, iterative methods for k-strict pseudocontractionsare far less developed than those for nonexpansive mappings. The reason, probably, is thatthe second term appearing in the previous definition impedes the convergence analysis foriterative algorithms used to find a fixed point of the k-strict pseudocontraction S. However,k-strict pseudocontractions have more powerful applications than nonexpansive mappingsdo in solving inverse problems. In the recent years the study of iterative methods like Mann’slike methods and CQ-methods has been extensively studied by many authors [1, 10–13] andthe references therein.

If C is a closed and convex subset of a Hilbert space H and F : C × C → R is abi-function we call equilibrium problem

Find x ∈ C s.t. F(x, y) ≥ 0, ∀y ∈ C, (1.2)

and we will indicate the set of solutions with EP(F).If A : C → H is a nonlinear mapping, we can choose F(x, y) = 〈Ax, y − x〉, so an

equilibrium point (i.e., a point of the set EP(F)) is a solution of variational inequality problem(VIP)

Find x ∈ C s.t.⟨Ax, y − x⟩ ≥ 0, ∀y ∈ C. (1.3)

We will indicate with V I(C,A) the set of solutions of VIP.The equilibrium problems, in its various forms, found application in optimization

problems, fixed point problems, convex minimization problems; in other words, equilibriumproblems are a unified model for problems arising in physics, engineering, economics, andso on (see [10]).

As in the case of nonexpansive mappings, also in the case of k-strict pseudocontractionmappings, in the recent years many papers concern the convergence of iterative methodsto a solutions of variational inequality problems or equilibrium problems; see example for,[10, 14–18].

Here we prove a strong convergence theorem of the proposed method and we obtain,as a particular case, approximation of solutions of systems of two equilibrium problems.

2. Preliminaries

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.We denote by PC the metric projection of H onto C. It is well known [19] that

⟨x − PC(x), PC(x) − y

⟩ ≥ 0, ∀x ∈ H and y ∈ C. (2.1)

Lemma 2.1. (see [20]) LetX be a Banach space with weakly sequentially continuous duality mappingJ , and suppose that (xn)n∈N converges weakly to x0 ∈ X, then for any x ∈ X,

lim infn→∞

‖xn − x0‖ ≤ lim infn→∞

‖xn − x‖. (2.2)

Moreover if X is uniformly convex, equality holds in (2.2) if and only if x0 = x.


Recall that a point u ∈ C is a solution of a VIP if and only if

u = PC(I − λA)u ∀λ > 0, that is, u ∈ V I(C,A)⇐⇒ u ∈ Fix(PC(I − λA)), ∀λ > 0.(2.3)

Definition 2.2. An operator A : C → H is said to be α-inverse strongly monotone operator ifthere exists a constant α > 0 such that

⟨Ax −Ay, x − y⟩ ≥ α∥∥Ax −Ay∥∥2 ∀x, y ∈ C. (2.4)

If α = 1 we say that A is firmly nonexpansive. Note that every α-inverse stronglymonotone operator is also 1/α Lipschitz continuous (see [21]).

Lemma 2.3. (see [2]). Let C be a nonempty closed convex subset of a real Hilbert space H and letS : C → C be a k-strict pseudocontractive mapping. Then St := tI + (1 − t)S with t ∈ [k, 1[ is anonexpansive mapping with Fix(St) = Fix(S).

3. Main Theorem

Theorem 3.1. Let C be a closed convex subset of a real Hilbert spaceH. Let

(i) A be an α-inverse strongly monotone mapping of C intoH,

(ii) B a β-inverse strongly monotone mapping of C intoH,

(iii) (Tn)n∈N and (Vn)n∈N two sequences of firlmy nonexpansive mappings from C toH.

Let S : C → C be a k-strict pseudocontraction Fix(S)/= ∅.Set Sk = kI + (1 − k)S and let us define the sequence (xn)n∈N as follows:

x1 ∈ C,C1 = C,

un = Tn(I − rnA)xn

zn = Vn(I − λnB)un,

yn = αnxn + (1 − αn)Skzn,

Cn+1 ={w ∈ Cn :

∥∥yn −w∥∥ ≤ ‖xn −w‖},xn+1 = PCn+1x1, ∀n ∈N,

(3.1)

where

(i) (αn)n∈N ⊂ [0, a] with a < 1;

(ii) (λn)n∈N ⊂ [b, c] ⊂ (0, 2β);

(iii) (rn)n∈N ⊂ [d, e] ⊂ (0, 2α).


Moreover suppose that

(i) F := Fix(S)⋂∩n Fix(Vn(I − λnB))

⋂∩n Fix(Tn(I − rnA))/= ∅;

(ii) (Tn(I − rnA))n∈N pointwise converges in C to an operator R and (Vn(I − λnB))n∈Npointwise converges in C to an operatorW ;

(iii) Fix(W) = ∩n Fix(Vn(I − λnB)) and Fix(R) = ∩n Fix(Tn(I − rnA)).

Then (xn)n∈N strongly converges to x∗ = PFx1.

Proof. We begin to observe that the mappings Tn(I − rnA) and Vn(I − λnB) are nonexpansivefor all n ∈ N since they are compositions of nonexpansive mappings (see [22, page 419]). Asa rule, if p ∈ F

∥∥un − p∥∥2 ≤ ∥∥xn − p∥∥2,

∥∥zn − p∥∥2 ≤ ∥∥un − p∥∥2 ≤ ∥∥xn − p∥∥2.

(3.2)

Now we divide the proof in more steps.

Step 1. Cn is closed and convex for each n ∈ N.Indeed Cn+1 is the intersection of Cn with the half space

{w ∈ H :

⟨w,xn − yn

⟩ ≤ L}, (3.3)

where L = (‖xn‖2 − ‖yn‖2)/2.

Step 2. F ⊆ Cn for each n ∈ N.For each w ∈ F we have

∥∥yn −w∥∥ = ‖αnxn + (1 − αn)Skzn −w‖≤ αn‖xn −w‖ + (1 − αn)‖zn −w‖= αn‖xn −w‖ + (1 − αn)‖Vn(I − λnB)un −w‖≤ αn‖xn −w‖ + (1 − αn)‖un −w‖= αn‖xn −w‖ + (1 − αn)‖Tn(I − rnA)xn −w‖≤ αn‖xn −w‖ + (1 − αn)‖xn −w‖= ‖xn −w‖.

(3.4)

So the claim immediately follows by induction.


Step 3. limn→+∞‖xn −x1‖ exists and (xn)n∈N is asymptotically regular, that is, limn→+∞‖xn+1 −xn‖ = 0.

Since xn = PCnx1, xn+1 = PCn+1x1, and Cn+1 ⊆ Cn, by (2.1) choosing y = xn+1, x = x1 andC = Cn, we have

0 ≤ 〈x1 − xn, xn − xn+1〉= 〈x1 − xn, xn − x1 + x1 − xn+1〉

≤ −‖x1 − xn‖2 + ‖x1 − xn‖‖x1 − xn+1‖,

(3.5)

that is, ‖xn − x1‖ ≤ ‖xn+1 − x1‖.By xn = PCnx1 and F ⊆ Cn, we have

‖x1 − xn‖ ≤ ‖x1 − PFx1‖. (3.6)

Then limn→+∞‖xn − x1‖ exists and (xn)n∈N is bounded. Moreover

‖xn+1 − xn‖2 = ‖xn+1 − x1 + x1 − xn‖2

= ‖xn+1 − x1‖2 + ‖xn − x1‖2 + 2〈xn+1 − x1, x1 − xn〉

= ‖xn+1 − x1‖2 + ‖xn − x1‖2 + 2〈xn+1 − xn, x1 − xn〉 − 2‖xn − x1‖2

≤ ‖xn+1 − x1‖2 − ‖xn − x1‖2 by (3.5),

(3.7)

and consequently limn→+∞‖xn+1 − xn‖ = 0.

Step 4. limn→+∞‖xn − yn‖ = 0 and limn→+∞‖xn − Skzn‖ = 0.By xn+1 ∈ Cn+1, it follows

∥∥yn − xn+1∥∥ ≤ ‖xn − xn+1‖,∥∥yn − xn∥∥ ≤ ∥∥yn − xn+1

∥∥ + ‖xn+1 − xn‖ ≤ 2‖xn+1 − xn‖ −→ 0.(3.8)

Moreover

∥∥yn − xn∥∥ = (1 − αn)‖xn − Skzn‖, (3.9)

and by boundedness of (αn)n∈N, it follows that limn→+∞‖xn − Skzn‖ = 0.


Step 5. limn→+∞‖Bun − Bw‖ = 0, for each w ∈ F.For w ∈ F, we have

∥∥yn −w∥∥2 ≤ αn‖xn −w‖2 + (1 − αn)‖Skzn −w‖2

≤ αn‖xn −w‖2 + (1 − αn)‖zn −w‖2

≤ αn‖xn −w‖2 + (1 − αn)‖Vn(I − λnB)un − Vn(I − λnB)w‖2

≤ αn‖xn −w‖2 + (1 − αn)‖(I − λnB)un − (I − λnB)w‖2

= αn‖xn −w‖2 + (1 − αn)(‖un −w‖2 + λ2

n‖Bun − Bw‖2 − 2λn〈Bun − Bw, un −w〉)

≤ αn‖xn −w‖2 + (1 − αn)(‖un −w‖2 − λn

(2β − λn

)‖Bun − Bw‖2)

≤ ‖xn −w‖2 + (1 − αn)λn(λn − 2β

)‖Bun − Bw‖2.

(3.10)

Consequently

(1 − αn)λn(2β − λn

)‖Bun − Bw‖2 ≤ ‖xn −w‖2 − ∥∥yn −w∥∥2

=(‖xn −w‖ − ∥∥yn −w∥∥)(‖xn −w‖ + ∥∥yn −w∥∥)≤ (∥∥xn − yn∥∥)(‖xn −w‖ + ∥∥yn −w∥∥),

(3.11)

and by Step 4, the assumptions on (αn)n∈N and (λn)n∈N, we obtain the claim of Step 5.

Step 6. limn→+∞‖un − zn‖ = 0.Since Vn is firmly nonexpansive, for any w ∈ F, we have

‖zn −w‖2 ≤ 〈(I − λnB)un − (I − λnB)w, zn −w〉

=14

{‖(I − λnB)un − (I − λnB)w + (zn −w)‖2

−‖(I − λnB)un − (I − λnB)w − (zn −w)‖2}

≤ 14

{‖un −w‖2 − λn

(2β − λn

)‖Bun − Bw‖2 + ‖zn −w‖2

−‖un − zn − λn(Bun − Bw)‖2}

≤ 14

{‖un −w‖2 + ‖zn −w‖2 − ‖un − zn − λn(Bun − Bw)‖2

}

=14

{‖un −w‖2 + ‖zn −w‖2 − ‖un − zn‖2

+2λn〈un − zn, Bun − Bw〉 − λ2n‖Bun − Bw‖2

}

(3.12)


which implies

3‖zn −w‖2 ≤ ‖un −w‖2 − ‖un − zn‖2 + 2λn〈un − zn, Bun − Bw〉

≤ ‖xn −w‖2 − ‖un − zn‖2 + 2λn‖un − zn‖ ‖Bun − Bw‖.(3.13)

Consequently

∥∥yn −w∥∥2 ≤ αn‖xn −w‖2 + (1 − αn)‖zn −w‖2

≤ ‖xn −w‖2 − (1 − αn)‖un − zn‖2 + 2(1 − αn)λn‖un − zn‖‖Bun − Bw‖(3.14)

which implies

(1 − αn)‖un − zn‖2 ≤ ‖xn −w‖2 − ∥∥yn −w∥∥2 + 2(1 − αn)λn‖un − zn‖ ‖Bun − Bw‖≤ (‖xn −w‖ − ∥∥yn −w∥∥)(‖xn −w‖ + ∥∥yn −w∥∥)+ 2(1 − αn)λn‖un − zn‖ ‖Bun − Bw‖

≤ (∥∥xn − yn∥∥)(‖xn −w‖ + ∥∥yn −w∥∥) + 2(1 − αn)λn‖un − zn‖ ‖Bun − Bw‖.(3.15)

By the assumptions on (αn)n∈N, Steps 4 and 6, and the boundedness of (xn)n∈N (yn)n∈N and(un)n∈N the claim follows.

Step 7. limn→+∞‖xn − un‖ = 0 and limn→+∞‖xn − Skxn‖ = 0.Since Tn is firmly nonexpansive, for each p ∈ ∩n Fix(Tn(I − rn)A), we have

∥∥un − p∥∥2 =∥∥Tn(I − rnA)xn − Tn(I − rnA)p

∥∥2

≤ ⟨un − p, (I − rnA)xn − (I − rnA)p⟩

=12

(∥∥(I − rnA)xn − (I − rnA)p∥∥2 +

∥∥un − p∥∥2

−∥∥(I − rnA)xn − (I − rnA)p − (un − p)∥∥2)

=12

(∥∥xn − p∥∥2 − rn(2α − rn)∥∥Axn −Ap∥∥2 +

∥∥un − p∥∥2

−∥∥xn − un − rn(Axn −Ap)∥∥2)

≤ 12

(∥∥xn − p∥∥2 +∥∥un − p∥∥2 − ‖xn − un‖2

−r2n

∥∥Axn −Ap∥∥2+ 2rn

⟨xn − un,Axn −Ap

⟩),

(3.16)


and consequently

∥∥un − p∥∥2 ≤(∥∥xn − p∥∥2 − ‖xn − un‖2 + 2rn‖xn − un‖

∥∥Axn −Ap∥∥). (3.17)

Then, for each w ∈ F, we have

∥∥yn −w∥∥2 ≤ αn‖xn −w‖2 + (1 − αn)‖un −w‖2

≤ ‖xn −w‖2 − (1 − αn)‖xn − un‖2

+ 2(1 − αn)rn‖xn − un‖ ‖Axn −Aw‖ by (3.17),

(3.18)

consequently

(1 − αn)‖xn − un‖2 ≤ ‖xn −w‖2 − ∥∥yn −w∥∥2 + 2(1 − αn)rn‖xn − un‖‖Axn −Aw‖≤ ∥∥xn − yn∥∥(‖xn −w‖ + ∥∥yn −w∥∥) + 2(1 − αn)rn‖xn − un‖‖Axn −Aw‖,

(3.19)

and by the assumptions on (αn)n∈N, Step 4 and the boundedness of (xn)n∈N and (yn)n∈N itfollows that ‖xn − un‖ → 0 as n → +∞. By Step 6 we note that also ‖xn − zn‖ → 0.

Finally

‖xn − Skxn‖ ≤ ‖xn − Skzn‖ + ‖Skzn − Skxn‖≤ ‖xn − Skzn‖ + ‖zn − xn‖≤ ‖xn − Skzn‖ + ‖zn − un‖ + ‖un − xn‖,

(3.20)

and by previous steps, it follows that ‖xn − Skxn‖ → 0 as n → +∞.

Step 8. The set of weak cluster points of (xn)n∈N is contained in F.We will use three times the Opial’s Lemma 2.1.Let p be a weak cluster point of (xn)n∈N and let (xnj )j∈N be a subsequence of (xn)n∈N

such that xnj ⇀ p.We prove that p ∈ Fix(S) = Fix(Sk). We suppose for absurd that p /=Skp. By Opial’s

Lemma 2.1 and ‖xn − Skxn‖ → 0 as n → ∞, we obtain

lim infj→+∞

∥∥∥xnj − p∥∥∥ < lim inf

j→+∞

∥∥∥xnj − Skp∥∥∥

= lim infj→+∞

∥∥∥xnj − Skxnj − Skxnj − Skp∥∥∥ ≤ lim inf

j→+∞

[∥∥∥xnj − Skxnj∥∥∥ +∥∥∥Skxnj − Skp

∥∥∥]

= lim infj→+∞

∥∥∥xnj − p∥∥∥

(3.21)



Since Fix(R) = ∩n Fix(Tn(I − rnA)) it is enough to prove that p ∈ Fix(R). Now if p /=Rpwe note that

lim infj→+∞

∥∥∥xnj − p∥∥∥ < lim inf

j→+∞

∥∥∥xnj − Rp∥∥∥

≤ lim infj→+∞

[∥∥∥xnj − Tnj(I − rnjA

)xnj

∥∥∥

+∥∥∥Tnj

(I − rnjA

)xnj − Tnj

(I − rnjA

)p∥∥∥ +∥∥∥Tnj

(I − rnjA

)p − Rp

∥∥∥]

≤ lim infj→+∞

[∥∥∥xnj − unj∥∥∥ +∥∥∥xnj − p

∥∥∥ +∥∥∥Tnj

(I − rnjA

)p − Rp

∥∥∥]

= lim infj→+∞

∥∥∥xnj − p∥∥∥.

