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Volume 26, Number 8 July 1st, 2019 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fifteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA. Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $800, Electronic OPEN ACCESS. Individual:Print $400. For any other part of the world add $160 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2019 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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Editorial Board

Associate Editors of Journal of Computational Analysis and Applications

Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators. Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities. J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago, IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis.

Martin Bohner Department of Mathematics and Statistics, Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology. Jerry L. Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations George Cybenko Thayer School of Engineering

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Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks Sever S. Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001, AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities, Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics. Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR-06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications Saber N. Elaydi Department Of Mathematics Trinity University 715 Stadium Dr. San Antonio, TX 78212-7200 210-736-8246 e-mail: [email protected] Ordinary Differential Equations, Difference Equations J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected] Partial Differential Equations, Semigroups of Operators H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany

011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales, control theory and their applications Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics Tian-Xiao He Department of Mathematics and Computer Science P.O. Box 2900, Illinois Wesleyan University Bloomington, IL 61702-2900, USA Tel (309)556-3089 Fax (309)556-3864 [email protected] Approximations, Wavelet, Integration Theory, Numerical Analysis, Analytic Combinatorics Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators) Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected]

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Computational Mathematics Jong Kyu Kim Department of Mathematics Kyungnam University Masan Kyungnam,631-701,Korea Tel 82-(55)-249-2211 Fax 82-(55)-243-8609 [email protected] Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, ODE, PDE, Functional Equations. Robert Kozma Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA [email protected] Neural Networks, Reproducing Kernel Hilbert Spaces, Neural Percolation Theory Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston, RI 02881,USA [email protected] Differential and Difference Equations Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected]

Burkhard Lenze Fachbereich Informatik Fachhochschule Dortmund University of Applied Sciences Postfach 105018 D-44047 Dortmund, Germany e-mail: [email protected] Real Networks, Fourier Analysis, Approximation Theory Hrushikesh N. Mhaskar Department Of Mathematics California State University Los Angeles, CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks

Ram N. Mohapatra Department of Mathematics University of Central Florida Orlando, FL 32816-1364 tel.407-823-5080 [email protected] Real and Complex Analysis, Approximation Th., Fourier Analysis, Fuzzy Sets and Systems Gaston M. N'Guerekata Department of Mathematics Morgan State University Baltimore, MD 21251, USA tel: 1-443-885-4373 Fax 1-443-885-8216 Gaston.N'[email protected] [email protected] Nonlinear Evolution Equations, Abstract Harmonic Analysis, Fractional Differential Equations, Almost Periodicity & Almost Automorphy M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations, Optimization, Signal Analysis Mubenga N. Nkashama Department OF Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 205-934-2154 e-mail: [email protected] Ordinary Differential Equations, Partial Differential Equations Vassilis Papanicolaou Department of Mathematics National Technical University of Athens Zografou campus, 157 80 Athens, Greece tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability

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Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, [email protected] Functional Equations Svetlozar (Zari) Rachev, Professor of Finance, College of Business, and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 11794-3775 tel: +1-631-632-1998, [email protected] Alexander G. Ramm Mathematics Department Kansas State University Manhattan, KS 66506-2602 e-mail: [email protected] Inverse and Ill-posed Problems, Scattering Theory, Operator Theory, Theoretical Numerical Analysis, Wave Propagation, Signal Processing and Tomography Tomasz Rychlik Polish Academy of Sciences Instytut Matematyczny PAN 00-956 Warszawa, skr. poczt. 21 ul. Śniadeckich 8 Poland [email protected] Mathematical Statistics, Probabilistic Inequalities Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected] Approximation Theory, Banach spaces, Classical Analysis T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St.

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Existence and global attractiveness of pseudo

almost periodic solutions to impulsive partial

stochastic neutral functional differential equations

Zuomao Yan∗ and Fangxia Lu

January 5, 2018

Abstract: In this paper, we introduce a new concept of p-mean piecewisepseudo almost periodic for a stochastic process and establish a new compositiontheorem about pseudo almost periodic functions under non-Lipschitz conditions.Using this composition theorem, the analytic semigroup theory and fixed pointstrategy with stochastic analysis theory, we also study the existence and theglobal attractiveness for p-mean piecewise pseudo almost periodic mild solutionsfor impulsive partial neutral stochastic neutral functional differential equations.Moreover, an example is given to illustrate the general theorems.

2000 MR Subject Classification: 34A37; 60H10; 35B15; 34F05Keywords: Impulsive partial stochastic functional differential equations;

Pseudo almost periodic functions; Composition theorem; Analytic semigrouptheory; Fixed point

1 Introduction

The concept of pseudo almost periodic functions introduced initially by Zhang[1] is an important generalization of the classical almost periodic functions. Sincethen, there has been an intense interest in studying several extensions of thisconcept such as asymptotic pseudo almost periodic functions and Stepanov-likepseudo almost periodic functions. Some contributions on pseudo almost peri-odic type solutions to abstract differential equations have recently been made[2-8] and the references therein. On the other hand, it should be pointed outthat noise or stochastic perturbation is unavoidable and omnipresent in na-ture as well as that in man-made systems. Therefore, we must import thestochastic effects into the investigation of differential systems. The concept ofalmost periodicity is of great importance in probability for investigating stochas-tic processes. In fact, the existence of almost periodic, asymptotically almostperiodic and pseudo almost periodic solutions for stochastic differential systemshas been thoroughly investigated; see [9-18] and reference therein. In particu-lar, Bezandry and Diagana [19,20] introduced the concepts of p-mean pseudopseudo almost periodicity, and studied the existence of p-mean pseudo almost

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 26, NO.8, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

1343 YAN ET AL 1343-1386

periodic mild solutions to partial stochastic differential equations. Diop et al.[21] obtained the existence, uniqueness and global attractiveness of an p-meanpseudo almost periodic solution for stochastic evolution equation driven by afractional Brownian motion.

The theory of impulsive differential equations is an important branch of dif-ferential equations, which has an extensively physical background [22]. There-fore, it seems interesting to study the various types of impulsive differentialequations. The asymptotic properties of solutions of impulsive differential equa-tions have been considered by many authors. For example, Henrıquez et al. [23],Liu and Zhang [24], Stamov et al. [25-27] discussed the piecewise almost pe-riodic solutions of impulsive differential equations. Liu and Zhang [28], Cherif[29] established the existence and stability of piecewise pseudo almost peri-odic solutions to abstract impulsive differential equations. Bainov et al. [30]concerned with the asymptotic equivalence of impulsive differential equations.However, besides impulse effects and delays, stochastic effects likewise exist inreal systems. In recent years, several interesting results on impulsive partialstochastic systems have been reported in [31-33] and the references therein.Further, Zhang [34] obtained the existence and uniqueness of almost periodicsolutions for a class of impulsive stochastic differential equations with delay bymean of the Banach contraction principle. In [35], the authors investigated theexistence and stability of square-mean piecewise almost periodic solutions fornonlinear impulsive stochastic differential equations by using Schauder’s fixedpoint theorem. Neutral differential equations arise in many areas of appliedmathematics. For this reason, those equations have been of a great interestduring the last few decades. The literature relative to partial neutral stochas-tic differential equations is quite extensive; for more on this topic and relatedapplications we refer the reader to [36]. Similarly, for more on impulsive partialneutral stochastic functional differential equations we refer to [32,33,37,38]. Inthis paper, we study the existence and global attractiveness of p-mean piecewisepseudo almost periodic mild solutions to the following impulsive partial neutralstochastic neutral functional differential equations:

d[x(t)− h(t, xt)] = [Ax(t) + g(t, xt)]dt + f(t, xt)dW (t), (1)t ∈ R, t 6= ti, i ∈ Z,

∆x(ti) = x(t+i )− x(t−i ) = Ii(x(ti)), i ∈ Z, (2)

where A is the infinitesimal generator of an exponentially stable analytic semi-group T (t)t≥0 on a Hilbert space Lp(P, H) and W (t) is a two-sided stan-dard one-dimensional Brownian motion defined on the filtered probability space(Ω,F , P,Ft), where Ft = σW (u) −W (v);u, v ≤ t. The history xt ∈ D withq > 0, where xt being defined by xt(θ) = x(t + θ) for each θ ∈ [−q, 0]) andD = ψ : [−q, 0] → Lp(P, H), ψ continuous everywhere except for a finite num-ber of points at which ψ(s−) and ψ(s+) exist and ψ(s−) = ψ(s). The functionsh, g, f, Ii, ti satisfy suitable conditions which will be established later. The no-

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1344 YAN ET AL 1343-1386

tations x(t+i ), x(t−i ) represent the right-hand side and the left-hand side limitsof x(·) at ti, respectively.

To the best of our knowledge, the existence and global attractiveness ofp-mean piecewise pseudo almost periodic mild solutions for for nonlinear im-pulsive stochastic system (1)-(2) is an untreated original topic, which in fact isthe main motivation of the present paper. Although the papers [34,35] studiedthe piecewise almost periodic mild solution of impulsive stochastic differentialequations, besides the fact that [34,35] applies to the results under the Lipschitzconditions, the class of impulsive stochastic systems is also different from theone studied here. Further, many dynamical control systems arising from real-istic models can be described as impulsive partial neutral stochastic functionaldifferential systems. So it is natural to extend the concept of pseudo almostperiodicity to dynamical systems represented by these impulsive systems. Inthe paper, we will introduce the notion of p-mean piecewise pseudo almost peri-odic for stochastic processes, which, in turn generalizes all the above-mentionedconcepts, in particular, the notion of piecewise almost periodic. Then we willestablish a new composition theorem for p-mean pseudo almost periodic func-tions under non-Lipschitz conditions. As an application, we study and obtainthe existence and exponential stability of p-mean piecewise pseudo almost peri-odic mild solutions to system (1)-(2) by using the analytic semigroup theory andKrasnoselskii fixed point theorem with stochastic analysis theory. Such a resultgeneralizes most of known results on the existence of almost periodic solutionsof type system (1)-(2). It includes some results of almost periodic and pseudoalmost periodic solutions to stochastic differential equations without impulse.Moreover, the results are also new for deterministic systems with impulse.

The paper is organized as follows. In Section 2, we introduce some notationsand necessary preliminaries. In Section 3, we give the existence of p-meanpiecewise pseudo almost periodic mild solutions for (1)-(2). In Section 4, weestablish the global attractiveness of p-mean piecewise pseudo almost periodicmild solutions for (1)-(2). Finally, an example is given to illustrate our resultsin Section 5.

2 Preliminaries

Throughout the paper, N, Z,R and R+ stand for the set of natural numbers,integers, real numbers, positive real numbers, respectively. We assume that(H, ‖ · ‖), (K, ‖ · ‖K) are real separable Hilbert spaces and (Ω,F , P ) is sup-posed to be a filtered complete probability space. Define Lp(P, H), for p ≥ 1to be the space of all H-valued random variables V such that E ‖ V ‖p=

∫Ω‖

V ‖p dP < ∞. Then Lp(P, H) is a Banach space when it is equipped withits natural norm ‖ · ‖p defined by ‖ V ‖p= (

∫Ω

E ‖ V ‖p dP )1/p < ∞ foreach V ∈ Lp(P, H). Let C(R, Lp(P, H)), BC(R, Lp(P, H)) stand for the collec-tion of all continuous functions from R into Lp(P, H), the Banach space of allbounded continuous functions from R into Lp(P, H), equipped with the supnorm, respectively. We let L(K, H) be the space of all linear bounded operators

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from K into H, equipped with the usual operator norm ‖ · ‖L(K,H); in par-ticular, this is simply denoted by L(H) when K = H. Furthermore, L0

2(K, H)denotes the space of all Q-Hilbert-Schmidt operators from K to H with thenorm ‖ ψ ‖2

L02= Tr(ψQψ∗) < ∞ for any ψ ∈ L(K, H).

Definition 2.1 ([19]). A stochastic process x : R → Lp(P, H) is said to becontinuous provided that for any s ∈ R,

limt→s

E ‖ x(t)− x(s) ‖p= 0.

Definition 2.2 ([19]). A stochastic process x : R → Lp(P, H) is said to bestochastically bounded provided that

limN→∞

lim supt∈R

P ‖ x(t) ‖> N = 0.

Let T be the set consisting of all real sequences tii∈Z such that γ =infi∈Z(ti+1 − ti) > 0, limi→∞ ti = ∞, and limi→−∞ ti = −∞. For tii∈Z ∈ T,let PC(R, Lp(P, H)) be the space consisting of all stochastically bounded piece-wise continuous functions f : R → Lp(P, H) such that f(·) is stochasticallycontinuous at t for any t /∈ tii∈Z and f(ti) = f(t−i ) for all i ∈ Z; letPC(R × Lp(P, K), Lp(P, H)) be the space formed by all stochastically piece-wise continuous functions f : R × Lp(P, K) → Lp(P, H) such that for anyx ∈ Lp(P, K), f(·, x) ∈ PC(R, Lp(P, H)) and for any t ∈ R, f(t, ·) is stochasti-cally continuous at x ∈ Lp(P, K).Definition 2.3 ([19]). A function f ∈ C(R, Lp(P, H)) is said to be p-meanalmost periodic if for each ε > 0, there exists an l(ε) > 0, such that everyinterval J of length l(ε) contains a number τ with the property that E ‖ f(t +τ) − f(t) ‖p< ε for all t ∈ R. Denote by AP (R, Lp(P, H)) the set of suchfunctions.Definition 2.4 (Compare with [22]). A sequence xn is called p-mean almostperiodic if for any ε > 0, there exists a relatively dense set of its ε-periods, i.e.,there exists a natural number l = l(ε), such that for k ∈ Z, there is at least onenumber q in [k, k + l], for which inequality E ‖ xn+q − xn ‖p< ε holds for alln ∈ N. Denote by AP (Z, Lp(P, H))) the set of such sequences.

Define l∞(Z,Lp(P, H)) = x : Z → Lp(P, H) :‖ x ‖= supn∈Z(E ‖ x(n) ‖p

)1/p < ∞, and

PAP0(Z, Lp(P, H))

=

x ∈ l∞(Z, Lp(P, H)) : limn→∞

12n

n∑

j=−n

E ‖ x(n) ‖p dt = 0

.

Definition 2.5. A sequence xnn∈Z∈ l∞(Z, X) is called p-mean pseudo

almost periodic if xn = x1n + x2

n, where x1n ∈ AP (Z, Lp(P, H)), x2

n ∈ PAP0(Z,Lp(P, H)). Denote by PAP (Z, Lp(P, H)) the set of such sequences.Definition 2.6 (Compare with [22]). For tii∈Z ∈ T, the function f ∈PC(R, Lp(P, H)) is said to be p-mean piecewise almost periodic if the followingconditions are fulfilled:

4


1346 YAN ET AL 1343-1386

(i) tji = ti+j − ti, j ∈ Z, is equipotentially almost periodic, that is, for anyε > 0, there exists a relatively dense set Qε of R such that for each τ ∈ Qε

there is an integer q ∈ Z such that |ti+q − ti − τ | < ε for all i ∈ Z.

(ii) For any ε > 0, there exists a positive number δ = δ(ε) such that if thepoints t′ and t′′ belong to a same interval of continuity of ϕ and |t′−t′′| < δ,then E ‖ f(t′)− f(t′′) ‖p< ε.