(3.22)

This leads to a contraddiction again. By the hypotheses and Step 7 the claim follows. By thesame idea and using Step 6, we prove that p ∈ Fix(W) = ∩n Fix(Vn(I − λnB)).

Step 9. xn → x∗ = PFx1.Since x∗ = PFx1 ∈ Cn and xn = PCnx1, we have

‖x1 − xn‖ ≤ ‖x1 − x∗‖. (3.23)

Let (xnj )j∈N be a subsequence of (xn)n∈N such that xnj ⇀ p. By Step 8, p ∈ F. Thus

‖x1 − x∗‖ ≤∥∥x1 − p

∥∥ ≤ lim infj→+∞

∥∥∥x1 − xnj∥∥∥

≤ lim supj→+∞

∥∥∥x1 − xnj∥∥∥ ≤ ‖x1 − x∗‖.

(3.24)

Therefore we have

‖x1 − x∗‖ =∥∥x1 − p

∥∥ = limj→+∞

∥∥∥x1 − xnj∥∥∥. (3.25)

Since H has the Kadec-Klee property, then xnj → p as j → +∞.Moreover, by ‖x1 − x∗‖ = ‖x1 − p‖ and by the uniqueness of the projection PFx1, it

follows that p = x∗ = PFx1.Thence every subsequence (xnj )j∈N converges to x∗ as j → +∞ and consequently

xn → x∗, as n → +∞.

Remark 3.2. Let us observe that one can choose (Tn)n∈N and (Vn)n∈N as sequences of γn-inverse strongly monotone operators and ηn-inverse strongly monotone operators providedγn ≥ 1, ηn ≥ 1 for all n ∈ N.


The hypotheses (ii) and (iii) in the main Theorem 3.1 seem very strong but, in thesequel, we furnish two cases in which (ii) and (iii) are satisfied.

Let us remember that the metric projection on a convex closed set PC is a firmlynonexpansive mapping (see [19]) so we claim that have the following proposition.

Proposition 3.3. If (rn)n∈N ⊂ (0,∞) is such that limnrn = r > 0 and A an α-inverse stronglymonotone, then PC(I − rnA) realizes conditions (ii) and (iii) with R = PC(I − rA).

Proof. To prove (ii) we note that for each x ∈ C,

‖PC(I − rnA)x − PC(I − rA)x‖ ≤ ‖(I − rnA)x − (I − rA)x‖ ≤ |rn − r|‖Ax‖. (3.26)

Moreover, (iii) follows directly by (2.2).

Now we consider the mixed equilibrium problem

Find x ∈ C : f(x, y)+ h(x, y)+⟨Ax, y − x⟩ ≥ 0, ∀y ∈ C. (3.27)

In the sequel we will indicate with MEP(f, h,A) the set of solution of our mixed equilibriumproblem. If A = 0 we denote MEP(f, h, 0) with MEP(f, h).

We notice that for h = 0 andA = 0 the problem is the well-known equilibrium problem[23–25]. If h = 0 and A is an α-inverse strongly monotone operator we have the equilibriumproblems studied firstly in [26] and then in [18, 22, 27]. If h(x, y) = ϕ(y) −ϕ(x) and A = 0 werefound the mixed equilibrium problem studied in [16, 28, 29].

Definition 3.4. A bi-function g : C×C → R is monotone if g(x, y)+g(y, x) ≤ 0 for all x, y ∈ C.A function G : C → R is upper hemicontinuous if

lim supt→ 0

G(tx + (1 − t)y) ≤ G(y). (3.28)

Next lemma examines the case in which A = 0.

Lemma 3.5. Let C be a convex closed subset of a Hilbert spaceH.Let f : C × C → R be a bi-function such that

(f1) f(x, x) = 0 for all x ∈ C;(f2) f is monotone and upper hemicontinuous in the first variable;

(f3) f is lower semicontinuous and convex in the second variable.

Let h : C × C → R be a bi-function such that

(h1) h(x, x) = 0 for all x ∈ C;(h2) h is monotone and weakly upper semicontinuous in the first variable;

(h3) h is convex in the second variable.


Moreover let us suppose that

(H) for fixed r > 0 and x ∈ C, there exists a bounded set K ⊂ C and a ∈ K such that for allz ∈ C \K, −f(a, z) + h(z, a) + (1/r)〈a − z, z − x〉 < 0,

for r > 0 and x ∈ H let Tr : H → C be a mapping defined by

Trx ={z ∈ C : f

(z, y)+ h(z, y)+

1r

⟨y − z, z − x⟩ ≥ 0, ∀y ∈ C

}, (3.29)

called resolvent of f and h.Then

(1) Trx /= ∅;(2) Trx is a single value;

(3) Tr is firmly nonexpansive;

(4) MEP(f, h) = Fix(Tr) and it is closed and convex.

Proof. Let x0 ∈ H. For any y ∈ C define

Gr,x0y ={z ∈ C : −f(y, z) + h(z, y) + 1

r

⟨y − z, z − x⟩ ≥ 0

}. (3.30)

We will prove that, by KKM’s lemma, ∩y∈CGr,x0y is nonempty.First of all we claim that Gr,x0 is a KKM’s map. In fact if there exists {y1, . . . , yN} ⊂ C

such that y =∑

i αiyi (with∑

i αi = 1) does not appartiene to Gr,x0yi for any i = 1, . . . ,N then

−f(yi, y) + h(y, yi) + 1r

⟨yi − y, y − x0

⟩< 0, ∀i. (3.31)

By the convexity of f and h and the monotonicity of f , we obtain that

0 = f(y, y)+ h(y, y)+

1r

⟨y − y, y − x0

⟩

≤∑i

αif(y, yi

)+∑i

αih(y, yi

)+

1r

∑i

αi⟨yi − y, y − x0

⟩

≤ −∑i

αif(yi, y

)+∑i

αih(y, yi

)+

1r

∑i

αi⟨yi − y, y − x0

⟩

=∑i

αi

[−f(yi, y) + h(y, yi) + 1

r

⟨yi − y, y − x0

⟩]< 0,

(3.32)

that is absurd.


Now we prove that Gr,x0

w= Gr,x0 . We recall that, by the weak lower semicontinuity of

‖ · ‖2, the relation

lim supm

⟨y − zm, zm − x0

⟩ ≤ ⟨y − z, z − x0⟩

(3.33)

holds. Let z ∈ Gr,x0yw

and let (zm)m be a sequence in Gr,x0y such that zm ⇀ z.We want to prove that

−f(y, z) + h(z, y) + 1r

⟨y − z, z − x0

⟩ ≥ 0. (3.34)

Since f is lower semicontinuous and convex in the second variable and h is weakly uppersemicontinuous in the first variable, then

0 ≤ lim supm

[−f(y, zm) + h(zm, y) + 1

r

⟨y − z, z − x0

⟩]

≤ lim supm

(−f(y, zm)) + lim supm

h(zm, y

)+

1r

lim supm

⟨y − z, z − x0

⟩

≤ −lim infm

f(y, zm

)+ lim sup

mh(zm, y

)+

1r

lim supm

⟨y − z, z − x0

⟩

≤ −f(y, z) + h(z, y) + 1r

⟨y − z, z − x0

⟩.

(3.35)

Now we observe that Gr,x0yw= Gr,x0y is weakly compact for at least a point y ∈ C. In

fact by hypothesis (H) there exist a bounded K ⊂ C and a ∈ K, such that for all z ∈ C \ Kit results z/∈Gr,x0a. Then Gr,x0 a ⊂ K, that is, it is bounded. It follows that Gr,x0a is weaklycompact. Then by KKM’s lemma ∩y∈CGr,x0y is nonempty. However if z ∈ ∩y∈CGr,x0 then

−f(y, z) + h(z, y) + 1r

⟨y − z, z − x0

⟩ ≥ 0, ∀y ∈ C. (3.36)

As in [24, Lemma 3], since f is upper hemicontinuous and convex in the first variable andmonotone, we obtain that (3.36) is equivalent to claim that z is such that

f(z, y)+ h(z, y)+

1r

⟨y − z, z − x0

⟩ ≥ 0, ∀y ∈ C, (3.37)

that is, z ∈ Tr(x0). This prove (1). To prove (2) and (3) we consider z1 ∈ Trx1 and z2 ∈ Trx2.They satisfy the relations

f(z1, z2) + h(z1, z2) +1r〈z2 − z1, z1 − x1〉 ≥ 0,

f(z2, z1) + h(z2, z1) +1r〈z1 − z2, z2 − x2〉 ≥ 0.

(3.38)


By the monotonicity of f and h, summing up both the terms,

0 ≤ 1r[〈z2 − z1, z1 − x1〉 − 〈z2 − z1, z2 − x2〉]

=1r[〈z2 − z1, z1 − x1 − z2 + x2〉]

=1r

[−‖z2 − z1‖2 + 〈z2 − z1, x2 − x1〉

](3.39)

so we conclude

‖z2 − z1‖2 ≤ 〈z2 − z1, x2 − x1〉 (3.40)

that means simultaneously that z1 = z2 if x1 = x2 and Tr is firmly nonexpansive.To prove (4), it is enough to follow (iii) and (iv) in [25, Lemma 2.12].

Remark 3.6. We note that if h = 0, our lemma reduces to [25, Lemma 2.12]. The coercivitycondition (H) is fulfilled.

Moreover our lemma is more general than [16, Lemma 2.2]. In fact

(i) our hypotheses on f are weaker (f weak upper semicontinuous implies f upperhemicontinuous);

(ii) if ϕ satisfies the condition in Lemma 2.2 , choosing h(x, y) = ϕ(y) − ϕ(x) one hasthat h is concave and upper semicontinuous in the first variable and convex andlower semicontinous in the second variable;

(iii) the coercivity condition (H) by the equivalence of (3.36) and (3.37) is the same.

Lemma 3.7. Let us suppose that (f1)–(f3), (h1)–(h3) and (H) hold. Let x, y ∈ H, r1, r2 > 0. Then

∥∥Tr2y − Tr1x∥∥ ≤ ∥∥y − x∥∥ +

∣∣∣∣r2 − r1

r2

∣∣∣∣∥∥Tr2y − y

∥∥. (3.41)

Proof. By Lemma 3.5, defining u1 = Tr1x and u2 := Tr2y, we know that

f(u2, z) + h(u2, z) +1r2〈z − u2, u2 − y〉 ≥ 0, ∀z ∈ C,

f(u1, z) + h(u1, z) +1r1〈z − u1, u1 − x〉 ≥ 0, ∀z ∈ C.

(3.42)

In particular,

f(u2, u1) + h(u2, u1) +1r2

⟨u1 − u2, u2 − y

⟩ ≥ 0,

f(u1, u2) + h(u1, u2) +1r1〈u2 − u1, u1 − x〉 ≥ 0.

(3.43)


Hence, summing up this two inequalities and using the monotonicity of f and h,

⟨u2 − u1,

u1 − xr1

− u2 − yr2

⟩≥ 0. (3.44)

We derive from (3.44) that

⟨u2 − u1, u1 − u2 − x + u2 − r1

r2

(u2 − y

)⟩ ≥ 0, (3.45)

and so

−‖u2 − u1‖2 +⟨u2 − u1,

(u2 − y

)(1 − r1

r2

)+(y − x)

⟩≥ 0. (3.46)

Then,

‖u2 − u1‖2 ≤ ‖u2 − u1‖(∥∥y − x∥∥ +

∣∣∣∣1 − r1

r2

∣∣∣∣∥∥u2 − y

∥∥), (3.47)

and thus the claim holds.

Proposition 3.8. Let us suppose that f and h are two bi-functions satisfying the hypotheses ofLemma 3.5. Let Tr be the resolvent of f and h. Let A be an α-inverse strongly monotone operator.Let us suppose that (rn)n∈N ⊂ (0,∞) is such that limnrn = r > 0. Then Trn(I − rnA) realize (ii) and(iii) in Theorem 3.1.

Proof. Let x be in a bounded closed convex subset K of C. To prove (i) it is enough to observethat by Lemma 3.7

‖Trn(I − rnA)x − Tr(I − rA)x‖ ≤ |rn − r|‖Ax‖ + |rn − r|r‖Tr(I − rA)x − (I − rA)x‖. (3.48)

When n → ∞, by boundedness of the terms that do not depend on n, we obtain (ii).To prove (iii) let W = Tr(I − rA) the pointwise limit of Trn(I − rnA). It is necessary

to prove only that Fix(W) ⊂ ∩n Fix(Trn(I − rnA)). Let x ∈ Fix(W). We want to prove thatx ∈ MEP(f, h,A). Let wn = Trn(I − rnA)x. Thus, by definition of Trn , wn is the unique pointsuch that

f(wn, y

)+ h(wn, y

)+

1rn

⟨y −wn,wn − (I − rnA)x

⟩ ≥ 0, ∀y. (3.49)

By monotonicity of f and h this implies

h(wn, y

)+

1rn

⟨y −wn,wn − (I − rnA)x

⟩ ≥ f(y,wn

). (3.50)


Passing to the limit on n, by (f3) and (h2) we obtain

h(x, y)+⟨y − x,Ax⟩ ≥ f(y, x), ∀y. (3.51)

Let now u = ty + (1 − t)x with t ∈ [0, 1]. Then by the convexity of f and h

0 = f(u, u) + h(u, u) ≤ t[f(u, y) + h(u, y)] + (1 − t)[f(u, x) + h(u, x)]≤ t[f(u, y) + h(u, y)] + 〈u − x,Ax〉= t[f(u, y)+ h(u, y)+⟨y − x,Ax⟩].

(3.52)

Passing t → 0+ we obtain by (f1) and (h1)

f(x, y)+ h(x, y)+⟨Ax, y − x⟩ ≥ 0. (3.53)

That is, x ∈MEP(f, h,A). At this point we observe that from the definitions of MEP(f, h,A)and Trn , one has MEP(f, h,A) = Fix(Trn(I − rnA)).

By Propositions 3.3 and 3.8 we can exhibit iterative methods to approximate fixedpoints of the k-strict pseudo contraction that are also

(1) solution of a system of two variational inequalities VI(C,A) and VI(C,B) (Vn = Tn =PC);

(2) solution of a system of two mixed equilibrium problems (Tn = Trn and Vn = Tλn);

(3) solution of a mixed equilibrium problem and a variational inequality (Tn = Trn andVn = PC).

However when the properties of the mapping Tn and Vn are well known, one canprove convergence theorems like Theorem 3.1 without use of Opial’s lemma.

In next theorem our purpose is to prove a strong convergence theorem to approximatea fixed point of S that is also a solution of a mixed equilibrium problem and a solution of avariational inequality V I(C,B). One can note that we relax the hypotheses on the convergenceof the sequences (rn)n∈N and (λn)n∈N.

Theorem 3.9. Let C be a closed convex subset of a real Hilbert spaceH, let f, h : C ×C → R be twobi-functions satisfying (f1)–(f3),(h1)–(h3), and (H). Let S : C → C be a k-strict pseudocontraction.

LetA be an α-inverse strongly monotone mapping ofC intoH and let B be a β-inverse stronglymonotone mapping of C intoH.

Let us suppose that F = Fix(S) ∩MEP(f, h,A) ∩ V I(C,B)/= ∅.


Set Sk = kI + (1 − k)S, one defines the sequence (xn)n∈N as follows:

x1 ∈ C,C1 = C,

f(un, y

)+ h(un, y

)+

1rn

⟨y − un, un − xn

⟩+⟨Axn, y − un

⟩ ≥ 0,

zn = PC(I − λnB)un,yn = αnxn + (1 − αn)Skzn,

Cn+1 ={w ∈ Cn :

∥∥yn −w∥∥ ≤ ‖xn −w‖},xn+1 = PCn+1x1, ∀n ∈N,

(3.54)

where

(i) (αn)n∈N ⊂ [0, a] with a < 1;

(ii) (λn)n∈N ⊂ [b, c] ⊂ (0, 2β);

(iii) (rn)n∈N ⊂ [d, e] ⊂ (0, 2α).

Then (xn)n∈N strongly converges to x∗ = PFx1.

Proof. First of all we observe that by Lemma 3.5 we have that un = Trn(I − rnA)xn. We canfollow the proof of Theorem 3.1 from Steps 1–7. We prove only the following.