(iii) For every ε > 0, there exists a relatively dense set Ω(ε) in R such that ifτ ∈ Ω(ε), then

E ‖ f(t + τ)− f(t) ‖p< ε

for all t ∈ R satisfying the condition |t − ti| > ε, i ∈ Z. The number τ iscalled ε-translation number of f.

We denote by APT (R, Lp(P, H)) the collection of all the p-mean piecewisealmost periodic functions. Obviously, the space APT (R, Lp(P, H)) endowedwith the sup norm defined by ‖ f ‖∞= supt∈R(E ‖ f(t) ‖p)1/p for any f ∈APT (R, Lp(P, H)) is a Banach space. Let UPC(R, Lp(P, H)) be the space ofall stochastic functions f ∈ PC(R, Lp(P, H)) such that f satisfies the condition(ii) in Definition 2.6.Definition 2.7. The function f ∈ PC(R × Lp(P, K), Lp(P, H)) is said to bep-mean piecewise almost periodic in t ∈ R uniform in x ∈ Lp(P, K) if for everycompact subset K ⊆ Lp(P, K), f(·, x) : x ∈ K is uniformly bounded, andgiven ε > 0, there exists a relatively dense subset Ωε such that

E ‖ f(t + τ, x)− f(t, x) ‖p< ε

for all x ∈ K, τ ∈ Ωε, and t ∈ R satisfying |t − ti| > ε. Denote by APT (R ×Lp(P, K), Lp(P, H)) the set of all such functions.

Similarly as the proof of [22, Lemma 35], one hasLemma 2.1. Assume that f ∈ APT (R, Lp(P, H)), the sequence xii∈Z ∈AP (Z, Lp(P, H)), and tji, j ∈ Z are equipotentially almost periodic. Then,for each ε > 0, there exist relatively dense sets Ωε of R and Ωε of Z such that

(i) E ‖ f(t + τ)− f(t) ‖p< ε for all t ∈ R, |t− ti| > ε, τ ∈ Ωε and i ∈ Z.

(ii) E ‖ xi+q − xi ‖p< ε for all q ∈ Ωε and i ∈ Z.

(iii) E ‖ xqi − τ ‖p< ε for all q, τ ∈ Ωε and i ∈ Z.

We need to introduce the new space of functions defined for each q > 0 by

PC0T (R, Lp(P, H), q)

=

f ∈ PC(R, Lp(P, H)) : limt→∞

(sup

θ∈[t−q,t]

E ‖ f(θ) ‖p

)= 0

,

5


1347 YAN ET AL 1343-1386

PAP 0T (R, Lp(P, H), q) =

f ∈ PC(R, Lp(P, H)) :

limr→∞

12r

∫ r

−r

(sup

θ∈[t−q,t]

E ‖ f(θ) ‖p

)dt = 0

,

PAP 0T (R× Lp(P, K), Lp(P, H), q)

=

f ∈ PC(R× Lp(P, K), Lp(P, H)) :

limr→∞

12r

∫ r

−r

(sup

θ∈[t−q,t]

E ‖ f(θ, x) ‖p

)dt = 0

uniformly with respect to x ∈ K,

where K is an arbitrary compact subset of Lp(P, K)

.

Similar to [4], one hasLemma 2.2. The spaces PAP 0

T (R, Lp(P, H), q) and PAP 0T (R×Lp(P, K), Lp(P,

H), q) endowed with the uniform convergence topology are Banach spaces.Definition 2.8. A function f ∈ PC(R, Lp(P, H)) is said to be p-mean piecewisepseudo almost periodic if it can be decomposed as f = f1 + f2, where f1 ∈APT (R, Lp(P, H)) and f2 ∈ PAP 0

T (R, Lp(P, H), q). Denoted by PAPT (R, Lp(P,H), q) the set of all such functions.

PAPT (R, Lp(P, H), q) is a Banach space with the sup norm ‖ · ‖∞ .Similar to [1,28], one has

Remark 2.1. (i) PAP 0T (R, Lp(P, H), q) is a translation invariant set of PC(R,

Lp(P, H))). (ii) PC0T (R, Lp(P, H), q) ⊂ PAP 0

T (R, Lp(P, H), q).Lemma 2.3. Let fnn∈N ⊂ PAP 0

T (R, Lp(P, H), q) be a sequence of functions.If fn converges uniformly to f, then f ∈ PAP 0

T (R, Lp(P, H), q).One can refer to Lemma 2.5 in [6] for the proof of Lemma 2.3.

Definition 2.9. A function f ∈ PC(R × Lp(P, K), Lp(P, H)) is said to be p-mean piecewise pseudo almost periodic if it can be decomposed as f = f1 + f2,where f1 ∈ APT (R×Lp(P, K), Lp(P, H)) and f2 ∈ PAP 0

T (R×Lp(P, K), Lp(P,H), q).

Denoted by PAPT (R× Lp(P, K), Lp(P, H), q) the set of all such functions.We need the following composition of p-mean pseudo almost periodic pro-

cesses.Lemma 2.4. Assume f ∈ PAPT (R × Lp(P, K), Lp(P, H), q). Suppose thatf(t, x) satisfies

E ‖ f(t, x)− f(t, y) ‖p≤ Λ(E ‖ x− y ‖p) (3)

for all t ∈ R, x, y ∈ Lp(P, K), where Λ is a concave and continuous nonde-creasing function from R+ to R+ such that Λ(0) = 0,Λ(s) > 0 for s > 0and

∫0+

dsΛ(s) = +∞. Here, the symbol

∫0+ stands for limε→0+

∫ +∞ε

. If φ(·) ∈PAPT (R, Lp(P, K), q) then f(·, φ(·)) ∈ PAPT (R, Lp(P, H), q).

6


1348 YAN ET AL 1343-1386

Proof. Assume that f = f1 + f2, φ = φ1 + φ2, where f1 ∈ APT (R× Lp(P, K),Lp(P, H)), f2 ∈ PAP 0

T (R×Lp(P, K), Lp(P, H), q), φ1 ∈ APT (R, Lp(P, H)), andφ2 ∈ PAP 0

T (R, Lp(P, H), q). Consider the decomposition

f(t, φ(t)) = f1(t, φ1(t)) + [f(t, φ(t))− f(t, φ1(t))] + f2(t, φ1(t)).

Since f1(·, φ1(·)) ∈ APT (R, Lp(P, H)), it remains to prove that both [f(·, φ(·))−f(·, φ1(·))] and f2(·, φ1(·)) belong to PAP 0

T (R, Lp(P, H), q). Indeed, using (3),it follows that

12r

∫ r

−r

(sup

θ∈[t−q,t]

E ‖ f(θ, φ(θ))− f(θ, φ1(θ)) ‖p

)dt

≤ 12r

∫ r

−r

(sup

θ∈[t−q,t]

Λ(E ‖ φ(θ)− φ1(θ) ‖p))

dt

=12r

∫ r

−r

(sup

θ∈[t−q,t]

Λ(E ‖ φ2(θ) ‖p))

dt,

noting that Λ is a concave and continuous nondecreasing function and Λ(0) = 0,we deduce that Λ(E ‖ φ2(θ) ‖p) ≤ Λ(supθ∈[t−q,t] E ‖ φ2(θ) ‖p), and

12r

∫ r

−r

(sup

θ∈[t−q,t]

Λ(E ‖ φ2(θ) ‖p))

dt

≤ 12r

∫ r

−r

Λ(

supθ∈[t−q,t]

E ‖ φ2(θ) ‖p

)dt

≤ Λ(

12r

∫ r

−r

(sup

θ∈[t−q,t]

E ‖ φ2(θ) ‖p

)dt

)→ 0 as r →∞,

which implies that [f(·, φ(·))− f(·, φ1(·))] ∈ PAP 0T (R, Lp(P, H), q).

Since φ1(R) is relatively compact in Lp(P, K) and f1 is uniformly continuouson sets of the form R × K where K ⊂ Lp(P, K) is compact subset, for ε > 0there exists ξ ∈ (0, ε) such that

E ‖ f1(t, z)− f1(t, z) ‖p≤ ε, z, z ∈ φ1(R)

with |z−z| < ξ. Now, fix z1, ..., zn ∈ φ1(R) such that φ1(R) ⊂ ⋃nj=1 Bξ(zj , L

p(P,

K)). Obviously, the sets Dj = φ−11 (Bξ(zj)) form an open covering of R, and

therefore using the sets B1 = D1, B2 = D2\D1 and Bj = Dj\⋃j−1

k=1 Dk oneobtains a covering of R by disjoint open sets. For t ∈ Bj , φ1(t) ∈ Bξ(zj),

E ‖ f2(t, φ1(t)) ‖p

≤ 3p−1E ‖ f(t, φ1(t))− f(t, zj) ‖p

+3p−1E ‖ −f1(t, φ1(t)) + f1(t, zj) ‖p +3p−1E ‖ f2(t, zj) ‖≤ 3p−1Λ(E ‖ φ1(t)− zj ‖p) + 3p−1ε + 3p−1E ‖ f2(t, zj) ‖≤ 3p−1Λ(ε) + 3p−1ε + 3p−1E ‖ f2(t, zj) ‖ .

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1349 YAN ET AL 1343-1386

Now using the previous inequality it follows that

12r

∫ r

−r

(sup

θ∈[t−q,t]

E ‖ f1(θ, φ1(θ)) ‖p

)dt

≤ 12r

n∑

j=1

∫

Bj∩[−r,r]

(sup

θ∈[t−q,t]

E ‖ f1(θ, φ1(θ)) ‖p

)dt

≤ 3p−1 12r

n∑

j=1

∫

Bj∩[−r,r]

[sup

j=1,...,n

(sup

θ∈[t−q,t]∩Bj

×E ‖ f(θ, φ1(θ))− f(θ, zj) ‖p

)]dt

+3p−1 12r

n∑

j=1

∫

Bj∩[−r,r]

[sup

j=1,...,n

(sup

θ∈[t−q,t]∩Bj

×E ‖ f1(θ, φ1(θ))− f1(θ, zj) ‖p

)]dt

+3p−1 12r

n∑

j=1

∫

Bj∩[−r,r]

[sup

j=1,...,n

(sup

θ∈[t−q,t]∩Bj

E ‖ f2(θ, zj) ‖p

)]dt

≤ 3p−1 12r

∫ r

−r

[Λ(ε) + ε]dt

+3p−1n∑

j=1

12r

∫ r

−r

(sup

θ∈[t−q,t]

E ‖ f2(θ, zj) ‖p

)dt.

In view of the above it is clear that f2(·, φ1(·)) belongs to PAP 0T (R, Lp(P, H), q).

This completes the proof.Lemma 2.5. Assume the sequence of vector-valued functions Iii∈Z is pseudoalmost periodic, and there is a concave nondecreasing function from R+ to R+

such that Λi(0) = 0,Λi(s) > 0 for > 0 and∫0+

dsΛi(s)

= +∞,

E ‖ Ii(x)− Ii(y) ‖p≤ Λi(E ‖ x− y ‖p)

for all x, y ∈ Lp(P, K), i ∈ Z. If φ ∈ PAPT (R, Lp(P, H), q)∩UPC(R, Lp(P, H))such that R(φ) ⊂ Lp(P, K), then Ii(φ(ti)) is pseudo almost periodic.Proof. Assume that φ = φ1 + φ2, where φ1 ∈ APT (R, Lp(P, H)), φ2 ∈PAP 0

T (R, Lp(P, H), q). Fix φ ∈ PAPT (R, Lp(P, H), q) ∩ UPC(R, Lp(P, H)),first we show φ(ti) is pseudo almost periodic. One can refer to Lemma 37in [22] that the sequence φ(ti) is almost periodic. Next we need to show thatφ(ti) ∈ PAP0(Z, Lp(P, H)). By the hypothesis, φ, φ1 ∈ UPC(R, Lp(P, H)), soφ2 ∈ UPC(R, Lp(P, H)). Let 0 < ε < 1, there exists 0 < ξ < min1, γ suchthat for t ∈ (ti − ξ, ti), i ∈ Z, we have

E ‖ φ2(t) ‖p≤ (1− ε)E ‖ φ2(ti) ‖p, i ∈ Z.

Since tji , i ∈ Z, j = 0, 1, ... are equipotentially almost periodic, t1i is an almostperiodic sequence. Here we assume a bound of t1i is Mt and |ti| ≥ |t−i|;

8


1350 YAN ET AL 1343-1386

therefore,

12ti

∫ ti

−ti

(sup

θ∈[t−q,t]

E ‖ φ2(θ) ‖p

)dt

≥ 12ti

i∑

j=−i+1

∫ tj

tj−ξ

(sup

θ∈[t−q,t]

E ‖ φ2(θ) ‖p

)dt

≥ 12ti

i∑

j=−i+1

ξ(1− ε)E ‖ φ2(tj) ‖p

≥ ξ(1− ε)Mt

12ti

i∑

j=−i+1

E ‖ φ2(tj) ‖p .

Since φ2 ∈ PAP 0T (R, Lp(P, H), q), it follows from the inequality above that

φ2(ti) ∈ PAP0(Z, Lp(P, H)). Hence, φ(ti) is pseudo almost periodic.Now, we show Ii(φ(ti)) is pseudo almost periodic. Let

I(t, x) = (t− n)In(x), n ≤ t < n + 1, n ∈ Z,

ϑ(t) = (t− n)φn(tn), n ≤ t < n + 1, n ∈ Z.

Since In, φ(tn) are two pseudo almost periodic sequences, Refer to Lemma1.7.12. in [39], we get that I ∈ PAP (R × Lp(P, K), Lp(P, H)), ϑ ∈ PAP (R,Lp(P, K)). For every t ∈ R, there exists a number n ∈ Z such that |t− n| ≤ 1,we have for x1, x2 ∈ Lp(P, K),

E ‖ I(t, x1)− I(t, x2) ‖p

≤ E ‖ In(x1)− In(x2) ‖p

≤ Λn(E ‖ x1 − x2 ‖p).

Similar to the proof of Lemma 2.4, I(·, ϑ(·)) ∈ PAP (R, Lp(P, H)). Again, sim-ilarly as the proof of Lemma 1.7.12 in [39], we have that I(i, ϑ(i)) is a pseudoalmost periodic sequence, that is, Ii(φ(ti)) is pseudo almost periodic. Thiscompletes the proof.

Let 0 ∈ ρ(A), then it is possible to define the fractional power Aα, for0 < α ≤ 1, as a closed linear operator on its domain D(Aα). Furthermore, thesubspace D(Aα) is dense in H and the expression ‖ x ‖α=‖ Aαx ‖, x ∈ D(Aα),defines a norm on D(Aα). Hereafter we denote by Hα the Banach space D(Aα)with norm ‖ x ‖α . Throughout the rest of this paper, we denote by ‖ · ‖α,∞the sup norm of the space PAPT (R, Lp(P, Hα)).Lemma 2.6 ([40]). Let 0 < α ≤ β ≤ 1. Then the following properties hold:

(a) Hβ is a Banach space and Hβ → Hα is continuous.

(b) The function s → AβT (s) is continuous in the uniform operator topologyon (0,∞) and there exists Mβ > 0 such that ‖ AβT (t) ‖≤ Mβe−δtt−β foreach t > 0.

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1351 YAN ET AL 1343-1386

(c) For each x ∈ D(Aβ) and t ≥ 0, AβT (t)x = T (t)Aβx.