Step 10. The set of weak cluster points of (xn)n∈N is contained in F.Let p be a cluster point of xn; we begin to prove that p ∈MEP(f, h,A). We know that

f(un, y

)+ h(un, y

)+⟨Axn, y − un

⟩+

1rn


⟩ ≥ 0, ∀y ∈ C, (3.55)

and by (f2)

h(un, y

)+⟨Axn, y − un

⟩+

1rn


⟩ ≥ f(y, un), ∀y ∈ C. (3.56)

Let (xnj )j∈N be a subsequence of (xn)n∈N weakly convergent to p, then by Step 7 unj ⇀ p asj → +∞. Let ρt := ty + (1 − t)p, t ∈]0, 1]. Then by (3.56)

⟨ρt − unj , Aρt

⟩=⟨ρt − unj , Aρt −Axnj

⟩+⟨Axnj , ρt − unj

⟩

≥⟨ρt − unj , Aρt −Axnj

⟩+ f(y, unj

)− h(unj , y

)− 1rnj

⟨y − unj , unj − xnj

⟩

=⟨ρt − unj , Aρt −Aunj

⟩+⟨ρt − unj , Aunj −Axnj

⟩

+ f(y, unj

)− h(unj , y

)− 1rnj


⟩

≥⟨ρt − unj , Aunj −Axnj

⟩+ f(y, unj

)− h(unj , y

)− 1rnj


⟩.

(3.57)


Since A is Lipschitz continuous and ‖unj −xnj‖ → 0 as j → +∞, we have ‖Aunj −Axnj‖ → 0as j → +∞.

By condition (f3), for x ∈ H fixed, the function f(x, ·) is lower semicontinuos andconvex, and thus weakly lower semicontinuous [30].

Since ‖xn−un‖ → 0, as n → ∞ and by the assumption on rn we obtain (unj−xnj )/rnj →0. Then we obtain by (h2)

⟨ρt − p,Aρt

⟩ ≥ f(y, p) − h(p, y). (3.58)

Using (f1), (f3), (h1), (h3) we obtain

0 = f(ρt, ρt

)+ h(ρt, ρt

) ≤ tf(ρt, y) + (1 − t)f(ρt, p) + th(ρt, y) + (1 − t)h(ρt, p)≤ tf(ρt, y) + th(ρt, y) + (1 − t)(f(ρt, p) − h(p, ρt))≤ tf(ρt, y) + th(ρt, y) + (1 − t)⟨ρt − p,Aρt⟩= t(f(ρt, y

)+ h(ρt, y

)+ (1 − t)⟨y − p,Aρt⟩).

(3.59)

Consequently

f(ρt, y

)+ h(ρt, y

)+ (1 − t)⟨y − p,Aρt⟩ ≥ 0 (3.60)

by (f2) and (h2), as t → 0, we obtain p ∈MEP(f, h,A).Now we prove that p ∈ V I(C,B).We define the maximal monotone operator

Tx =

{Bx +NCx, if x ∈ C,∅, se x /∈C,

(3.61)

where NCx is the normal cone to C at x, that is,

NCx = {w ∈ H : 〈x − u,w〉 ≥ 0, ∀u ∈ C}. (3.62)

Since zn ∈ C, by the definition of NC we have

⟨x − zn, y − Bx

⟩ ≥ 0. (3.63)

But zn = PC(I − λnB)un, then

〈x − zn, zn − (I − λnB)un〉 ≥ 0, (3.64)

and hence

⟨x − zn, zn − un

λn+ Bun

⟩≥ 0. (3.65)


By (3.63), (3.65), and by the β-inverse monotonicity of B, we obtain

⟨x − znj , y

⟩≥⟨x − znj , Bx

⟩

≥⟨x − znj , Bx

⟩−⟨x − znj ,

znj − unjλnj

+ Bunj

⟩

=⟨x − znj , Bx − Bznj

⟩+⟨x − znj , Bznj − Bunj

⟩

−⟨x − znj ,

znj − unjλnj

⟩.

(3.66)

By ‖xn − zn‖ → 0 as n → +∞ (immediately consequence of Steps 6 and 7), it follows thatznj ⇀ p as j → +∞. Then

⟨x − p, y⟩ ≥ 0, (3.67)

moreover, since T is a maximal operator, 0 ∈ Tp, that is, p ∈ V I(C,B).Finally, to prove that p ∈ Fix(S) = Fix(Sk) we follow Step 8 as in Theorem 3.1.Since also Step 9 can be followed as in Theorem 3.1, we obtain the claim.

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Research ArticleA New System of Generalized Nonlinear MixedVariational Inclusions in Banach Spaces

Jian Wen Peng

College of Mathematics and Computer Science, Chongqing Normal University,Chongqing Sichuan 400047, China

Correspondence should be addressed to Jian Wen Peng, [email protected]

Received 5 July 2009; Accepted 14 September 2009


Copyright q 2010 Jian Wen Peng. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We introduce and study a new system of generalized nonlinear mixed variational inclusions in realq-uniformly smooth Banach spaces. We prove the existence and uniqueness of solution and theconvergence of some new n-step iterative algorithms with or without mixed errors for this systemof generalized nonlinear mixed variational inclusions. The results in this paper unify, extend, andimprove some known results in literature.

1. Introduction

Variational inclusion problems are among the most interesting and intensively studied classesof mathematical problems and have wide applications in the fields of optimization andcontrol, economics and transportation equilibrium, as well as engineering science. For thepast years, many existence results and iterative algorithms for various variational inequalityand variational inclusion problems have been studied. For details, see [1–25] and thereferences therein.

Recently, some new and interesting problems, which are called to be system ofvariational inequality problems, were introduced and studied. Pang [1], Cohen and Chaplais[2], Bianchi [3], and Ansari and Yao [4] considered a system of scalar variational inequalities,and Pang showed that the traffic equilibrium problem, the spatial equilibrium problem,the Nash equilibrium, and the general equilibrium programming problem can be modeledas a system of variational inequalities. Ansari et al. [5] considered a system of vectorvariational inequalities and obtained its existence results. Allevi et al. [6] considered a systemof generalized vector variational inequalities and established some existence results withrelative pseudomonoyonicity. Kassay and Kolumban [7] introduced a system of variationalinequalities and proved an existence theorem by the Ky Fan lemma. Kassay et al. [8] studied


Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem. Peng [9], Peng and Yang [10] introduced a system ofquasivariational inequality problems and proved its existence theorem by maximal elementtheorems. Verma [11–15] introduced and studied some systems of variational inequalitiesand developed some iterative algorithms for approximating the solutions of system ofvariational inequalities in Hilbert spaces. J. K. Kim and D. S. Kim [16] introduced and studieda new system of generalized nonlinear quasivariational inequalities in Hilbert spaces. Cho etal. [17] introduced and studied a new system of nonlinear variational inequalities in Hilbertspaces. They proved some existence and uniqueness theorems of solutions for the system ofnonlinear variational inequalities.

As generalizations of system of variational inequalities, Agarwal et al. [18] introduceda system of generalized nonlinear mixed quasivariational inclusions and investigatedthe sensitivity analysis of solutions for this system of generalized nonlinear mixedquasivariational inclusions in Hilbert spaces. Peng and Zhu [19] introduce a new systemof generalized nonlinear mixed quasivariational inclusions in q-uniformly smooth Banachspaces and prove the existence and uniqueness of solutions and the convergence of severalnew two-step iterative algorithms with or without errors for this system of generalizednonlinear mixed quasivariational inclusions. Kazmi and Bhat [20] introduced a systemof nonlinear variational-like inclusions and proved the existence of solutions and theconvergence of a new iterative algorithm for this system of nonlinear variational-likeinclusions. Fang and Huang [21], Verma [22], and Fang et al. [23] introduced and studieda new system of variational inclusions involving H-monotone operators, A-monotoneoperators and (H,η)-monotone operators, respectively. Yan et al. [24] introduced and studieda system of set-valued variational inclusions which is more general than the model in [21].Peng and Zhu [25] introduced and studied a system of generalized mixed quasivariationalinclusions involving (H,η)-monotone operators which contains those mathematical modelsin [11–16, 21–24] as special cases.

Inspired and motivated by the results in [1–25], the purpose of this paper is tointroduce and study a new system of generalized nonlinear mixed quasivariational inclusionswhich contains some classes of system of variational inclusions and systems of variationalinequalities in the literature as special cases. Using the resolvent technique for them-accretivemappings, we prove the existence and uniqueness of solutions for this system of generalizednonlinear mixed quasivariational inclusions. We also prove the convergence of some new n-step iterative sequences with or without mixed errors to approximation the solution for thissystem of generalized nonlinear mixed quasivariational inclusions. The results in this paperunifies, extends, and improves some results in [11–16, 19] in several aspects.

2. Preliminaries

Throughout this paper we suppose that E is a real Banach space with dual space, norm andthe generalized dual pair denoted by E∗, ‖ · ‖ and 〈·, ·〉, respectively, 2E is the family of all thenonempty subsets of E, dom(M) denotes the domain of the set-valued map M : E → 2E,and the generalized duality mapping Jq : E → 2E

∗is defined by

Jq(x) ={f∗ ∈ E∗ :

⟨x, f∗

⟩=∥∥f∗∥∥ · ‖x‖,∥∥f∗∥∥ = ‖x‖q−1

}, ∀x ∈ E, (2.1)


where q > 1 is a constant. In particular, J2 is the usual normalized duality mapping. It isknown that, in general, Jq(x) = ‖x‖2J2(x), for all x /= 0, and Jq is single-valued if E∗ is strictlyconvex.

The modulus of smoothness of E is the function ρE : [0,∞) → [0,∞) defined by

ρE(t) = sup{

12(∥∥x + y

∥∥ +∥∥x − y∥∥) − 1 : ‖x‖ ≤ 1,

∥∥y∥∥ ≤ t}. (2.2)

A Banach space E is called uniformly smooth if

limt→ 0

ρE(t)t

= 0. (2.3)

E is called q-uniformly smooth if there exists a constant c > 0, such that

ρE(t) ≤ ctq, q > 1. (2.4)

Note that Jq is single-valued if E is uniformly smooth.Xu [26] and Xu and Roach [27] proved the following result.

Lemma 2.1. Let E be a real uniformly smooth Banach space. Then, E is q-uniformly smooth if andonly if there exists a constant cq > 0, such that for all x, y ∈ E,

∥∥x + y∥∥q ≤ ‖x‖q + q⟨y, Jq(x)⟩ + cq∥∥y∥∥q. (2.5)

Definition 2.2 (see [28]). Let M : dom(M) ⊆ E → 2E be a multivalued mapping:

(i) M is said to be accretive if, for any x, y ∈ dom(M), u ∈ M(x), v ∈ M(y), thereexists jq(x − y) ∈ Jq(x − y) such that

⟨u − v, jq

(x − y)⟩ ≥ 0; (2.6)

(ii) M is said to be m-accretive if M is accretive and (I + ρM)(dom(M)) = E holds forevery (equivalently, for some) ρ > 0, where I is the identity operator on E.

Remark 2.3. It is well known that, if E =H is a Hilbert space, then M : dom(M) ⊆ E → 2E ism-accretive if and only if M is maximal monotone (see, e.g., [29]).

We recall some definitions needed later.

Definition 2.4 (see [28]). Let the multivalued mapping M : dom(M) ⊆ E → 2E be m-accretive, for a constant ρ > 0, the mapping RM

ρ : E → dom(M) which is defined by

RMρ (u) =

(I + ρM

)−1(u), u ∈ E, (2.7)

is called the resolvent operator associated with M and ρ.


Remark 2.5. It is well known that RMρ is single-valued and nonexpansive mapping (see [28]).

Definition 2.6. Let E be a real uniformly smooth Banach space, and let T : E → E be a single-valued operator. However, T is said to be

(i) r-strongly accretive if there exists a constant r > 0 such that

⟨Tx − Ty, Jq

(x − y)⟩ ≥ r∥∥x − y∥∥q, ∀x, y ∈ E, (2.8)

or equivalently,

⟨Tx − Ty, J2

(x − y)⟩ ≥ r∥∥x − y∥∥2

, ∀x, y ∈ E; (2.9)

(ii) s-Lipschitz continuous if there exists a constant s > 0 such that

∥∥T(x) − T(y)∥∥ ≤ s∥∥x − y∥∥, ∀x, y ∈ E. (2.10)

Remark 2.7. If T is r-strongly accretive, then T is r-expanding, that is,

∥∥T(x) − T(y)∥∥ ≥ r∥∥x − y∥∥, ∀x, y ∈ E. (2.11)

Lemma 2.8 (see [30]). Let {an}, {bn}, {cn} be three real sequences, satisfying

an+1 ≤ (1 − tn)an + bn + cn, ∀n ≥ 0, (2.12)

where tn ∈ (0, 1),∑∞

n=0 tn =∞, for all n ≥ 0, bn = ◦(tn),∑∞

n=0 cn <∞. Then an → 0.

3. System of Generalized Nonlinear Mixed Variational Inequalities

In this section, we will introduce a new system of generalized nonlinear mixed variationalinclusions which contains some classes of system of variational inclusions and systems ofvariational inequalities in literature as special cases.

In what follows, unless other specified, we always suppose that θ is a zero element inE, and for each i = 1, 2, . . . , n, Ti and Si : E → E are single-valued mappings, Mi : E → 2E

is an m-accretive operator. We consider the following problem: find (x∗1, x∗2, . . . , x

∗n) ∈ En such

that

θ ∈ ρ1T1x∗2 + ρ1S1x

∗2 + x

∗1 − x∗2 + ρ1M1

(x∗1),

θ ∈ ρ2T2x∗3 + ρ2S2x

∗3 + x

∗2 − x∗3 + ρ2M2

(x∗2),

...

θ ∈ ρn−1Tn−1x∗n + ρn−1Sn−1x

∗n + x

∗n−1 − x∗n + ρn−1Mn−1

(x∗n−1

),

θ ∈ ρnTnx∗1 + ρnSnx∗1 + x∗n − x∗1 + ρnMn(x∗n),

(3.1)


which is called the system of generalized nonlinear mixed variational inclusions, where ρi >0 (i = 1, 2, . . . , n) are constants.

In what follows, there are some special cases of the problem (3.1).(i) If n = 2, then problem (3.1) reduces to the system of nonlinear mixed

quasivariational inclusions introduced and studied by Peng and Zhu [19].If E = H is a Hilbert space and n = 2, then problem (3.1) reduces to the system of

nonlinear mixed quasivariational inclusions introduced and studied by Agarwal et al. [18].(ii) If E =H is a Hilbert space, and for each i = 1, 2, . . . , n,Mi(x) = ∂φi(x) for all x ∈ H,

where φi : H → R ∪ {+∞} is a proper, convex, lower semicontinuous functional, and ∂φidenotes the subdifferential operator of φi, then problem (3.1) reduces to the following systemof generalized nonlinear mixed variational inequalities, which is to find (x∗1, x

∗2, . . . , x

∗n) ∈ Hn

such that

⟨ρ1T1x

∗2 + ρ1S1x

∗2 + x

∗1 − x∗2, x − x∗1

⟩ ≥ ρ1φ1(x∗1) − ρ1φ1(x), ∀x ∈ H,

⟨ρ2T2x

∗3 + ρ2S2x

∗3 + x

∗2 − x∗3, x − x∗2

⟩ ≥ ρ2φ2(x∗2) − ρ2φ2(x), ∀x ∈ H,

...⟨ρn−1Tn−1x

∗n + ρn−1Sn−1x

∗n + x

∗n−1 − x∗n, x − x∗n−1

⟩ ≥ ρn−1φn−1(x∗n−1

) − ρn−1φn−1(x), ∀x ∈ H,⟨ρnTnx

∗1 + ρnSnx

∗1 + x

∗n − x∗1, x − x∗n

⟩ ≥ ρnφn(x∗n) − ρnφn(x), ∀x ∈ H,(3.2)

where ρi > 0 (i = 1, 2, . . . , n) are constants.(iii) If n = 2, then (3.2) reduces to the problem of finding (x∗1, x

∗2) ∈ H ×H such that

⟨ρ1T1x

∗2 + ρ1S1x

∗2 + x

∗1 − x∗2, x − x∗1

⟩ ≥ ρ1φ1(x∗1) − ρ1φ1(x), ∀x ∈ H,

⟨ρ2T2x

∗1 + ρ2S2x

∗1 + x

∗2 − x∗1, x − x∗2

⟩ ≥ ρ2φ2(x∗2) − ρ2φ2(x), ∀x ∈ H.

(3.3)

Moreover, if φ1 = φ2 = ϕ, then problem (3.3) becomes the system of generalizednonlinear mixed variational inequalities introduced and studied by J. K. Kim and D. S. Kimin [16].