(d) A−β is a bounded linear operator in H with D(Aβ) = Im(A−β).

Next, we introduce a useful compactness criterion on PC(R, Lp(P, H), q).Let h : R → R+ be a continuous function such that h(t) ≥ 1 for all t ∈ R

and h(t) →∞ as |t| → ∞. Define

PC0h(R, Lp(P, H), q)

=

f ∈ PC(R, Lp(P, H)) : lim|t|→∞

(sup

θ∈[t−q,t]

E ‖ f(θ) ‖p

h(θ)

)= 0

endowed with the norm ‖ f ‖h= supt∈R(supθ∈[t−q,t]E‖f(θ)‖p

h(θ)), it is a Banach

space.Lemma 2.7. A set B ⊆ PC0

h(R, Lp(P, H), q) is relatively compact if and only

if it verifies the following conditions:

(i) lim|t|→∞

(supθ∈[t−q,t]E‖f(t)‖p

h(t)) = 0 uniformly for f ∈ B.

(ii) B(t) = f(t) : f ∈ B is relatively compact in Lp(P, H) for every t ∈ R.

(iii) The set B is equicontinuous on each interval (ti, ti+1)(i ∈ Z).

One can refer to Lemma 4.1 in [28] for the proof of Lemma 2.7.Lemma 2.8 (Krasnoselskii’s Fixed Point Theorem [41]). Let D be a closed,bounded, and convex subset of a Banach space X. Let Ψ1 and Ψ2 be operators,defined on D satisfying the conditions:

(a) Ψ1x + Ψ2y ∈ D when x, y ∈ D.

(b) The operator Ψ1 is a contraction.

(c) The operator Ψ2 is continuous and Ψ2(D) is contained in a compact set.

Then the equation Ψ1x + Ψ2x = x has a solution on D.

3 Existence

In this section, we investigate the existence of p-mean piecewise pseudo almostperiodic mild solution for system (1)-(2). We begin introducing the followingsconcepts of mild solutions.Definition 3.1. An Ft -progressively measurable process x : [σ, σ+b) → H, b >0 is called a mild solution of system (1)-(2) on [σ, σ + b), if xσ = ϕ ∈ D, thefunction s → AT (t − s)h(s, xs) is integrable on [0, t) for every σ < t < σ + b,and σ 6= ti, i ∈ Z,

x(t) = T (t− σ)[ϕ(σ)− h(σ, ϕ)] + h(t, xt) +∫ t

σ

AT (t− s)h(s, xs)ds

10


1352 YAN ET AL 1343-1386

+∫ t

σ

T (t− s)g(s, xs)ds +∫ t

σ

T (t− s)f(s, xs)dW (s)

+∑

σ<ti<t

T (t− ti)Ii(x(ti)), t ∈ [σ, σ + b). (4)

Additionally, we make the following hypotheses:

(H1) A is the infinitesimal generator of a exponentially stable analytic semi-group (T (t))t≥0 on Lp(P, H) such that for all t ≥ 0, ‖ T (t) ‖≤ Me−δt withM, δ > 0. Moreover, T (t) is compact for t > 0.

(H2) There exist constants β, L > 0 such that 0 < β < 1, the function h ∈PAPT (R×D, Lp(P, Hβ), q), and

E ‖ Aβh(t1, ψ1)−Aβh(t2, ψ2) ‖p≤ L[|t1 − t2|+ ‖ ψ1 − ψ2 ‖pD],

t1, t2 ∈ R, ψ1, ψ2 ∈ D,

E ‖ Aβh(t, ψ) ‖p≤ L(‖ ψ ‖pD +1), t ∈ R, ψ ∈ D.

(H3) The functions g ∈ PAPT (R×D, Lp(P, H), q), f ∈ PAPT (R×D, Lp(P, L02),

q), and for each t ∈ R, ψ1, ψ2 ∈ D,

E ‖ g(t, ψ1)− g(t, ψ2) ‖p +E ‖ f(t, ψ1)− f(t, ψ2) ‖pL0

2

≤ Λ(E ‖ ψ1 − ψ2 ‖pD),

where Λ is a concave and continuous nondecreasing function from R+ toR+ such that Λ(0) = 0,Λ(s) > 0 for s > 0 and

∫0+

dsΛ(s) = +∞.

(H4) For any ρ1 > 0, there exist a constant µ > 0 and nondecreasing continuousfunction Θ : R+ → R+ such that, for all t ∈ R, and ψ ∈ D with E ‖ x ‖p

D>µ,

E ‖ g(t, ψ) ‖p +E ‖ f(t, ψ) ‖pL0

2≤ ρ1Θ(E ‖ ψ ‖p

D).

(H5) The functions Ii ∈ PAP (Z, Lp(P, H)), and for each t ∈ R, x1, x2 ∈Lp(P, H), i ∈ Z,

E ‖ Ii(x1)− Ii(x2) ‖p≤ Λi(E ‖ x1 − x2 ‖p),

where Λi are concave and continuous nondecreasing functions from R+ toR+ such that Λi(0) = 0, Λi(s) > 0 for s > 0 and

∫0+

dsΛi(s)

= +∞.

(H6) For any ρ2 > 0, there exist a constant µ > 0 and nondecreasing continuousfunction Θi : R+ → R+, i ∈ Z, such that, for all t ∈ R, and x ∈ Lp(P, H)with E ‖ x ‖p> µ,

E ‖ Ii(x) ‖p≤ ρ2Θi(E ‖ x ‖p).

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1353 YAN ET AL 1343-1386

To study the system (1)-(2) we need the following results.

Lemma 3.1. Assume that (H1) holds. If h ∈ PAPT (R, Lp(P, Hβ), q) and if His the function defined by

H(t) :=∫ t

−∞AT (t− s)h(s)ds

for each t ∈ R, then H ∈ PAPT (R, Lp(P, H), q).Proof. Since h ∈ PAPT (R, Lp(P, Hβ), q), there exist h1 ∈ APT (R, Lp(P, Hβ))and h2 ∈ PAP 0

T (R, Lp(P, Hβ), q), such that h = h1 + h2. Then H(t) can bedecomposed as

H(t) =∫ t

−∞AT (t− s)h1(s)ds +

∫ t

−∞AT (t− s)h2(s)ds =: H1(t) + H2(t).

Next we show that H1(t) ∈ APT (R, Lp(P, H)) and H2(t) ∈ PAP 0T (R, Lp(P, H),

q). Thus, the following verification procedure is divided into three steps.Step 1. H1 ∈ UPC(R, Lp(P, H)).Let t′, t′′ ∈ (ti, ti+1), i ∈ Z, t′′ < t′. By (H1), for any ε > 0, there exists

0 < ξ < ( ε2h1

)1/pβ such that 0 < t′ − t′′ < ξ, we have

‖ T (t′ − t′′)− I ‖p≤ δ1ε

2h1

,

where h1 = 2p−1Mp1−β(1− p(1−β)

p−1 )1−p ‖ h1 ‖pβ,∞, δ1 = (Γ(1− p(1−β)

p−1 ))p−1δ−pβ .Using Holder’s inequality, we have

E ‖ H1(t′)−H1(t′′) ‖p

≤ 2p−1E

wwww∫ t′′

−∞AT (t′′ − s)[T (t′ − t′′)− I]h1(s)ds

wwwwp

+2p−1E

wwww∫ t′

t′′AT (t′ − s)h1(s)ds

wwwwp

≤ 2p−1Mp1−β ‖ T (t′ − t′′)− I ‖p

×(∫ t′′

−∞(t′′ − s)−

pp−1 (1−β)e−δ(t′′−s)ds

)p−1

×(∫ t′′

−∞e−δ(t′′−s)E ‖ Aβh1(s) ‖p ds

)

+2p−1Mp1−β

(∫ t′

t′′(t′ − s)−

pp−1 (1−β)e−δ(t′−s)ds

)p−1

×(∫ t′

t′e−δ(t′−s)E ‖ Aβg1(s) ‖p ds

)

≤ 2p−1Mp1−β ‖ T (t′ − t′′)− I ‖p

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1354 YAN ET AL 1343-1386

×(

Γ(1− p(1− β)p− 1

)δp(1−β)

p−1 −1

)p−1 1δ

sups∈R

E ‖ h1(s) ‖pβ

+2p−1Mp1−β

(∫ t′

t′′(t′ − s)−

pp−1 (1−β)

)p−1

(t′ − t′′) sups∈R

E ‖ h1(s) ‖pβ

< 2p−1Mp1−β ‖ h1 ‖p

β,∞δ1ε

2h1

(Γ(1− p(1− β)

p− 1))p−1

δ−pβ

+2p−1Mp1−β

(1− p(1− β)

p− 1

)1−p

‖ h1 ‖pβ,∞

[(ε

2h1

) 1pβ

]pβ

<ε

2+

ε

2= ε.

Consequently, H1 ∈ UPC(R, Lp(P, H)).Step 2. H1 ∈ APT (R, Lp(P, H)).Let ti < t ≤ ti+1. For ε > 0, let Ωε be a relatively dense set of R formed by

ε-periods of F. For τ ∈ Ωε and 0 < η < minε, γ/2, we have

E ‖ H1(t + τ)−H1(t) ‖p

≤ E

wwww∫ t

−∞A1−βT (t− s)[Aβh1(s + τ)−Aβh1(s)]ds

wwwwp

≤ Mp1−β

(∫ t

−∞(t− s)−

pp−1 (1−β)e−δ(t−s)ds

)p−1

×(∫ t

−∞e−δ(t−s)E ‖ Aβh1(s + τ)−Aβh1(s) ‖p ds

)

≤ Mp1−β

(∫ t

−∞(t− s)−


)p−1

×[ i−1∑

j=−∞

∫ tj+1−η

tj+η

e−δ(t−s)E ‖ Aβh1(s + τ)−Aβh1(s) ‖p ds

+i−1∑

j=−∞

∫ tj+η

tj


+i−1∑

j=−∞

∫ tj+1

tj+1−η


+∫ t

ti


].

Since h1 ∈ APT (R, Lp(P, Hβ)), one has

E ‖ Aβh1(t + τ)−Aβh1(t) ‖p< ε

for all t ∈ [tj + η, tj+1 − η], j ∈ Z, j ≤ i, and t − s ≥ t − ti + ti − (tj+1 − η) ≥

13


1355 YAN ET AL 1343-1386

t− ti + γ(i− 1− j) + η. Then,

i−1∑

j=−∞

∫ tj+1−η

tj+η

e−δ(t−s)E ‖ Aβh1(s + τ)−Aβh1(s) ‖p

≤ εi−1∑

j=−∞

∫ tj+1−η

tj+η

e−δ(t−s)ds

≤ ε

δ

i−1∑

j=−∞e−δ(t−tj+1+η)

≤ ε

δ

i−1∑

j=−∞e−δγ(i−j−1)

≤ ε

δ(1− e−δγ),

i−1∑

j=−∞

∫ tj+η

tj


≤ 2p−1 sups∈R

E ‖ Aβh1(s) ‖pi−1∑

j=−∞

∫ tj+η

tj

e−δ(t−s)ds

≤ 2p−1 ‖ h1 ‖pβ,∞ εeδη

i−1∑

j=−∞e−δ(t−tj)

≤ 2p−1 ‖ h1 ‖pβ,∞ εeδηe−δ(t−ti)

i−1∑

j=−∞e−δγ(i−j)

≤ 2p−1 ‖ h1 ‖pβ,∞ eδγ/2ε

1− e−δγ.

Similarly, one has

i−1∑

j=−∞

∫ tj+1

tj+1−η

e−δ(t−s)E ‖ Aβh1(s + τ)−Aβh1(s) ‖p ds ≤ M1ε,

∫ t

ti

e−δ(t−s)E ‖ Aβh1(s + τ)−Aβh1(s) ‖p ds ≤ M2ε,

where M1, M2 are some positive constants. Therefore, we get that E ‖ H1(t +τ)−H1(t) ‖p≤ N1ε for a positive constant N1. Hence, H1 ∈ APT (R, Lp(P, H)).

Step 3. H2 ∈ PAP 0T (R, Lp(P, H), q).

In fact, for r > 0, one has

12r

∫ r

−r

supθ∈[t−q,t]

E ‖ H2(θ) ‖p dt

14


1356 YAN ET AL 1343-1386

=12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ θ

−∞AT (θ − s)h2(s)ds

wwwwp

dt

=12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ ∞

0

A1−βT (s)Aβh2(θ − s)ds

wwwwp

dt

≤ Mp1−β

12r

∫ r

−r

(∫ ∞

0

s−(1−β)e−δsds

)p−1

×∫ ∞

0

e−δs supθ∈[t−q,t]

E ‖ Aβh2(θ − s) ‖p dsdt

= Mp1−β

(∫ ∞

0

s−(1−β)e−δsds

)p−1

×∫ ∞

0

e−δsds12r

∫ r

−r

supθ∈[t−q,t]

E ‖ Aβh2(θ − s) ‖p dt.

Since h2 ∈ PAP 0T (R, Lp(P, Hβ), q), it follows that h2(· − s) ∈ PAP 0

T (R, Lp(P,Hβ), q) for each s ∈ R by Remark 2.1, hence

12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ θ

−∞AT (θ − s)h2(s)ds

wwwwp

dt → 0 as r →∞

for all s ∈ R. Using the Lebesgue’s dominated convergence theorem, we haveH2 ∈ PAP 0

T (R, Lp(P, H), q). This completes the proof.Lemma 3.2. Assume that (H1) holds. If g ∈ PAPT (R, Lp(P, H), q) and if Gis the function defined by

G(t) :=∫ t

−∞T (t− s)g(s)ds

for each t ∈ R, then G ∈ PAPT (R, Lp(P, H), q).Proof. Since g ∈ PAPT (R, Lp(P, H), q), there exist g1 ∈ APT (R, Lp(P, H))and g2 ∈ PAP 0

T (R, Lp(P, H), q), such that g = g1 + g2. Then G(t) can bedecomposed as

G(t) =∫ t

−∞T (t− s)g1(s)ds +

∫ t

−∞T (t− s)g2(s)ds =: G1(t) + G2(t).

Next we show that G1(t) ∈ APT (R, Lp(P, H)) and G2(t) ∈ PAP 0T (R, Lp(P, H),

q). Thus, the following verification procedure is divided into three steps.Step 1. G1 ∈ UPC(R, Lp(P, H)).Let t′, t′′ ∈ (ti, ti+1), i ∈ Z, t′′ < t′. By (H1), for any ε > 0, there exists

0 < ξ < ( ε2g1

)1/p such that 0 < t′ − t′′ < ξ, we have

‖ T (t′ − t′′)− I ‖p≤ δ1ε

2g1,

15


1357 YAN ET AL 1343-1386

where g1 = 2p−1Mp ‖ g1 ‖p∞, δ2 = δ−p. Using Holder’s inequality, we have

E ‖ G1(t′)−G1(t′′) ‖p

≤ 2p−1E

wwww∫ t′′

−∞T (t′′ − s)[T (t′ − t′′)− I]g1(s)ds

wwwwp

+2p−1E

wwww∫ t′

t′′T (t′ − s)g1(s)ds

wwwwp

≤ 2p−1Mp ‖ T (t′ − t′′)− I ‖p

(∫ t′′

−∞e−δ(t′′−s)ds

)p−1

×(∫ t′′

−∞e−δ(t′′−s)E ‖ g1(s) ‖p ds

)

+2p−1Mp

(∫ t′

t′′e−δ(t′−s)ds

)p−1(∫ t′

t′e−δ(t′−s)E ‖ g1(s) ‖p ds

)

≤ 2p−1Mp ‖ T (t′ − t′′)− I ‖p 1δp

sups∈R

E ‖ g1(s) ‖p

+2p−1Mp(t′ − t′′)p sups∈R

E ‖ g1(s) ‖p

< 2p−1Mp ‖ g1 ‖p∞

δ2ε

2g1

1δp

+ 2p−1Mp ‖ g1 ‖p∞

[(ε

2g1

) 1p]p

<ε

2+

ε

2= ε.