(iv) For i = 1, 2, . . . , n, if φi = δki (the indicator function of a nonempty closed convexsubset Ki ⊂ H) and Ti = 0, then (3.2) reduces to the problem of finding (x∗1, x

∗2, . . . , x

∗n) ∈∏n

i=1Ki, such that

⟨ρ1S1x

∗2 + x

∗1 − x∗2, x − x∗1

⟩ ≥ 0, ∀x ∈ K1,⟨ρ2S2x

∗3 + x

∗2 − x∗3, x − x∗2

⟩ ≥ 0, ∀x ∈ K1,

...⟨ρn−1Sn−1x

∗n + x

∗n−1 − x∗n, x − x∗n−1

⟩ ≥ 0, ∀x ∈ Kn−1,⟨ρnSnx

∗1 + x

∗n − x∗1, x − x∗n

⟩ ≥ 0, ∀x ∈ Kn.

(3.4)


Problem (3.4) is called the system of nonlinear variational inequalities. Moreover, if n =2, then problem (3.4) reduces to the following system of nonlinear variational inequalities,which is to find (x∗1, x

∗2) ∈ K1 ×K2 such that

⟨ρ1S1x

∗2 + x

∗1 − x∗2, x − x∗1

⟩ ≥ 0, ∀x ∈ K1,⟨ρ2S2x

∗1 + x

∗2 − x∗1, x − x∗2

⟩ ≥ 0, ∀x ∈ K2.(3.5)

If S1 = S2 and K1 = K2 = K, then (3.5) reduces to the problem introduced andresearched by Verma [11–13].

Lemma 3.1. For any given x∗i ∈ E(i = 1, 2, . . . , n), (x∗1, x∗2, . . . , x

∗n) is a solution of the problem (3.1)

if and only if

x∗1 = RM1ρ1

[x∗2 − ρ1

(T1x

∗2 + S1x

∗2)],

x∗2 = RM2ρ2

[x∗3 − ρ2

(T2x

∗3 + S2x

∗3)],

...

x∗n−1 = RMn−1ρn−1

[x∗n − ρn−1(Tn−1x

∗n + Sn−1x

∗n)],

x∗n = RMnρn

[x∗1 − ρn

(Tnx

∗1 + Snx

∗1

)],

(3.6)

where RMiρi = (I + ρiMi)

−1 is the resolvent operators ofMi for i = 1, 2, . . . , n.

Proof. It is easy to know that Lemma 3.1 follows from Definition 2.4 and so the proof isomitted.

4. Existence and Uniqueness

In this section, we will show the existence and uniqueness of solution for problems (3.1).

Theorem 4.1. Let E be a real q-uniformly smooth Banach spaces. For i = 1, 2, . . . , n, let Si : E → Ebe strongly accretive and Lipschitz continuous with constants ki and μi, respectively, let Ti : E → Ebe Lipschitz continuous with constant νi, and letMi : E → E be anm-accretive mapping. If for eachi = 1, 2, . . . , n,

0 < q

√1 − qρiki + cqρqi μ

q

i + ρiνi < 1, (4.1)

then (3.1) has a unique solution (x∗1, x∗2, . . . , x

∗n) ∈ En.


Proof. First, we prove the existence of the solution. Define a mapping F : E → E as follows:

F(x) = RM1ρ1

[x2 − ρ1(T1x2 + S1x2)

],

x2 = RM2ρ2

[x3 − ρ2(T2x3 + S2x3)

],

...

xn−1 = RMn−1ρn−1

[xn − ρn−1(Tn−1xn + Sn−1xn)

],

xn = RMnρn

[x − ρn(Tnx + Snx)

].

(4.2)

For i = 1, 2, . . . , n, since RMiρi is a nonexpansive mapping, Si is strongly accretive and

Lipschitz continuous with constants ki and μi, respectively, and Ti is Lipschitz continuouswith constant νi, for any x, y ∈ E, we have

∥∥F(x) − F(y)∥∥=∥∥∥RM1

ρ1

[x2 − ρ1(T1x2 + S1x2)

] − RM1ρ1

[y2 − ρ1

(T1y2 + S1y2

)]∥∥∥≤ ∥∥(x2 − y2

) − ρ1((T1x2 + S1x2) −

(T1y2 + S1y2

))∥∥≤ ∥∥(x2 − y2

) − ρ1(S1x2 − S1y2

)∥∥ + ρ1∥∥T1x2 − T1y2

∥∥≤ q

√∥∥x2 − y2∥∥q − qρ1

⟨S1x2 − S1y2, Jq

(x2 − y2

)⟩+ cqρ

q

1

∥∥S1x2 − S1y2∥∥q + ρ1ν1

∥∥x2 − y2∥∥

=(

q

√1 − qρ1k1 + cqρ

q

1μq

1 + ρ1ν1

)∥∥x2 − y2∥∥

=(

q

√1 − qρ1k1 + cqρ

q

1μq

1 + ρ1ν1

)∥∥∥RM2ρ2

[x3 − ρ2(T2x3 + S2x3)

] − RM2ρ2

[y3 − ρ2

(T2y3 + S2y3

)]∥∥∥

≤(

q

√1 − qρ1k1 + cqρ

q

1μq

1 + ρ1ν1

)∥∥x3 − y3 − ρ2[(S2x3 − S2y3

)+(T2x3 − T2y3

)]∥∥

≤(

q

√1 − qρ1k1 + cqρ

q

1μq

1 + ρ1ν1

)∥∥x3 − y3 − ρ2(S2x3 − S2y3

)∥∥ + ρ2∥∥T2x3 − T2y3

∥∥

≤(

q

√1 − qρ1k1 + cqρ

q

1μq

1 + ρ1ν1

)

×[

q

√∥∥x3 − y3∥∥q − qρ2

⟨S2x3 − S2y3, Jq

(x3 − y3

)⟩+ cqρ

q

2

∥∥S2x3 − S2y3∥∥q + ρ2ν2

∥∥x3 − y3∥∥]

≤(

q

√1 − qρ1k1 + cqρ

q

1μq

1 + ρ1ν1

)(q

√1 − qρ2k2 + cqρ

q

2μq

2 + ρ2ν2

)∥∥x3 − y3∥∥

≤ · · · ≤n−1∏i=1

(q


q

i + ρiνi)∥∥xn − yn∥∥


=n−1∏i=1

∥∥∥RMnρn

[x − ρn(Tnx + Snx)

] − RMnρn

[y − ρn

(Tny + Sny

)]∥∥∥

≤n−1∏i=1

(q


q

i + ρiνi)[∥∥x − y − ρn(Snx) − Sny∥∥ + ρn

∥∥Tnx − Tny∥∥]

≤n−1∏i=1

(q


q

i + ρiνi)

×[

q

√∥∥x − y∥∥q − qρn⟨Snx − Sny, Jq(x − y)⟩ + cqρqn∥∥Snx − Sny∥∥q + ρnνn∥∥x − y∥∥]

≤n∏i=1

(q


q

i + ρiνi)∥∥x − y∥∥.

(4.3)

It follows from (4.1) that

0 <n∏i=1

(q


q

i + ρiνi)< 1. (4.4)

Thus, (4.3) implies that F is a contractive mapping and so there exists a point x∗1 ∈ E suchthat

x∗1 = F(x∗1). (4.5)

Let

x∗i = RMiρi

[x∗i+1 − ρi

(Ti(x∗i+1

)+ Si

(x∗i+1

))], i = 1, 2, . . . , n − 1

x∗n = RMnρn

[x∗1 − ρn

(Tn

(x∗1)+ Sn

(x∗1))] (4.6)

then by the definition of F, we have

x∗1 = RM1ρ1

[x∗2 − ρ1

(T1x

∗2 + S1x

∗2)],

x∗2 = RM2ρ2

[x∗3 − ρ2

(T2x

∗3 + S2x

∗3)],

...



∗n + Sn−1x

∗n)],

x∗n = RMnρn

[x∗1 − ρn(Tnx1

∗ + Snx1∗)],

(4.7)

that is, (x∗1, x∗2, . . . , x

∗n) is a solution of problem (3.1).


Then, we show the uniqueness of the solution. Let (x1, x2, . . . , xn) be another solutionof problem (3.1). It follows from Lemma 3.1 that

x1 = RM1ρ1

[x2 − ρ1(T1x2 + S1x2)

],

x2 = RM2ρ2

[x3 − ρ2(T2x3 + S2x3)

],

...

xn−1 = RMn−1ρn−1

[xn − ρn−1(Tn−1xn + Sn−1xn)

],

xn = RMnρn

[x1 − ρn(Tnx1 + Snx1)

].

(4.8)

As the proof of (4.3), we have

∥∥x∗1 − x1∥∥ ≤ n∏

i=1

(q


q

i + ρiνi)∥∥x∗1 − x1

∥∥. (4.9)

It follows from (4.1) that

0 <n∏i=1

(q


q

i + ρiνi)< 1. (4.10)

Hence,

x∗1 = x1, (4.11)

and so for i = 2, 3, . . . , n, we have

x∗i = xi. (4.12)


Remark 4.2. (i) If E is a 2-uniformly smooth space, and there exist ρi > 0 (i = 1, 2, . . . , n) suchthat

0 < ρi < min

{2(ki − νi)c2μ

2i − ν2

i

,1νi

}, νi < c2μi. (4.13)

Then (4.1) holds. We note that the Hilbert spaces and Lp (or lq) spaces (2 ≤ q < ∞) are2-uniformly smooth.

(ii) Let n = 2, by Theorem 4.1, we recover [19, Theorem 3.1]. So Theorem 4.1 unifies,extends, and improves [19, Theorem 3.1, Corollaries 3.2 and 3.3], [16, Theorems 2.1–2.4] andthe main results in [13].


5. Algorithms and Convergence

This section deals with an introduction of some n-step iterative sequences with or withoutmixed errors for problem (3.1) that can be applied to the convergence analysis of the iterativesequences generated by the algorithms.

Algorithm 5.1. For any given point x0 ∈ E, define the generalized N-step iterative sequences{x1,k}, {x2,k}, . . . , {xn,k} as follows:

x1,k+1 = (1 − αk)x1,k + αkRM1ρ1

[x2,k − ρ1(T1x2,k + S1x2,k)

]+ αku1,k +wk,

x2,k = RM2ρ2

[x3,k − ρ2(T2x3,k + S2x3,k)

]+ u2,k,

...

xn−1,k = RMn−1ρn−1

[xn,k − ρn−1(Tn−1xn,k + Sn−1xn,k)

]+ un−1,k,

xn,k = RMnρn

[x1,k − ρn(Tnx1,k + Snx1,k)

]+ un,k,

(5.1)

where x1,1 = x0, {αk} is a sequence in [0, 1], and {ui,k} ⊂ E (i = 1, 2, . . . , n), {wk} ⊂ E are thesequences satisfying the following conditions:

∞∑k=1

αk = +∞;∞∑k=1

‖wk‖ < +∞; limk→∞

‖ui,k‖ = 0, i = 1, 2, . . . , n. (5.2)

Theorem 5.2. Let Ti, Si, and Mi be the same as in Theorem 4.1, and suppose that thesequences {x1,k}, {x2,k}, . . . , {xn,k} are generated by Algorithm 5.1. If the condition (4.1) holds, then(x1,k, x2,k, . . . , xn,k) converges strongly to the unique solution (x∗1, x

∗2, . . . , x

∗n) of the problem (3.1).

Proof. By the Theorem 4.1, we know that problem (3.1) has a unique solution (x∗1, x∗2, . . . , x

∗n),

it follows from Lemma 3.1 that

x∗1 = RM1ρ1

[x∗2 − ρ1

(T1x

∗2 + S1x

∗2)],

x∗2 = RM2ρ2

[x∗3 − ρ2

(T2x

∗3 + S2x

∗3)],

...



∗n + Sn−1x

∗n)],

x∗n = RMnρn

[x∗1 − ρn

(Tnx

∗1 + Snx

∗1

)].

(5.3)


By (5.1) and (5.3), we have

∥∥x1,k+1 − x∗1∥∥

=∥∥∥(1 − αk)x1,k + αkRM1

ρ1

[x2,k − ρ1(T1x2,k + S1x2,k)

]+ αku1,k +wk − x∗1

∥∥∥≤ (1 − αk)

∥∥x1,k − x∗1∥∥ + αk

∥∥∥RM1ρ1

[x2,k − ρ1(T1x2,k + S1x2,k)

] − RM1ρ1

[x∗2 − ρ1

(T1x

∗2 + S1x

∗2)]∥∥∥

+ αk‖u1,k‖ + ‖wk‖≤ (1 − αk)

∥∥x1,k − x∗1∥∥ + αk

∥∥(x2,k − x∗2) − ρ1

[(T1x2,k + S1x2,k) −

(T1x

∗2 + S1x

∗2)]∥∥

+ αk‖u1,k‖ + ‖wk‖.(5.4)

For i = 1, 2, . . . , n, since Si is strongly monotone and Lipschitz continuous withconstants ki and μi, respectively, and Ti is Lipschitz continuous with constant νi, we get fori = 1, 2, . . . , n − 1,

∥∥(xi+1,k − x∗i+1

) − ρi[(Tixi+1,k + Sixi+1,k) −(Tix

∗i+1 + Six

∗i+1

)]∥∥≤ ∥∥(xi+1,k − x∗i+1

) − ρi(Sixi+1,k − Six∗i+1

)∥∥ + ρi∥∥Tixi+1,k − Tix∗i+1

∥∥≤ q

√∥∥xi+1,k − x∗i+1

∥∥q − qρi⟨Sixi+1,k − Six∗i+1, Jq(xi+1,k − x∗i+1

)⟩+ cqρ

q

i

∥∥Sixi+1,k − Six∗i+1

∥∥q+ ρiνi

∥∥xi+1,k − x∗i+1

∥∥≤ ξi

∥∥xi+1,k − x∗i+1

∥∥,(5.5)

where ξi = q


q

i + ρiνi.It follows from (5.4) and (5.5) that

∥∥x1,k+1 − x∗1∥∥ ≤ (1 − αk)

∥∥x1,k − x∗1∥∥ + αkξ1

∥∥x2,k − x∗2∥∥ + αk‖u1,k‖ + ‖wk‖. (5.6)

By (5.1), (5.3), and (5.5), we have

∥∥x2,k − x∗2∥∥

=∥∥∥RM2

ρ2

[x3,k − ρ2(T2x3,k + S2x3,k)

] − RM2ρ2

[x∗3 − ρ2

(T2x

∗3 + S2x

∗3)]

+ u2,k

∥∥∥≤ ∥∥(x3,k − x∗3

) − ρ2[(T2x3,k + S2x3,k) −

(T2x

∗3 + S2x

∗3)]∥∥ + ‖u2,k‖

≤ ξ2∥∥x3,k − x∗3

∥∥ + ‖u2,k‖,


∥∥x3,k − x∗3∥∥

=∥∥∥RM3

ρ3

[x4,k − ρ3(T3x4,k + S3x4,k)

] − RM3ρ3

[x∗4 − ρ3

(T3x

∗4 + S3x

∗4

)]+ u3,k

∥∥∥≤ ∥∥(x4,k − x∗4

) − ρ3[(T3x4,k + S3x4,k) −

(T3x

∗4 + S3x

∗4

)]∥∥ + ‖u3,k‖≤ ξ3

∥∥x4,k − x∗4∥∥ + ‖u3,k‖, . . . ,∥∥xn−1,k − x∗n−1

∥∥=∥∥∥RMn−1

ρn−1


] − RMn−1ρn−1


∗n + Sn−1x

∗n)]+ un−1,k

∥∥∥≤ ∥∥(xn,k − x∗n) − ρn−1[(Tn−1xn,k + Sn−1xn,k) − (Tn−1x

∗n + Sn−1x

∗n)]

∥∥ + ‖un−1,k‖≤ ξn−1‖xn,k − x∗n‖ + ‖un−1,k‖,

(5.7)

since Sn is strongly accretive and Lipschitz continuous with constants kn and μn, respectively,and Tn is Lipschitz continuous with constant νn, we get

‖xn,k − x∗n‖

=∥∥∥RMn

ρn


] − RMnρn

[x∗1 − ρn

(Tnx

∗1 + Snx

∗1

)]+ un,k

∥∥∥≤ ∥∥(x1,k − x∗1

) − ρn[(Tnx1,k + Snx1,k) −(Tnx

∗1 + Snx

∗1

)]∥∥ + ‖un,k‖

≤ q

√∥∥x1,k − x∗1∥∥q − qρn⟨Snx1,k − Snx∗1, Jq

(x1,k − x∗1

)⟩+ cqρ

qn

∥∥Snx1,k − Snx∗1∥∥q

+ ρn∥∥Tnx1,k − Tnx∗1

∥∥≤ ξn

∥∥x1,k − x∗1∥∥ + ‖un,k‖,

(5.8)

where ξn = q

√1 − qρnkn + cqρqnμqn + ρnνn.