Consequently, G1 ∈ UPC(R, Lp(P, H)).Step 2. G1 ∈ APT (R, Lp(P, H)).Let ti < t ≤ ti+1. For ε > 0, let Ωε be a relatively dense set of R formed by


E ‖ G1(t + τ)−G1(t) ‖p

≤ E

wwww∫ t

−∞T (t− s)[g1(s + τ)− g1(s)]ds

wwwwp

≤ Mp

(∫ t

−∞e−δ(t−s)ds

)p−1

×(∫ t

−∞e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p ds

)

≤ Mp

(∫ t


)p−1

×[ i−1∑

j=−∞

∫ tj+1−η

tj+η

e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p ds

+i−1∑

j=−∞

∫ tj+η

tj

e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p ds

16


1358 YAN ET AL 1343-1386

+i−1∑

j=−∞

∫ tj+1

tj+1−η

e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p ds

+∫ t

ti

e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p ds

].

Since g1 ∈ APT (R, Lp(P, H)), one has

E ‖ g1(t + τ)− g1(t) ‖p< ε

for all t ∈ [tj + η, tj+1 − η], j ∈ Z, j ≤ i, and t − s ≥ t − ti + ti − (tj+1 − η) ≥t− ti + γ(i− 1− j) + η. Then,

i−1∑

j=−∞

∫ tj+1−η

tj+η

e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p

≤ εi−1∑

j=−∞

∫ tj+1−η

tj+η

e−δ(t−s)ds

≤ ε

δ

i−1∑

j=−∞e−δ(t−tj+1+η)

≤ ε

δ

i−1∑

j=−∞e−δγ(i−j−1)

≤ ε

δ(1− e−δγ),

i−1∑

j=−∞

∫ tj+η

tj

e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p ds

≤ 2p−1 sups∈R

E ‖ g1(s) ‖pi−1∑

j=−∞

∫ tj+η

tj

e−δ(t−s)ds

≤ 2p−1 ‖ g1 ‖p∞ εeδη

i−1∑

j=−∞e−δ(t−tj)

≤ 2p−1 ‖ g1 ‖p∞ εeδηe−δ(t−ti)

i−1∑

j=−∞e−δγ(i−j)

≤ 2p−1 ‖ g1 ‖p∞ eδγ/2ε

1− e−δγ.

Similarly, one has

i−1∑

j=−∞

∫ tj+1

tj+1−η

e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p ds ≤ M3ε,

17


1359 YAN ET AL 1343-1386

∫ t

ti

e−δ(t−s)E ‖ g1(s + τ)− g1(s) ‖p ds ≤ M4ε,

where M3, M4 are some positive constants. Therefore, we get that E ‖ G1(t +τ)−G1(t) ‖p≤ N2ε for a positive constant N2. Hence, G1 ∈ APT (R, Lp(P, H)).

Step 3. G2 ∈ PAP 0T (R, Lp(P, H), q).

In fact, for r > 0, one has

12r

∫ r

−r

supθ∈[t−q,t]

E ‖ G2(θ) ‖p dt

=12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ θ

−∞T (θ − s)g2(s)ds

wwwwp

dt

=12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ ∞

0

T (s)g2(θ − s)ds

wwwwp

dt

≤ Mp 12r

∫ r

−r

(∫ ∞

0

e−δsds

)p−1

×∫ ∞

0

e−δs supθ∈[t−q,t]

E ‖ g2(θ − s) ‖p dsdt

= Mp

(∫ ∞

0

e−δsds

)p−1 ∫ ∞

0

e−δsds

× 12r

∫ r

−r

supθ∈[t−q,t]

E ‖ g2(θ − s) ‖p dt.

Since g2 ∈ PAP 0T (R, Lp(P, Hβ), q), it follows that g2(·−s) ∈ PAP 0

T (R, Lp(P, H),q) for each s ∈ R by Remark 2.1, hence

12r

∫ r

−r

supθ∈[t−p,t]

E

wwww∫ θ

−∞T (θ − s)g2(s)ds

wwwwp

dt → 0 as r →∞

for all s ∈ R. Using the Lebesgue’s dominated convergence theorem, we haveG2 ∈ PAP 0

T (R, Lp(P, H), q). This completes the proof.Lemma 3.3. Assume that (H1) holds. If f ∈ PAPT (R, Lp(P, L0

2), q) and if Fis the function defined by

F (t) :=∫ t

−∞T (t− s)f(s)ds

for each t ∈ R, then F ∈ PAPT (R, Lp(P, H), q).Proof. Since f ∈ PAPT (R, Lp(P, L0

2)), there exist f1 ∈ APT (R, Lp(P, L02)) and

f2 ∈ PAP 0T (R, Lp(P, L0

2), q), such that f = f1 + f2. Hence,

F (t) =∫ t

−∞T (t− s)f1(s)dW (s)

+∫ t

−∞T (t− s)f2(s)dW (s) =: F1(t) + F2(t).

18


1360 YAN ET AL 1343-1386

Next we show that F1(t) ∈ APT (R, Lp(P, H)) and F2(t) ∈ PAP 0T (R, Lp(P, H), q).

Thus, the following verification procedure is divided into three steps.Step 1. F1 ∈ UPC(R, Lp(P, H)).Let t′, t′′ ∈ (ti, ti+1), i ∈ Z, t′′ < t′. By (H4), for any ε > 0, there exists

0 < ξ < ( ε2f1

)p/2(p−1) such that 0 < t′ − t′′ < ξ, we have for p > 2,

‖ T (t′ − t′′)− I ‖p≤ δ3ε

2f1

,

where f1 = 2p−1MpCp ‖ f1 ‖p∞, δ3 = ( pδ

p−2 )(p−2)/2 pδ2 . Using Holder’s inequality

and the Ito integral [42], we have

E ‖ F1(t′)− F1(t′′) ‖p

≤ 2p−1E

wwww∫ t′′

−∞T (t′′ − s)[T (t′ − t′′)− I]f1(s)dW (s)

wwwwp

+2p−1E

wwww∫ t′

t′′T (t′ − s)f1(s)dW (s)

wwwwp

≤ 2p−1MpCpE

[ ∫ t′′

−∞e−2δ(t′′−s) ‖ T (t′ − t′′)− I ‖2

× ‖ f1(s) ‖2L02

ds

]p/2

+2p−1MpCpE

[ ∫ t′

t′′e−2δ(t′−s) ‖ f1(s) ‖2L0

2ds

]p/2

≤ 2p−1MpCp ‖ T (t′ − t′′)− I ‖p

(∫ t′′

−∞e−

pp−2 δ(t′′−s)ds

) p−2p

×(∫ t′′

−∞e−

p2 δ(t′′−s)ds

)sups∈R

‖ f1(s) ‖pL0

2

+2p−1MpCp

(∫ t′

t′′e−

pp−2 δ(t′−s)ds

) p−2p

×(∫ t′

t′′e−

p2 δ(t′−s)ds

)sups∈R

‖ f1(s) ‖pL0

2

≤ 2p−1MpCp ‖ f1 ‖p∞

δ3ε

2f1

(pδ

p− 2

) p−2p pδ

2

+2p−1MpCp ‖ f1 ‖p∞

[(ε

2f1

)p/2(p−1)]2(p−2)/p

<ε

2+

ε

2= ε.