It follows from (5.6)–(5.8) that

∥∥x1,k+1 − x∗1∥∥

≤ (1 − αk)∥∥x1,k − x∗1

∥∥ + αkξ1∥∥x2,k − x∗2

∥∥ + αk‖u1,k‖ + ‖wk‖≤ (1 − αk)

∥∥x1,k − x∗1∥∥ + αkξ1

[ξ2∥∥x3,k − x∗3

∥∥ + ‖u2,k‖]+ αk‖u1,k‖ + ‖wk‖

≤ (1 − αk)∥∥x1,k − x∗1

∥∥ + αkξ1ξ2∥∥x3,k − x∗3

∥∥ + αkξ1‖u2,k‖ + αk‖u1,k‖ + ‖wk‖≤ (1 − αk)

∥∥x1,k − x∗1∥∥ + αkξ1ξ2ξ3

∥∥x4,k − x∗4∥∥ + αkξ1ξ2‖u3,k‖ + αkξ1‖u2,k‖ + αk‖u1,k‖ + ‖wk‖

≤ · · · ≤ (1 − αk)∥∥x1,k − x∗1

∥∥ + αkξ1ξ2 · · · ξn−1‖xn,k − x∗n‖ + αkξ1ξ2 · · · ξn−2‖un−1,k‖+ · · · + αkξ1‖u2,k‖ + αk‖u1,k‖ + ‖wk‖


≤ (1 − αk)∥∥x1,k − x∗1

∥∥ + αkξ1ξ2 · · · ξn∥∥x1,k − x∗1

∥∥ + αkξ1ξ2 · · · ξn−1‖un,k‖ + αkξ1ξ2 · · · ξn−2‖un−1,k‖+ · · · + αkξ1‖u2,k‖ + αk‖u1,k‖ + ‖wk‖

= [1 − αk(1 − ξ1ξ2 · · · ξn)]∥∥x1,k − x∗1

∥∥

+ αk(1 − ξ1ξ2 · · · ξn) 11 − ξ1ξ2 · · · ξn (‖u1,k‖ + ξ1‖u2,k‖ + · · · + ξ1ξ2 · · · ξn−1‖un,k‖) + ‖wk‖.

(5.9)

Let

ak =∥∥x1,k − x∗1

∥∥, tk = αk(1 − ξ1ξ2 · · · ξn), ck = ‖wk‖,

bk =1

1 − ξ1ξ2 · · · ξn (‖u1,k‖ + ξ1‖u2,k‖ + · · · + ξ1ξ2 · · · ξn−1‖un,k‖).(5.10)

Then (5.9) can be written as follows:

ak+1 ≤ (1 − tk)ak + bktk + ck. (5.11)

From the assumption (5.2), we know that {ak}, {bk}, {tk}, {ck} satisfy the conditions ofLemma 2.8.

Thus ak → 0 (k → ∞), that is, ‖x1,k − x∗1‖ → 0 (k → ∞). It follows from (5.6)–(5.8)that ‖xn,k − x∗n‖ → 0 (k → ∞), ‖xn−1,k − x∗n−1‖ → 0 (k → ∞), . . . , ‖x2,k − x∗2‖ → 0 (k →∞).So xi,k → x∗i (k → ∞) for i = 1, 2, . . . , n. That is, (x1,k, x2,k, . . . , xn,k) converges strongly to theunique solution (x∗1, x

∗2, . . . , x

∗n) of (3.1).

For i = 1, 2, . . . , n, let ui,k = 0 and wk = 0, by Algorithm 5.1 and Theorem 5.2, it is easyto obtain the following Algorithm 5.3 and Theorem 5.4.

Algorithm 5.3. For any given point x0 ∈ E, define the generalized N-step iterative sequences{x1,k}, {x2,k}, . . . , {xn,k} as follows:

x1,k+1 = (1 − αk)x1,k + αkRM1ρ1

[x2,k − ρ1(T1x2,k + S1x2,k)

],

x2,k = RM2ρ2

[x3,k − ρ2(T2x3,k + S2x3,k)

],

...

xn−1,k = RMn−1ρn−1


],

xn,k = RMnρn


],

(5.12)

where x1,1 = x0, {αk} is a sequence in [0, 1], satisfying∑∞

k=1 αk = +∞.

Theorem 5.4. Let Ti, Si, and Mi be the same as in Theorem 4.1, and suppose that the sequences{x1,k}, {x2,k}, . . . , {xn,k} are generated by Algorithm 5.3. If (4.1) holds, then (x1,k, x2,k, . . . , xn,k)converges strongly to the unique solution (x∗1, x

∗2, . . . , x

∗n) of (3.1).


Remark 5.5. Theorem 5.4 unifies and generalizes [19, Theorems 4.3 and 4.4] and the mainresults in [11, 12]. So Theorem 5.2 unifies, extends, and improves the corresponding resultsin [11–14, 16, 19].

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grants10771228 and 10831009) and the Research Project of Chongqing Normal University (Grant08XLZ05).

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Research ArticleSome Convergence Theorems of a Sequence inComplete Metric Spaces and Its Applications

M. A. Ahmed and F. M. Zeyada

Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

Correspondence should be addressed to M. A. Ahmed, [email protected]

Received 20 June 2009; Accepted 7 September 2009


Copyright q 2010 M. A. Ahmed and F. M. Zeyada. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The concept of weakly quasi-nonexpansive mappings with respect to a sequence is introduced.This concept generalizes the concept of quasi-nonexpansive mappings with respect to a sequencedue to Ahmed and Zeyada (2002). Mainly, some convergence theorems are established and theirapplications to certain iterations are given.

1. Introduction

In 1916, Tricomi [1] introduced originally the concept of quasi-nonexpansive for realfunctions. Subsequently, this concept has studied for mappings in Banach and metric spaces(see, e.g., [2–7]). Recently, some generalized types of quasi-nonexpansive mappings in metricand Banach spaces have appeared. For example, see Ahmed and Zeyada [8], Qihou [9–11]and others.

Unless stated to the contrary, we assume that (X, d) is a metric space. Let T : D ⊆ X →X be any mapping and let F(T) be the set of all fixed points of T . If F : X → R where R isthe set of all real numbers and if c ∈ R, set Lc := {x ∈ X : F(x) ≤ c}. We use the symbol μto denote the usual Kuratowski measure of noncompactness. For some properties of μ, seeZeidler [12, pages 493–495]. For a given x0 ∈ D, the Picard iteration (xn) is determined by:

(I) xn = T(xn−1) = Tn(x0), n ∈N

where N is the set of all positive integers.If X is a normed space, D is a convex set, and T : D → D, Ishikawa [13] gave the

following iteration:

(II) xn = Tα,β(xn−1) = Tnα,β(x0), Tα,β = (1 − α)I + αT[(1 − β)I + βT],


Research ArticleFixed Points of Single- and Set-ValuedMappings in Uniformly Convex Metric Spaceswith No Metric Convexity

Rafa Espınola,1 Aurora Fernandez-Leon,1 and Bozena Piatek2

1 Departamento de Analisis Matematico, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain2 Institute of Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland

Correspondence should be addressed to Rafa Espınola, [email protected]

Received 20 April 2009; Accepted 28 May 2009


Copyright q 2010 Rafa Espınola et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We study the existence of fixed points and convergence of iterates for asymptotic pointwisecontractions in uniformly convex metric spaces. We also study the existence of fixed points for set-valued nonexpansive mappings in the same class of spaces. Our results do not assume convexity ofthe metric which makes a big difference when studying the existence of fixed points for set-valuedmappings.

1. Introduction

This paper is motivated by the recent paper [1]. In [1] the authors study different questionsrelated to fixed points of asymptotic pointwise contractive/nonexpansive mappings inCAT(0) spaces. CAT(0) spaces are studied in [1] as a very significant example within theclass of uniformly convex metric spaces (the reader can consult [2] for details on CAT(0)spaces). In our present paper we propose to consider similar questions on uniformly convexmetric spaces under the mildest additional conditions we may impose. More precisely, wewill work with uniformly convex metric spaces with either a monotone modulus of convexityin the sense first given in [3] or a lower semicontinuous from the right modulus of convexity(see Section 2 for proper definitions). For a recent survey on the existence of fixed pointsin geodesic spaces, the reader may check [4], for recent achievements on related topics thereader may also check [5].

The notion of asymptotic pointwise contractions was introduced in [6]. Then it wasalso studied in [7] where, by means of ultrapower techniques, different results about the


existence of fixed points and convergence of iterates were proved. In [8] new proofs werepresented but this time after applying only elementary techniques. Very recently, in [1], thesetechniques were applied in CAT(0), where the authors attend to the Bruhat-Tits inequality forCAT(0) spaces in order to obtain such results. In the present paper we show that actually mostof those results still hold for general uniformly convex metric spaces under mild conditionson the modulus of convexity. In Section 3 we focus on single-valued mappings and, inparticular, on mappings which are asymptotically pointwise contractive/nonexpansive tostudy the existence of fixed points, convergence of Picard’s iterates, and the structure oftheir sets of fixed points. As a technical result we need to show that bounded sequencesin these spaces have a unique asymptotic center which, as a by-product, leads to Kirk’s FixedPoint Theorem. In Section 4 we study different problems regarding set-valued mappingsin these spaces. The main technical difficulty to achieve similar results to those shown in[1] is that now we cannot count on the existence of fixed points for nonexpansive set-valued mappings for the kind of spaces we deal with. Finding fixed point for set-valuednonexpansive mappings in uniformly convex metric spaces was first studied by Shimizu andTakahashi [9], where the existence of fixed points was guaranteed under stronger conditionson the modulus of convexity and the additional condition of metric convexity of the space.The fact that we do not have that the metric are convex will make the problem morecomplicated and this will take us to impose new conditions on the modulus of convexitywhich we will relate with the geometry of the space.

2. Basic Definitions and Results

We introduce next some basic definitions.

Definition 2.1. Let (X, d) be a metric space. A mapping T : X → X is called a pointwisecontraction if there exists a mapping α : X → [0, 1) such that

d(T(x), T

(y)) ≤ α(x)d(x, y) (2.1)

for any y ∈ X.

It is proved in [8] (see also [6]) that if K is a weakly compact convex subset of aBanach space and T : K → K is a pointwise contraction, then T has a unique fixed point andthe sequence of the iterates of T converges to the fixed point for any x ∈ K. As it is pointed outin [1], the uniqueness of fixed points and convergence of iterates for these mappings directlyfollow if existence is guaranteed.

Definition 2.2. Let (X, d) be a metric space. Let T : X → X be a mapping, and let αn : X →[0,∞) for each n ∈ N be such that

d(Tn(x), Tn

(y)) ≤ αn(x)d(x, y) for any y ∈ X. (2.2)

Then

(i) T is called an asymptotic pointwise contraction if {αn} converges pointwise to α :X → [0, 1);


(ii) T is called an asymptotic pointwise nonexpansive mapping if lim supαn(x) ≤ 1 forany x ∈ X;

(iii) T is called a strongly asymptotic pointwise contraction if lim supαn(x) ≤ k, with0 < k < 1, for any x ∈ X.

In this paper we will mainly work with uniformly convex geodesic metric space. Sincethe definition of uniform convexity requires the existence of midpoints, the word geodesic isredundant and so, for simplicity, we will usually omit it.

Definition 2.3. A geodesic metric space (X, d) is said to be uniformly convex if for any r > 0 andany ε ∈ (0, 2] there exists δ ∈ (0, 1] such that for all a, x, y ∈ X with d(x, a) ≤ r, d(y, a) ≤ rand d(x, y) ≥ εr it is the case that

d(m,a) ≤ (1 − δ)r, (2.3)

where m stands for any midpoint of any geodesic segment [x, y]. A mapping δ : (0,+∞) ×(0, 2] → (0, 1] providing such a δ = δ(r, ε) for a given r > 0 and ε ∈ (0, 2] is called a modulusof uniform convexity.

Notice that this definition of uniform convex metric spaces is weaker than the oneused in [9] in two ways. First, we do not impose that the metric is convex and, second, ourmodulus of convexity does depend on the two variables r and εwhile it is assumed to dependonly on ε in [9].

Definition 2.4. Let (X, d) be a metric space, then the metric is said to be convex if for any x, yand z in X, and m a midpoint in between x and y,

d(z,m) ≤ 1/2(d(z, x) + d

(z, y

)). (2.4)

It is easy to see that uniformly convex metric spaces are uniquely geodesic, that is, foreach two points there is just one geodesic joining them. Therefore midpoints and geodesicsegments [x, y] joining two points are unique. In this case there is a natural way to defineconvexity. A subset C of a (uniquely) geodesic space is said to be convex if [x, y] ⊆ C for anyx, y ∈ C. For more about geodesic spaces the reader may check [2].

To obtain our results we will need to impose additional conditions on the modulusof convexity. Following [3, 10] we consider the notion of monotone modulus of convexity asfollows.

Definition 2.5. If a uniformly convex metric space X admits a modulus of convexity δ suchthat it decreases with r (for each fixed ε) then we say that δ is a monotone modulus ofconvexity for X.

In the same way we define a lower semicontinuous from the right modulus ofconvexity as follows.

Definition 2.6. If a uniformly convex metric space X admits a modulus of convexity δ suchthat it is lower semicontinuous from the right with respect to r (for each fixed ε) then we sayδ is a lower semicontinuous from the right modulus of convexity for X.


LetX be a metric space andF a family of subsets ofX. Then, following [1], we say thatF defines a convexity structure on X if it contains the closed balls and is stable by intersection.

Let X be a metric space and F a convexity structure on X. Given Φ : X → [0,∞), wesay that Φ is F-convex if {x : Φ(x) ≤ r} ∈ F for any r ≥ 0.

If we consider a bounded sequence {xn} in X, we are able to define a function r(·, xn),called type, such that for each x

r(x, xn) = lim supn→∞

d(xn, x). (2.5)

The asymptotic center of a bounded sequence with respect to a subset C of X is then defined as

AC({xn}) ={x ∈ X : r(x, xn) ≤ r

(y, xn

)for any y ∈ C

}. (2.6)

If the asymptotic center is taken with respect to X then it is simply denoted by A({xn}).

Definition 2.7. We say that a convexity structure is T -stable if types are F-convex.

In [1] the following definition of compactness for convexity structure was considered.

Definition 2.8. Given F a convexity structure, we will say that F is compact if any family(Aα)α∈Γ of elements of F has nonempty intersection provided ∩α∈F /= ∅ for any finite subsetF ⊂ Γ.

In our paper we will rather use the idea of compactness given in [11]. Notice that thissecond notion of compactness is weaker than the previous one.

Definition 2.9. Given F a convexity structure, we will say that F is nested compact if anydecreasing chain (Aα)α∈Γ of nonempty bounded elements of F has nonempty intersection.

A very important property given in [3] about complete uniformly convex metricspaces with monotone modulus of convexity is that decreasing sequences of nonemptybounded closed and convex subsets of these spaces have nonempty intersection. As aconsequence, we have that if F stands for the collection of nonempty closed and convexsubsets of a complete uniformly convex metric space with monotone modulus of convexity,then F is a nested compact convexity structure.

Remark 2.10. It is not hard to see that the same remains true if the monotone condition on themodulus is replaced by lower semicontinuity from the right.

3. Asymptotic Pointwise Contractions in Uniformly ConvexMetric Spaces

In this section we give different results for the above defined mappings in uniformly convexmetric spaces. Although, for expository reasons, our results will be usually proved only foruniformly convex metric spaces with a monotone modulus of convexity, they also hold whenthere is a lower semicontinuous modulus of convexity. Some indications about differences inboth cases will be given. We begin with a technical result.


Proposition 3.1. Let (X, d) be a complete uniformly convex metric space with a monotone (or lowersemicontinuous from the right) modulus of convexity δ(r, ε). Consider the family F of all nonemptyclosed and convex subsets of X. Then F defines a nested compact and T -stable convexity structure onX.

Proof. It only remains to be proved that F is T -stable. Let {xn} be a bounded sequence in Xand consider the type defined by {xn}. We need to show that Cr = {x : r(x, xn) ≤ r} ∈ F forany positive r. It is immediate to see that Cr is closed and nonempty. To see that Cr ∈ F is alsoconvex, consider x and y to be two different points in Cr . There is no restriction if we assumethat lim supd(y, xn) ≤ lim supd(x, xn) = r1 ≤ r. Let m be the midpoint of the segment [x, y]and take ε1 = d(x, y)/(r + 1), then, by uniform convexity, we have that

d(m,xn) ≤(1 − δ(max

{d(x, xn), d

(y, xn

)}, ε1

))max

{d(x, xn), d

(y, xn

)}< max

{d(x, xn), d

(y, xn

)},

(3.1)

and so,

lim supd(m,xn) ≤ lim sup max{d(x, xn), d

(y, xn

)}= r1 ≤ r. (3.2)

Hence, m ∈ Cr .

The following theorems were proved in [1] under the hypothesis of compactness onthe convexity structure. We state it, however, under the hypothesis of nested compactnesssince this is all it is actually required in the proofs given in [1].

Theorem 3.2. LetX be a bounded metric space. Assume that the convexity structureA(M) is nestedcompact. Let T : X → X be a pointwise contraction. Then T has a unique fixed point x0. Moreoverthe orbit {Tn(x)} converges to x0, for each x ∈ X.

Theorem 3.3. LetX be a bounded metric space. Assume that the convexity structureA(M) is nestedcompact. Let T : X → X be a strongly asymptotic pointwise contraction. Then T has a unique fixedpoint x0. Moreover the orbit {Tn(x)} converges to x0, for each x ∈ X.

Now the next corollary follows.

Corollary 3.4. The above theorems hold for complete bounded uniformly convex metric spaces witheither monotone or lower semicontinuous from the right modulus of convexity.

The following lemma is immediate.

Lemma 3.5. Let X be a metric space and F a nested compact convexity structure on X which isT -stable. Then for any type r(·, xn), there exists x0 ∈ X such that

r(x0, xn) = inf{r(x, xn) : x ∈ X}. (3.3)

As a direct consequence of Proposition 3.1 and the previous lemma we get thefollowing result for asymptotic pointwise contractions. We omit the details of its proof asit follows similar patterns as in [1, Theorem 4.2].


Theorem 3.6. Let (X, d) be a complete uniformly convex metric space with a monotone (or lowersemicontinuous from the right) modulus of convexity δ(r, ε). Suppose X is bounded. Then every T :X → X asymptotic pointwise contraction has a unique fixed point x0. Moreover, the orbit {Tn(x)}converges to x0 for each x ∈ X.

Next we show some consequences of Proposition 3.1 and Lemma 3.5. The casesfor monotone and lower semicontinuous from the right modulus of convexity are shownseparately as they require different proofs.

Corollary 3.7. Let X be a complete uniformly convex metric space with a monotone modulus ofconvexity and {xn} a bounded sequence inX. Then the set of asymptotic centers of {xn} is a singleton.

Proof. Let u and v be two different points in A({xn}), and let m be the midpoint of [u, v]. Letr = r(u, xn) = r(u, xn), c = r + 1, and ε1 = d(u, v)/c. By the uniform convexity, there existsN ∈ N such that for every n ≥N,

d(m,xn) ≤ (1 − δ(max{d(u, xn), d(v, xn)}, ε1))max{d(u, xn), d(v, xn)}≤ (1 − δ(c, ε1))max{d(u, xn), d(v, xn)}.

(3.4)

If we let n go to infinite, we obtain that r(m,xn) ≤ (1 − δ(c, ε1))r < r, which is clearly acontradiction.

Remark 3.8. This corollary has been first proved in [12, Proposition 3.3] for a certain class ofuniformly convex hyperbolic spaces with monotone modulus of convexity.

Now we show the lower semicontinuous case.

Corollary 3.9. Let X be a complete uniformly convex metric space with a lower semicontinuous fromthe right modulus of convexity and {xn} a bounded sequence in X. Then the set of asymptotic centersof {xn} is a singleton.

Proof. Let u and v be two different points in A({xn}) and let m be the midpoint of [u, v]. Letr = r(u, xn) = r(v, xn), ε = d(u, v)/(r + 1), and let us fix p ∈ N. Then max{d(u, xn), d(v, xn)} ≤r + p−1 for each n large enough. By the uniform convexity,

d(m,xn) ≤(

1 − δ(r + p−1, ε

))(r + p−1

)(3.5)

for the same n as above and finally

r(m,xn) ≤(

1 − δ(r + p−1, ε

))(r + p−1

). (3.6)

Now it suffices to observe that

δ(r + p−1, ε

)≥ 1

2δ(r, ε),

1 − δ(r + p−1, ε

)≤ 1 − 1

2δ(r, ε).

(3.7)


for p large enough. Combining it with (3.6) and taking limp→∞ we obtain r(m,xn) < r as inthe former corollary, and thus the contradiction.

Another consequence is Kirk Fixed Point Theorem in uniformly convex metric spaces.

Corollary 3.10. Let X be a complete uniformly convex geodesic metric space with a monotone(or lower semicontinuous from the right) modulus of convexity. Suppose X is bounded, then anynonexpansive mapping T : X → X has a fixed point.

Proof. Consider x ∈ X and {Tn(x)} the sequence of its iterates. Let ω be the only asymptoticcenter of {Tn(x)} in X. Then, by the nonexpansiveness of T , it follows that r(T(ω), Tn(x)) ≤r(ω, Tn(x)) and so, T(ω) = ω.

Now we present a counterpart for [1, Theorem 5.1].

Theorem 3.11. Let (X, d) be a complete uniformly convex metric space with a monotone (or lowersemicontinuous from the right) modulus of convexity δ(r, ε). Let C be a bounded closed convexnonempty subset of X. Then any T : C → C asymptotic pointwise nonexpansive mapping has afixed point, and the set of fixed points of T , Fix(T), is closed and convex.

Proof. Let x ∈ C and consider xn = Tn(x). From Corollary 3.7, we know that AC({xn}) is asingleton. Let ω be the only point in that set, that is, ω is such that r(ω, xn) = inf{r(u, xn) : u ∈C}. We want to show that {Tm(ω)} is a Cauchy sequence. Suppose this is not the case. Thenthere exists a separated subsequence {Tmi(ω)} of {Tm(ω)}, that is, there exists ε > 0 such thatd(Tmk(ω), Tmh(ω)) ≥ ε for every k /=h in N.

Let mkh be the midpoint of the segment [Tmk(ω), Tmh(ω)], c = diam(C) and ε1 = ε/c.The uniform convexity of the space, together with its monotone character, implies that forevery k and h in N

d(mkh, xn)≤(1−δ(max{d(Tmh(ω), xn), d(Tmk(ω), xn)}, ε1))max{d(Tmh(ω), xn), d(Tmk(ω), xn)}≤ (1 − δ(c, ε1))max{d(Tmh(ω), xn), d(Tmk(ω), xn)}.

(3.8)

Notice that, by definition of T ,

r(Tm(ω), xn) ≤ αm(ω)r(ω, xn). (3.9)

Then, if we let n go to infinity,

r(ω, xn) ≤ r(mkh, xn)

≤ (1 − δ(c, ε1))max{r(Tmk(ω), xn), r(Tmh(ω), xn)}≤ (1 − δ(c, ε1))max{αmk(ω)r(ω, xn), αmh(ω)r(ω, xn)}.

(3.10)

Since T is pointwise asymptotic nonexpansive, then

r(ω, xn) ≤ (1 − δ(c, ε1))r(ω, xn), (3.11)


and so r(ω, xn) = 0, which is a contradiction since, in virtue of (3.9), this implies that Tm(ω)converges to ω. Therefore, {Tm(ω)} is a Cauchy sequence and its limit, again by (3.9), is ω.Then, from the continuity of T , T(ω) = ω.

In consequence, Fix(T) is nonempty. Now, since T is continuous, Fix(T) is closed. Weshow next that Fix(T) is also convex. Let u, v be two different points in Fix(T) and w themidpoint of the segment [u, v]. We need to show that w ∈ Fix(T). Now, since T is pointwiseasymptotic nonexpansive,

d(u, Tn(w)) = d(Tn(u), Tn(w)) ≤ αn(w)d(u,w) =αn(w)d(u, v)

2(3.12)

and, equally,

d(v, Tn(w)) = d(Tn(v), Tn(w)) ≤ αn(w)d(v,w) =αn(w)d(u, v)

2. (3.13)

Therefore, for ε > 0, there exists n0 such that if n ≥ n0 then

Tn(w) ∈ B(u,d(u, v)

2+ ε

)∩ B

(v,d(u, v)

2+ ε

)= Dε, (3.14)

but, from the proof of Proposition 2.2 in [3], the diameters of the sets Dε tend to 0 as ε tendsto 0 and so lim Tn(w) = w, which proves w is a fixed point of T .

Remark 3.12. The proof for the lower semicontinuous case follows in a similar way butfollowing the reasoning of Corollary 3.9.

In [1] a demiclosed principle is also given for asymptotic pointwise nonexpansivemappings in CAT(0) spaces. Next we show that an equivalent result is also possible foruniformly convex metric spaces. Following [1] we define

{xn}⇀C ω if and only if r(ω, xn) = infx∈C

r(x, xn), (3.15)

where C is a closed and convex subset of a uniformly convex metric space containing thebounded sequence {xn}. Notice that this definition does not depend on the set C when thespace X is a complete CAT(0) space. This is due to the fact that the asymptotic center ofa bounded sequence of a complete CAT(0) space belongs to the closed convex hull of thesequence, which easily follows from the very well-known fact that the metric projection ontoclosed convex subsets of a complete CAT(0) space is nonexpansive (see [2] for details). Recallthat the existence and uniqueness of such a ω ∈ C in a complete uniformly convex metricspaces with monotone modulus of convexity is guaranteed by Corollary 3.7.

Proposition 3.13. Let (X, d) be a complete uniformly convex metric space with a monotone modulusof convexity δ(r, ε). Let C be a bounded closed convex nonempty subset of X. Let T : C → C anasymptotic pointwise nonexpansive mapping. Let {xn} ∈ C be an approximate fixed point sequence,that is, limn→∞d(xn, T(xn)) = 0, and such that xn ⇀ ω for a certain ω ∈ C. Then T(ω) = ω.


Proof. Since {xn} is an approximate fixed point sequence, then we have that

r(x, xn) = lim supn→∞

d(x, Tm(xn)) = r(x, Tm(xn)) (3.16)

for any m ≥ 1( see Note Added in Proof at the end of the paper) . In consequence, sincer(Tm(x), Tm(xn)) ≤ αm(x)r(x, xn) for x ∈ C, (3.9) holds for any x.

Therefore, particularizing for ω, we have that lim supm→∞r(Tm(ω), xn) = r(ω, xn).

Now we claim that Tm(ω) → ω as m → ∞. Suppose on the contrary that there exist an ε > 0and a subsequence {Tmk(ω)} of {Tm(ω)} such that d(Tmk(ω), ω) ≥ ε for every k ∈ N. Let ωmk

be the midpoint of the geodesic segment [Tmk(ω), ω], c = diam(C) and ε1 = ε/c. By uniformconvexity, for every k, we have that

d(ωmk , xn) ≤ (1 − δ(max{d(ω, xn), d(Tmk(ω), xn)}, ε1))max{d(ω, xn), d(Tmk(ω), xn)}≤ (1 − δ(c, ε1))max{d(ω, xn), d(Tmk(ω), xn)}.

(3.17)

If we consider the upper limit of the above inequality when n → ∞, we get

r(ω, xn) ≤ r(ωmk , xn) ≤ (1 − δ(c, ε1))max{r(ω, xn), r(Tmk(ω), xn)}. (3.18)

If we do the same when k → ∞, we finally obtain that r(ω, xn) ≤ (1 − δ(c, ε1))r(ω, xn).Therefore r(ω, xn) = 0, and the existence of fixed point follows the same as in Theorem 3.11.

Remark 3.14. The proof for the lower semicontinuous case follows in a similar way butfollowing the reasoning of Corollary 3.9.

4. Fixed Points of Set-Valued Mappings

In this section we present fixed points theorems for set-valued mappings defined onuniformly convex metric spaces. Results stated for uniformly convex metric space with amonotone modulus of convexity also hold if there is a lower semicontinuous from the rightmodulus of convexity. Proofs of this second case will be omitted as they are based on technicalresults already proved for both kinds of modulus in Section 3. The Hausdorff metric on theclosed and bounded parts of a metric space X is defined as follows. If U and V are boundedand closed subsets of a metric space X, then

H(U,V ) = inf{ε > 0 : U ⊆Nε(V ), V ⊆Nε(U)}, (4.1)

where Nε(V ) = {y ∈ X : dist(y, V ) = inf{d(y, x) : x ∈ V } < ε}. Let C be a subset of a metricspace X. A mapping T : C → 2X with nonempty bounded closed values is nonexpansive if

H(T(x), T

(y)) ≤ d(x, y) (4.2)

for all x, y ∈ C. Our main goal in this section is to study if given X is a bounded uniformlyconvex metric space with monotone modulus of convexity, then every nonexpansive


mapping T : X → 2X with nonempty and compact values has a fixed point, that is, a pointx ∈ X such that x ∈ T(x). This problem was first solved in the affirmative by Shimizu andTakahashi in [9] under the assumption that the metric is convex. If we lack this condition theproblem is much more complicated and we can offer only partial answers. Our first answerwill be achieved after imposing condition (i):

(i) there exists a point x ∈ X such that for each t ∈ (0, 1) there is a number s(t) ∈ (0, 1)such that for all y, z ∈ X:

d(u, v) ≤ s(t) d(y, z), (4.3)

where u, v stand for points from geodesic segments [x, y] and [x, z], respectively,and d(x, u) = t d(x, y), d(x, v) = t d(x, z).

Remark 4.1. Condition (i) can be seen as a kind of very weak hyperbolicity condition. In fact,it is immediate to see that hyperbolic uniformly convex spaces studied in [3, 10, 12] satisfycondition (i) as well as for any uniformly convex CAT(k) space with k ∈ R. Notice here thatCAT(k) spaces with k ≤ 0 are particular examples of hyperbolic uniformly convex spaces.

Theorem 4.2. Let (X, d) be a complete uniformly convex metric space with a monotone modulusof convexity. Suppose that X is bounded and the condition (i) holds true. Then each nonexpansiveset-valued mapping T : X → 2X \ {∅} with compact values has a fixed point.

Proof. Let us fix x0 ∈ X satisfying the condition (i). Now, from (i) and the fact that T(x) iscompact for any x, it follows that set-valued mappings Tt : X → 2X , t ∈ (0, 1) defined by

Tt(x) ={u ∈ X : ∃ y ∈ T(x) s.t. u ∈ [

x0, y], d(x0, u) = t d

(x0, y

)}(4.4)

are compact-valued contractions with constants s(t). Nadler’s Fixed Point Theorem for set-valued contractions implies that Tt has a fixed point for each t and therefore T has anapproximate fixed point sequence, that is, a sequence {xn} such that lim dist(xn, T(xn)) = 0.According to Corollary 3.7 there is a unique asymptotic center of {xn} in X. Now the rest ofthe proof follows the same patterns of the standard one for uniformly convex Banach spaces(see [13, Theorem 15.3, page 165]).

It is wellknown that a CAT(1) space needs not to be uniformly convex if its diameteris not smaller than π/2. Next we show, however, that the same above idea can be applied toCAT(1) spaces of radius smaller than π/2. Remember that for k > 0 the numberDk is definedas Dk = π/

√k and that the radius of a bounded subset C of X is given by

radX(C) = inf{r : there exists x ∈ X such that C ⊆ B(x, r)}. (4.5)

Theorem 4.3. Let k > 0 and let (X, d) be a complete CAT(k) space with rad(X) < Dk/2. Then eachnonexpansive mapping T : X → 2X \ {∅} with compact values has at least one fixed point.

Proof. Take x0 ∈ X in such a way that supy∈Xd(x0, y) < Dk/2. As it is shown in [14, Lemma3], it is enough to take s(t) = sin(tπ/2) for t ∈ (0, 1) to verify condition (i) with x = x0 in X. Ina similar manner as above we obtain an approximate fixed point sequence {xn}. On account


of rad({xn}) ≤ rad(X) and [15, Proposition 4.1] the asymptotic center of each subsequenceof {xn} is unique, and the rest of the proof is not different from the case of uniformly convexBanach spaces.

Next we consider two further conditions to guarantee the existence of fixed points fornonexpansive set-valued mappings with compact values in uniformly convex metric spaces.

(ii) There is a function f : N × R+ → R+ such that

limδ→ 0+

f(n, δ) = 0, n ∈ N, (4.6)

and for all x, y, z ∈ X and u ∈ [x, y], v ∈ [x, z] was chosen in such a way thatd(x, u) = n−1 d(x, y), d(x, v) = n−1 d(x, z), and d(u, v) ≤ δ, we have

d(y, z

) ≤ f(n, δ) (4.7)

(iii) limr→R+,ε→ 2−δ(r, ε) = 1, where δ(r, ε) is, as usual, a monotone modulus of convexityof the space.

Remark 4.4. Notice that, roughly speaking, conditions (i) and (ii) give opposite informationabout the geometry of the space. While condition (i) implies that geodesic emanating from asame point must separate and no matters how fast, condition (ii) imposes a superior boundabout how much two geodesics emanating from a same point are allowed to separate. It iseasy to see that any geodesic space admitting bifurcating geodesics cannot verify condition(ii). In particular, condition (ii) does not hold in R-trees. It easily follows from the definitionsthat geodesic spaces with curvature bounded below by a real number k (check [16] for adetailed exposition about these spaces) satisfy condition (ii).

Before stating our next result we need to introduce a definition.

Definition 4.5. A geodesic space X is said to have the geodesic extension property if anygeodesic segment in X is actually contained in a geodesic line, that is, in a geodesic γ : R →X.

Theorem 4.6. Let X be a complete uniformly convex metric space with a monotone modulus ofconvexity and the geodesic extension property. Moreover, suppose that X satisfies conditions (ii) and(iii) andC is a nonempty bounded closed and convex subset ofX. If T : C → 2C\{∅} is a nonexpansivemappings with compact values then T has at least one fixed point.

Proof. From the proof of Theorem 4.2, we know that if infx∈Cdist(x, T(x)) = 0 then theconclusion follows. So let us suppose that

d := infx∈C

d(x, T(x)) > 0 (4.8)

and fix ε > 0 small enough. Let x0 be chosen in such a way that d(x0, T(x0)) < d + ε and letus denote this distance by d1. Clearly, from the compactness of T(x0) it follows the existenceof x1 ∈ T(x0) for which d(x0, x1) = d1. Now assume that m0 is a midpoint of the metric


segment [x0, x1]. The conditions imposed on T imply that there is m1 ∈ T(m0) such thatd(x1, m1) ≤ d1/2 and d(m0, m1) = d2 ≥ d.

To estimate the distance between x0 andm1 let us consider the midpoint p0 of [m0, m1].Using the uniformly convexity of X we obtain that

d(x1, p0

) ≤(

1 − δ(d1

2,

2dd1

))d1

2=: δ1. (4.9)

If we denote by x3/2 the point of the geodesic ray γ containing [x0, x1] such that d(x1, x3/2) =d1/2 and x3/2 /=m0, then d(m1, x3/2) ≤ f(2, δ1) and

d(x0, m1) ≥ 32d − f(2, δ1). (4.10)

Repeating our reasoning for m0, m1, and p0, we obtain points p1 ∈ T(p0) and m3/2 ∈ Xsatisfying

d(p1, m3/2

) ≤ f(2, δ2), d(m0, m3/2) =32d2, (4.11)

where

δ2 =(

1 − δ(d2

2,

2dd2

))d2

2. (4.12)

If we denote by x2 the point in γ for which d(x0, x2) = 2d1 and d(x1, x2) = d1, then it is easyto see that d(x2, m3/2) ≤ f(3, δ1). Hence

d(x0, p1

) ≥ d(x0, x2) − d(x2, m3/2) − d(m3/2, p1

) ≥ 2d − f(3, δ1) − f(2, δ2). (4.13)

The previous procedure gives us induction sequences of points {xn/2} and numbers{δn} such that

d(x0, xn/2) =n

2d1, (4.14)

and for n fixed, a point y ∈ C for which

d(y, xn/2

) ≤ n−1∑k=2

f(k, δn−k). (4.15)

Next let us fixM > 0 and findN ∈ N such thatM < (N−2)(d/2). Obviously, accordingto (ii), one can choose δ > 0 for which

N∑k=1

f(k, δ) < d, (4.16)


and by (iii) select ε > 0 such small that (1 − δ(r, d/r))((d + ε)/2) < δ for all r ∈ (d/2,(d + ε)/2].

Choosing x0 ∈ C such that d(x0, T(x0)) < d+ε and repeating our iterative procedureN-times, one may notice that the sequence {dn}Nn=1 decreases so δn < δ for each n ∈ {1, . . . ,N}.Finally we find y ∈ C for which, on account of (4.15),

d(y, xN/2

) ≤ N∑n=1

f(n, δ),

d(x0, y

) ≥Nd

2−

N∑n=1

f(n, δ) > (N − 2)d

2> M,

(4.17)

contrary to the boundedness of C. Hence d = 0 and the result follows.

From Theorems 3.11 and 4.2 one may get the following generalization of [1, Theorem5.2]. We omit the proof as it is analog to the one given for CAT(0) spaces in [1]. We first needsome notations and definitions.

Let X be a uniformly convex metric space. Consider the mappings t : X → X andT : X → 2X \ {∅}, then t and T are said to be commuting mappings if t(y) ∈ T(t(x)) for ally ∈ T(x) and for all x ∈ X. A point z is called a center for the mapping t : X → X if for eachx ∈ X, d(z, t(x)) ≤ d(z, x). The set Z(t) denotes the set of all centers of the mappings t.

Theorem 4.7. Let (X, d) be a bounded and complete uniformly convex space with a monotonemodulus of convexity for which condition (i) holds. Suppose that t : X → X is pointwiseasymptotically nonexpansive and T : X → 2X \ {∅} a nonexpansive mapping with compact andconvex values. If t and T commute and satisfy the condition

T(x) ∩ Fix(t) ⊂ Z(t), x ∈ Fix(t), (4.18)

then there is z ∈ X such that z = t(z) ∈ T(z).

Remark 4.8. The same result remains true if the uniformly convex metric space is supposed tohave the geodesic extension property, and conditions (ii)-(iii) hold instead of condition (i).

We finish this work with a last remark about a different condition to obtain anotherversion of Theorem 4.6. Let X be a metric space. Then we say that X has the Steckin property iffor ε, d, and r fixed positive numbers there exist ξ = ξ(ε, d, r) > 0 such that if x, y ∈ X satisfyd(x, y) = r, then

diam[B(y, d + ξ

) \ B(x, d + r)]< ε. (4.19)

The Steckin property was introduced in [17] to obtain different results regarding theexistence of unique nearest and farthest points to closed subsets of normed linear spaces. Thisproperty has been studied by many authors since then being of special relevance in the studyof uniformly convex Banach spaces; see for instance [18, 19]. It is known that the Steckinproperty does not happen in geodesic spaces with bifurcating geodesics [19, 20] and it has


been shown to be related to the property of having curvature bounded below (see [16] fordetails about spaces with curvature bounded below); see [19, 20]. We next show that thisproperty, in addition to a similar condition to (ii), leads to the existence of fixed points forset-valued nonexpansive mappings. We first introduce condition (ii)′:

(ii)′ there is a function f : N × R+ → R+ such that

limδ→ 0+

f(n, δ) = 0, n ∈ N, (4.20)

and for all x, y, z ∈ X and u ∈ [x, y], v ∈ [x, z] was chosen in such a way thatd(x, u) = (2/n)d(x, y), d(x, v) = (2/n)d(x, z), and d(u, v) ≤ δ, we have

d(y, z

) ≤ f(n, δ). (4.21)

Theorem 4.9. Let X be a complete uniformly convex metric space with a monotone modulus ofconvexity and the geodesic extension property. Moreover, suppose that X satisfies conditions (ii)′ andthe Steckin property. LetC be a nonempty bounded closed and convex subset ofX. If T : C → 2C\{∅}is a nonexpansive mappings with compact values then T has at least one fixed point.

Proof. We will omit details for this proof as it follows the same patterns of the proof ofTheorem 4.6. We will just point out that the main difference happens when the modulus ofconvexity is used to estimate d(x0, m1) first (see (4.10)), and d(y, xn/2) later (see (4.15)), nowwe use property S to achieve similar estimations.

Note Added in Proof

To prove (3.16) in Proposition 3.13 we need to ask for something more, in particular, itsuffices if we assume that T is uniformly continuous. Under this assumption we can proveby induction that {Tm(xn)} is an approximate fixed point sequence for each m ∈ N. Indeed,let m = 1, ε > 0 and choose δ = δ(ε) > 0 the one given by the uniform continuity of T . Since{xn} is an approximate fixed point sequence, there exists n0 ∈ N such that d(xn, T(xn)) ≤ δfor every n ≥ n0. This implies that

d(T(xn), T2(xn)

)≤ ε (4.22)

for every n ≥ n0. Thus {d(T(xn), T2(xn))} → 0 which proves our claim. Suppose now that{Tm−1(xn)} is an approximate fixed point sequence. We want to see that {Tm(xn)} is so too.Given ε > 0 we fix δ > 0 as in the case m = 1. Since {Tm−1(xn)} is an approximate fixed pointsequence, there exists n0 ∈ N such that d(Tm−1(xn), Tm(xn)) ≤ δ for every n ≥ n0. This impliesthat

d(Tm(xn), Tm+1(xn)

)≤ ε (4.23)

for every n ≥ n0. Thus {Tm(xn)} is an approximate fixed point sequence.


Now, since

d(x, Tm(xn)) ≤ d(x, xn) + d(xn, Tm(xn)), d(x, xn) ≤ d(x, Tm(xn)) + d(Tm(xn), xn), (4.24)

by recalling that d(Tm(xn), xn) ≤∑k=m−1

k=0 d(Tk(xn), Tk+1(xn)), (3.16) follows.

Acknowledgments

The two first authors were partially supported by the Ministery of Science and Technologyof Spain, Grant BFM 2000-0344-CO2-01 and La Junta de Antalucıa Project FQM-127. Thiswork was carried out while the third author was visiting the University of Seville. Sheacknowledges the kind hospitality of the Departamento de Analisis Matematico. This workis dedicated to W. A. Kirk.

References

[1] N. Hussain and M. A. Khamsi, “On asymptotic pointwise contractions in Metric spaces,” NonlinearAnalysis: Theory, Methods & Applications. In press.


[3] U. Kohlenbach and L. Leustean, “Asymptotically nonexpansive mappings in uniformlyconvex hyperbolic spaces,” to appear in Journal of the European Mathematical Society,http://arxiv.org/abs/0707.1626.

[4] W. A. Kirk, “Some recent results in metric fixed point theory,” Journal of Fixed Point Theory andApplications, vol. 2, no. 2, pp. 195–207, 2007.

[5] M. Arav, F. E. Castillo Santos, S. Reich, and A. J. Zaslavski, “A note on asymptotic contractions,” FixedPoint Theory and Applications, vol. 2007, Article ID 39465, 6 pages, 2007.


[7] M. A. Khamsi, “On asymptotically nonexpansive mappings in hyperconvex metric spaces,”Proceedings of the American Mathematical Society, vol. 132, no. 2, pp. 365–373, 2004.


[9] T. Shimizu and W. Takahashi, “Fixed point theorems in certain convex metric spaces,” MathematicaJaponica, vol. 37, no. 5, pp. 855–859, 1992.

[10] L. Leustean, “A quadratic rate of asymptotic regularity for CAT(0)-spaces,” Journal of MathematicalAnalysis and Applications, vol. 325, no. 1, pp. 386–399, 2007.

[11] M. A. Khamsi and W. A. Kirk, An Introduction toMetric Spaces and Fixed Point Theory, Pure and AppliedMathematics, Wiley-Interscience, New York, NY, USA, 2001.

[12] L. Leustean, “Nonexpansive iterations in uniformly convex W-hyperbolic spaces,” to appear inContemporary Mathematics, http://arxiv.org/abs/0810.4117.


[14] B. Piatek, “Halpern iteration in spherical spaces,” submitted to publication.[15] R. Espınola and A. Fernandez-Leon, “CAT(k)-spaces, weak convergence and fixed points,” Journal of

Mathematical Analysis and Applications, vol. 353, no. 1, pp. 410–427, 2009.[16] D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, vol. 33 of Graduate Studies in

Mathematics, American Mathematical Society, Providence, RI, USA, 2001.[17] S. B. Steckin, “Approximation properties of sets in normed linear spaces,” Revue Roumaine de

Mathematiques Pures et Appliquees, vol. 8, pp. 5–18, 1963 (Russian).[18] F. S. de Blasi, J. Myjak, and P. L. Papini, “Porous sets in best approximation theory,” Journal of the

London Mathematical Society, vol. 44, no. 1, pp. 135–142, 1991.


[19] A. Kaewcharoen and W. A. Kirk, “Proximinality in geodesic spaces,” Abstract and Applied Analysis,vol. 2006, Article ID 43591, 10 pages, 2006.

[20] T. Zamfirescu, “On the cut locus in Alexandrov spaces and applications to convex surfaces,” PacificJournal of Mathematics, vol. 217, no. 2, pp. 375–386, 2004.


for each n ∈N, where α ∈ (0, 1) and β ∈ [0, 1). When β = 0, it yields that Tα,0 = (1−α)I +αT =Tα. Therefore, the iteration scheme (II) becomes

xn = Tα(xn−1) = Tnα (x0). (1.1)

This iteration is called Mann iteration [14].The concepts of quasi-nonexpansive mappings, with respect to a sequence and

asymptotically regular mappings at a point were given in metric spaces as follows.

Definition 1.1 (see [6]). T : D → X is said to be quasi-nonexpansive mapping if for eachx ∈ D and for every p ∈ F(T), d(T(x), p) ≤ d(x, p).

Definition 1.2 (see [8]). The map T : D → X is said to be quasi-nonexpansive with respect to(xn) ⊆ D if for all n ∈N ∪ {0} and for every p ∈ F(T), d(xn+1, p) ≤ d(xn, p).

Lemma 2.1 in [8] stated that quasi-nonexpansiveness converts to quasi-nonexpansiveness with respect to (Tn(x0)) (resp., (Tnα (x0)), (Tnα,β(x0))) for each x0 ∈ D.The reverse implication is not true (see, [8, Example 2.1]). Also, the authors [8] showed thatthe continuity of T : D → X leads to the closedness of F(T) and the converse is not true (see,[8, Example 2.2]).

Definition 1.3 (see [15]). The mapping T : X → X is called an asymptotically regular at apoint x0 ∈ X if limn→∞d(Tn(x0), Tn+1(x0)) = 0.

The following definition is given by Angrisani and Clavelli.

Definition 1.4 (see [16]). Let X be a topological space. The function F : X → R is said to bea regular-global-inf (r.g.i) at x ∈ X if F(x) > infX(F) implies that there exists ε > 0 such thatε < F(x) − infX(F) and a neighborhood Nx of x such that F(y) > F(x) − ε for each y ∈ Nx. Ifthis condition holds for each x ∈ X, then F is said to be an r.g.i on X.

Definition 1.5 (see [17]). Let D be a convex subset of a normed space X. A mapping T : D →D is called directionally nonexpansive if ‖T(x) − T(m)‖ ≤ ‖x −m‖ for each x ∈ D and for allm ∈ [x, T(x)] where [x, y] denotes the segment joining x and y; that is, [x, y] = {λx+(1−λ)y :0 ≤ λ ≤ 1}.

Our objective in this paper is to introduce the concept of weakly quasi-nonexpansivemappings with respect to a sequence. Mainly, we establish some convergence theorems of asequence in complete metric spaces. These theorems generalize and improve [8, Theorems2.1 and 2.2], of [7, Theorems 1.1 and 1.1′], [5, Theorem 3.1], and [6, Proposition 1.1].

2. Main Result

In this section, we introduce the concept of weak quasi-nonexpansiveness of a mapping withrespect to a sequence that generalizes quasi-nonexpansiveness of a mapping with respectto a sequence in [8]. We give a lemma and a counterexample to show the relation betweenour new concept; the previous one appeared in [8] and a monotonically decreasing sequence(d(xn, F(T))).


Definition 2.1. Let (X, d) be a metric space and let (xn) be a sequence in D ⊆ X. Assume thatT : D → X is a mapping with F(T)/=φ satisfying limn→∞d(xn, F(T)) = 0. Thus, for a givenε > 0 there is a n1(ε) ∈ N such that d(xn, F(T)) < ε for all n ≥ n1(ε). T is called weaklyquasi-nonexpansive with respect to (xn) ⊆ D if for each ε > 0 there exists a p(ε) ∈ F(T) suchthat for all n ∈N with n ≥ n1(ε), d(xn, p(ε)) < ε.

We state the following lemma without proof.

Lemma 2.2. Let (X, d) be a metric space and, (xn) be a sequence inD ⊆ X. Assume that T : D → Xis a mapping with F(T)/=φ satisfying limn→∞d(xn, F(T)) = 0. If T is quasi-nonexpansive withrespect to (xn), then

(A) T is weakly quasi-nonexpansive with respect to (xn);

(B) (d(xn, F(T))) is a monotonically decreasing sequence in [0,∞).

The following example shows that the converse of Lemma 2.2 may not be true.

Example 2.3. Let X = [0, 1] be endowed with the Euclidean metric d. We define the mapT : X → X by T(x) = (3/4)x2 + (1/4)x for each x ∈ X. Assume that xn = 1/n for all n ∈N − {1, 2, 3}. Then

F(T) = {0, 1}, limn→∞

d(xn, F(T)) = limn→∞

d

(1n, F(T)

)= 0. (2.1)

Given ε > 0, there exists n1(ε) ∈N − {1, 2, 3} such that for all n ∈N − {1, 2, 3} with n ≥ n1(ε),there exists p = 0 ∈ F(T),

d(xn, 0) =∣∣∣∣ 1n− 0

∣∣∣∣ < ε. (2.2)

Thus, T is weakly quasi-nonexpansive with respect to (xn). But, T is not quasi-nonexpansivewith respect to (xn) (Indeed, there exists 1 ∈ F(T) such that for all n ∈N−{1, 2, 3}, d(xn+1, 1) >d(xn, 1)). Furthermore, the sequence (d(xn, F(T))) = (1/n) is monotonically decreasing in[0,∞).

Before stating the main theorem, let us introduce the following lemma without proof.

Lemma 2.4. Let (X, d) be a metric space and let (xn) be a sequence inD ⊆ X. Assume that T : D →X is weakly quasi-nonexpansive with respect to (xn) with F(T)/=φ satisfying limn→∞d(xn, F(T)) =0. Then, (xn) is a Cauchy sequence.

Now, we give the main theorem without proof in the following way.

Theorem 2.5. Let (xn) be a sequence in a subset D of a metric space (X, d) and let T : D → X be amap such that F(T)/=φ. Then

(a) limn→∞d(xn, F(T)) = 0 if (xn) converges to a point in F(T);

(b) (xn) converges to a point in F(T) if limn→∞d(xn, F(T)) = 0, F(T) is a closed set, T isweakly quasi-nonexpansive with respect to (xn), and X is complete.


As corollaries of Theorem 2.5, we have the following.

Corollary 2.6. For each x0 ∈ D, let (Tn(x0)) be a sequence in a subsetD of a metric space (X, d) andlet T : D → X be a map such that F(T)/=φ. Then

(a) limn→∞d(Tn(x0), F(T)) = 0 if (Tn(x0)) converges to a point in F(T);

(b) (Tn(x0)) converges to a point in F(T) if limn→∞d(Tn(x0), F(T)) = 0, F(T) is a closedset, T is weakly quasi-nonexpansive with respect to (Tn(x0)) and X is complete.

Corollary 2.7. For each x0 ∈ D, let (Tnα (x0)) be a sequence in a subsetD of a normed space (X, ‖ · ‖)and let T : D → X be a map such that F(T)/=φ. Then

(a) limn→∞d(Tnα (x0), F(T)) = 0 if (Tnα (x0)) converges to a point in F(T);

(b) (Tnα (x0)) converges to a point in F(T) if limn→∞d(Tnα (x0), F(T)) = 0, F(T) is a closedset, T is weakly quasi-nonexpansive with respect to (Tnα (x0)), and X is a Banach space.

Corollary 2.8. For each x0 ∈ D, let (Tnα,β(x0)) be a sequence in a subsetD of a normed space (X, ‖ ·‖)and let T : D → X be a map such that F(T)/=φ. Then

(a) limn→∞d(Tnα,β(x0), F(T)) = 0 if (Tnα,β

(x0)) converges to a point in F(T);

(b) (Tnα,β

(x0)) converges to a point in F(T) if limn→∞d(Tnα,β(x0), F(T)) = 0, F(T) is a closedset, T is weakly quasi-nonexpansive with respect to (Tn

α,β(x0)), and X is a Banach space.

Remark 2.9. (I) Theorem 2.5 generalizes and improves [8, Theorem 2.1] since T is weaklyquasi-nonexpansive with respect to (xn) instead of T being quasi-nonexpansive with respectto (xn).

(II) Corollary 2.6 generalizes and improves [7, Theorem 1.1 page 462] for somereasons. These reasons are the following:

(1) the closedness of D is superfluous;

(2) F(T) is closed instead of T being continuous;

(3) X is a complete metric space instead of X is a Banach space;

(4) T is weakly quasi-nonexpansive with respect to (Tn(x0)) in lieu of T being quasi-nonexpansive.

(III) Corollary 2.7 (resp. Corollary 2.8) generalizes and improves [7, Theorem 1.1′ page469] (resp. of [5, Theorem 3.1 page 98]) since the reasons (1) and (2) in (II) hold and

(1)′ the convexity of D in Theorem 1.1′ is superfluous;

(2)′ T is weakly quasi-nonexpansive with respect to (Tnα (x0)) (resp. (Tnα,β(x0)) instead ofT being quasi-nonexpansive.

(IV) If we take T : D → X instead of T : X → X, F(T) is closed in lieu of T : X → Xbeing continuous and T is weakly quasi-nonexpansive with respect to (Tn(x0)) in lieu of Tbeing quasi-nonexpansive, then Corollary 2.6 generalizes and improves Kirk [6, Proposition1.1].

In the light of Lemma 2.2 and Example 2.3, we state the following theorem.


Theorem 2.10. Let (xn) be a sequence in a subsetD of a complete metric space (X, d) and T : D → Xbe a map such that F(T)/=φ is a closed set. Assume that

(i) T is weakly quasi-nonexpansive with respect to (xn);

(ii) (d(xn, F(T))) is a monotonically decreasing sequence in [0,∞);

(iii) limn→∞d(xn, xn+1) = 0;

(iv) if the sequence (yn) satisfies limn→∞d(yn, yn+1) = 0, then

lim infnd(yn, F(T)

)= 0 or lim sup

nd(yn, F(T)

)= 0. (2.3)

Then (xn) converges to a point in F(T).

Proof. From the boundedness from below by zero of the sequence (d(xn, F(T))) and(ii), we obtain that limn→∞d(xn, F(T)) exists. So, from (iii) and (iv), we have thatlim infnd(xn, F(T)) = 0 or lim supnd(xn, F(T)) = 0. Then limn→∞d(xn, F(T)) = 0 (see, [18,page 37]). Therefore, by Theorem 2.5(b), the sequence (xn) converges to a point in F(T).

Corollary 2.11. For each x0 ∈ D, let (Tn(x0)) be a sequence in a subsetD of a complete metric space(X, d) and let T : D → X be a map such that F(T)/=φ is a closed set. Assume that

(i) T is weakly quasi-nonexpansive with respect to (Tn(x0));

(ii) (d(Tn(x0), F(T))) is a monotonically decreasing sequence in [0,∞);

(iii) limn→∞d(Tn(x0), Tn+1(x0)) = 0;

(iv) if the sequence (yn) satisfies limn→∞d(yn, yn+1) = 0, then

lim infnd(yn, F(T)

)= 0 or lim sup

nd(yn, F(T)

)= 0. (2.4)

Then (Tn(x0)) converges to a point in F(T).

Corollary 2.12. For each x0 ∈ D, let (Tnα (x0)) be a sequence in a subset D of a Banach space X andlet T : D → X be a map such that F(T)/=φ is a closed set. Assume that

(i) T is weakly quasi-nonexpansive with respect to (Tnα (x0));

(ii) (d(Tnα (x0), F(T))) is a monotonically decreasing sequence in [0,∞);

(iii) limn→∞‖Tnα (x0) − Tn+1α (x0)‖ = 0;

(iv) if the sequence (yn) satisfies limn→∞‖yn − yn+1‖ = 0, then

lim infnd(yn, F(T)

)= 0 or lim sup

nd(yn, F(T)

)= 0. (2.5)

Then (Tnα (x0)) converges to a point in F(T).

Corollary 2.13. For each x0 ∈ D, let (Tnα,β

(x0)) be a sequence in a subset D of a Banach space X andlet T : D → X be a map such that F(T)/=φ is a closed set. Assume that


(i) T is weakly quasi-nonexpansive with respect to (Tnα,β(x0));

(ii) (d(Tnα,β(x0), F(T))) is a monotonically decreasing sequence in [0,∞);

(iii) limn→∞‖Tnα,β(x0) − Tn+1α,β

(x0)‖ = 0;

(iv) if the sequence (yn) satisfies limn→∞‖yn − yn+1‖ = 0, then

lim infnd(yn, F(T)

)= 0 or lim sup

nd(yn, F(T)

)= 0. (2.6)

Then (Tnα,β

(x0)) converges to a point in F(T).

Remark 2.14. From Lemma 2.2, we find that [8, Theorem 2.2] is a special case of Theorem 2.10.Also, Corollary 2.11 generalizes and improves [7, Theorem 1.2 page 464] for the same reasonsin Remark 2.9(II).

We establish another consequence of Theorem 2.5 as follows.

Theorem 2.15. Let (xn) be a sequence in a subset D of a complete metric space (X, d). Furthermore,let T : D → X be a mapping such that F(T)/=φ is a closed set. Assume that the conditions (i) and(ii) in Theorem 2.10 hold and

(iii)′ the sequence (xn) contains a convergent subsequence (xnj ) converging to x∗ ∈ D such that

there exists a continuous mapping S : D → D satisfying S(xnj ) = xnj+1 for all j ∈N andd(S(x∗), p) < d(x∗, p) for some p ∈ F(T).

Then x∗ ∈ F(T) and limn→∞xn = x∗.

Proof. From (ii), one can deduce that limn→∞d(xn, F(T)) exists, say equal r ∈ [0,∞). Supposethat x∗ does not belong to F(T). So, we have from (iii)′ that for some p ∈ F(T),

d(x∗, p

)>d

(S(x∗), p

)=d

(S

(limj→∞

xnj

), p

)=d

(limj→∞

S(xnj

), p

)=d

(limj→∞

xnj+1, p

)=d

(x∗, p

).

(2.7)

This contradiction implies that x∗ ∈ F(T). Then,

r = limn→∞

d(xn, F(T)) = limj→∞

d(xnj , F(T)

)= d

(limj→∞

xnj , F(T))

= d(x∗, F(T)) = 0. (2.8)

From Theorem 2.5(b), we obtain that limn→∞xn = x∗.

Corollary 2.16. For each x0 ∈ D, let (Tn(x0)) be a sequence in a subsetD of a complete metric space(X, d). Furthermore, let T : D → X be a mapping such that F(T)/=φ is a closed set. Assume that theconditions (i) and (ii) in Corollary 2.11 hold and

(iii)′ the sequence (Tn(x0)) contains a convergent subsequence (Tnj (x0)) converging to x∗ ∈ D


such that there exists a continuous mapping S : D → D satisfying S(Tnj (x0)) =Tnj+1(x0) for all j ∈N and d(S(x∗), p) < d(x∗, p) for some p ∈ F(T).

Then x∗ ∈ F(T) and limn→∞Tn(x0) = x∗.

Corollary 2.17. For each x0 ∈ D, let (Tnα (x0)) be a sequence in a subsetD of a complete metric space(X, d). Furthermore, let T : D → X be a mapping such that F(T)/=φ is a closed set. Assume that theconditions (i) and (ii) in Corollary 2.12 hold and

(iii)′ the sequence (Tnα (x0)) contains a convergent subsequence (Tnjα (x0)) converging to x∗ ∈

D such that there exists a continuous mapping S : D → D satisfying S(Tnjα (x0)) =

Tnj+1α (x0) for all j ∈N and d(S(x∗), p) < d(x∗, p) for some p ∈ F(T).

Then x∗ ∈ F(T) and limn→∞Tnα (x0) = x∗.

Corollary 2.18. For each x0 ∈ D, let (Tnα,β

(x0)) be a sequence in a subsetD of a complete metric space(X, d). Furthermore, let T : D → X be a mapping such that F(T)/=φ is a closed set. Assume that theconditions (i) and (ii) in Corollary 2.13 hold and

(iii)′ the sequence (Tnα,β

(x0)) contains a convergent subsequence (Tnjα,β

(x0)) converging to x∗ ∈D such that there exists a continuous mapping S : D → D satisfying S(T

njα,β(x0)) =

Tnj+1α,β

(x0) for all j ∈N and d(S(x∗), p) < d(x∗, p) for some p ∈ F(T).Then x∗ ∈ F(T) and limn→∞Tnα,β(x0) = x∗.

Remark 2.19. Theorem 1.3 in [7] is a special case of Corollary 2.16 for the same reasons inRemark 2.9(II) and for the generalization of the conditions (1.6) and (1.7) in [7, Theorem 1.3]to the condition (iii)′ in Corollary 2.16.

From [17, Corollary 2.4] and Theorem 2.5(b), one can prove the following theorem.

Theorem 2.20. Let T : X → X be a mapping of a complete metric space (X, d) satisfying

(i) d(T(x), T2(x)) ≤ hd(x, T(x)) for some h ∈ (0, 1) and for all x ∈ X;

(ii) μ(T(Lc)) ≤ kμ(Lc) for some k < 1 and for all c > 0;

(iii) F is an r.g.i. on X;

(iv) (xn) is a sequence in X such that limn→∞d(xn, Txn) = 0 and T is weakly quasi-nonexpansive with respect to (xn).


Corollary 2.21. Let T : X → X be a mapping of a complete metric space (X, d) satisfying

(i) d(T(x), T2(x)) ≤ hd(x, T(x)) for some h ∈ (0, 1) and for all x ∈ X;



(iv) (Tn(x0)) is a sequence satisfying limn→∞d(Tn(x0), Tn+1(x0)) = 0 for each x0 ∈ X and Tis weakly quasi-nonexpansive with respect to (Tn(x0)).



Corollary 2.22. Let T : X → X be a mapping of a Banach space (X, d) satisfying

(i) ‖T(x) − T2(x)‖ ≤ h‖x − T(x)‖ for some h ∈ (0, 1) and for all x ∈ X;



(iv) (Tnα (x0)) is a sequence in X such that limn→∞‖Tnα (x0) − TTnα (x0)‖ = 0 for each x0 ∈ Xand T is weakly quasi-nonexpansive with respect to (Tnα (x0)).


Corollary 2.23. Let T : X → X be a mapping of a Banach space (X, d) satisfying

(i) ‖T(x) − T2(x)‖ ≤ h‖x − T(x)‖ for some h ∈ (0, 1) and for all x ∈ X;



(iv) (Tnα,β

(x0)) is a sequence inX such that limn→∞‖Tnα,β(x0)−TTnα,β(x0)‖ = 0 for each x0 ∈ Xand T is weakly quasi-nonexpansive with respect to (Tnα,β(x0)).

Then (Tnα,β


Theorem 2.24. LetD be a bounded closed convex subset of a Banach space X. Suppose that T : D →D satisfies

(i) T is directionally nonexpansive on D;


(iii) F is an r.g.i. on D;

(iv) (xn) ⊆ D satisfies limn→∞‖xn−Txn‖ = 0 and T is weakly quasi-nonexpansive with respectto (xn).


Proof. The conclusion is obtained by combining [17, Theorem 3.3] and Theorem 2.5(b).

Corollary 2.25. LetD be a bounded closed convex subset of a Banach spaceX. Suppose that T : D →D satisfies




(iv) (Tn(x0)) for each x0 ∈ D satisfies limn→∞‖Tn(x0) − Tn+1(x0)‖ = 0 and T is weaklyquasi-nonexpansive with respect to (Tn(x0)).







(iv) (Tnα (x0)) for each x0 ∈ D satisfies limn→∞‖Tnα (x0) − TTnα (x0)‖ = 0 and T is weaklyquasi-nonexpansive with respect to (Tnα (x0)).






(iv) (Tnα,β(x0)) for each x0 ∈ D satisfies limn→∞‖Tnα,β(x0) − TTnα,β(x0)‖ = 0 and T is weaklyquasi-nonexpansive with respect to (Tn

α,β(x0)).

Then (Tnα,β


Remark 2.28. It is worth to mention that Corollaries 2.12, 2.13, 2.17, 2.18, 2.21–2.23, 2.25–2.27are new results.

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