19


1361 YAN ET AL 1343-1386

For p = 2. Let ε > 0, there exists 0 < ξ < ε2f1

such that 0 < t′− t′′ < ξ, we have

‖ T (t′ − t′′)− I ‖2≤ 2δε

f1

,

where f1 = 2M2 ‖ f1 ‖2∞ . Similar to the above discussion, one has

E ‖ F1(t′)− F1(t′′) ‖2

≤ 2M2 ‖ T (t′ − t′′)− I ‖2(∫ t′′

−∞e−2δ(t′′−s)ds

)sups∈R

‖ f1(s) ‖2L02

+2M2

(∫ t′

t′′e−2δ(t′−s)ds

)sups∈R

‖ f1(s) ‖2L02

≤ 2M2 ‖ f1 ‖2∞2δε

2f1

(∫ t′′

−∞e−2δ(t′′−s)ds

)+ 2Mp ‖ f1 ‖2∞

(ε

2f1

)

<ε

2+

ε

2= ε.

Consequently, F1 ∈ UPC(R, Lp(P, H)).Step 2. F1 ∈ APT (R, Lp(P, H)).Let ti < t ≤ ti+1. For ε > 0, let Ωε be a relatively dense set of R formed by


E ‖ F1(t + τ)− F1(t) ‖p

= E

wwww∫ t

−∞T (t− s)[f1(s + τ)− f1(s)]dW (s)

wwwwp

≤ CpE

[ ∫ t

−∞‖ T (t− s) ‖2‖ f1(s + τ)− f1(s) ‖2L0

2ds

]p/2

≤ CpMpE

[ ∫ t

−∞e−2δ(t−s) ‖ f1(s + τ)− f1(s) ‖2L0

2ds

]p/2

≤ CpMp

(∫ t

−∞e−

pp−2 δ(t−s)ds

) p−2p

×[ i−1∑

j=−∞

∫ tj+1−η

tj+η

e−p2 δ(t−s)E ‖ f1(s + τ)− f1(s) ‖p

L02

ds

+i−1∑

j=−∞

∫ tj+η

tj

e−p2 δ(t−s)E ‖ f1(s + τ)− f1(s) ‖p

L02

ds

+i−1∑

j=−∞

∫ tj+1

tj+1−η

e−p2 δ(t−s)E ‖ f1(s + τ)− f1(s) ‖p

L02

ds

+∫ t

ti

e−p2 δ(t−s)E ‖ f1(s + τ)− f1(s) ‖p

L02

ds

].

20


1362 YAN ET AL 1343-1386

Since f1 ∈ APT (R, Lp(P, L02)), one has

E ‖ f1(t + τ)− f1(t) ‖pL0

2< ε

for all t ∈ [tj + η, tj+1 − η] and j ∈ Z, j ≤ i. Then,

i−1∑

j=−∞

∫ tj+1−η

tj+η

e−p2 δ(t−s)E ‖ f1(s + τ)− f1(s) ‖p

L02

ds

≤ εi−1∑

j=−∞

∫ tj+1−η

tj+η

e−p2 δ(t−s)ds

≤ 2δp

i−1∑

j=−∞e−

p2 δ(t−tj+1+η)

≤ 2ε

δp

i−1∑

j=−∞e−

p2 δγ(i−j−1)

≤ 2ε

δp(1− e−δγ),

i−1∑

j=−∞

∫ tj+η

tj

e−p2 δ(t−s)E ‖ f1(s + τ)− f1(s) ‖p

L02

ds

≤ 2p−1 sups∈R

E ‖ f1(s) ‖pL0

2

i−1∑

j=−∞

∫ tj+1+η

tj

e−p2 δ(t−s)ds

≤ 2p−1 sups∈R

E ‖ f1(s) ‖pL0

2εe

p2 δηe−

p2 δ(t−ti)

i−1∑

j=−∞e−

p2 δα(i−j)

≤ 2p−1 sups∈R

E ‖ f1(s) ‖pL0

2εe

p2 δηe−

p2 δ(t−ti)

i−1∑

j=−∞e−

p2 δγ(i−j)

≤ 2p−1 ‖ f1 ‖p∞ eδγ/4ε

1− e−p2 δγ

.

Similarly, one has

i−1∑

j=−∞

∫ tj+1

tj+1−η

e−p2 δ(t−s)E ‖ f1(s + τ)− f1(s) ‖p

L02

ds ≤ M5ε,

∫ t

ti

e−p2 δ(t−s)E ‖ f1(s + τ)− f1(s) ‖p

L02

ds ≤ M6ε,

where M5, M6 are some positive constants. Therefore, we get that E ‖ F1(t +τ)− F1(t) ‖p≤ N3ε for a positive constant N3. For p = 2, we have

E ‖ F1(t + τ)− F1(t) ‖2

21


1363 YAN ET AL 1343-1386

≤ M2E

∫ t

−∞e−2δ(t−s) ‖ f1(s + τ)− f1(s) ‖2L0

2ds

≤ M2

[ i−1∑

j=−∞

∫ tj+1−η

tj+η

e−2δ(t−s)E ‖ f1(s + τ)− f1(s) ‖2L02

ds

+i−1∑

j=−∞

∫ tj+η

tj

e−2δ(t−s)E ‖ f1(s + τ)− f1(s) ‖2L02

ds

+i−1∑

j=−∞

∫ tj+1

tj+1−η

e−2δ(t−s)E ‖ f1(s + τ)− f1(s) ‖2L02

ds

+∫ t

ti

e−2δ(t−s)E ‖ f1(s + τ)− f1(s) ‖2L02

ds

].

Similarly, we get that E ‖ F1(t+ τ)−F1(t) ‖2≤ N4ε for a positive constant N4.Hence, F1 ∈ APT (R, Lp(P, H)).

Step 3. F2 ∈ PAP 0T (R, Lp(P, H), q).

In fact, for r > 0, one has for p > 2,

12r

∫ r

−r

supθ∈[t−q,t]

E ‖ F2(θ) ‖p dt

=12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ θ

−∞T (θ − s)f2(s)dW (s)

wwwwp

dt

=12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ ∞

0

T (s)f2(θ − s)dW (s)wwww

p

dt

≤ Cp12r

∫ r

−r

supθ∈[t−q,t]

E

[ ∫ ∞

0

e−2s ‖ f2(θ − s) ‖2L02

ds

]p/2

dt

≤ MpCp12r

∫ r

−r

(∫ ∞

0

e−p

p−2 δsds

) p−2p

×∫ ∞

0

e−p2 δs sup

θ∈[t−q,t]

E ‖ f2(θ − s) ‖pL0

2dsdt

= MpCp

(∫ ∞

0

e−p−2

p δsds

) p−2p

∫ ∞

0

e−p2 δsds

× 12r

∫ r

−r

supθ∈[t−q,t]

E ‖ f2(θ − s) ‖pL0

2dt.

For p = 2, we have

12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ θ

−∞T (θ − s)f2(s)dW (s)

wwww2

dt

≤ M2 12r

∫ r

−r

∫ ∞

0

e−2s supθ∈[t−q,t]

E ‖ f2(θ − s) ‖2L02

dsdt

22


1364 YAN ET AL 1343-1386

= M2

(∫ ∞

0

e−2δsds

)12r

∫ r

−r

supθ∈[t−q,t]

E ‖ f2(θ − s) ‖pL0

2dt.

Since f2 ∈ PAP 0T (R, Lp(P, L0

2), q), it follows that f2(·−s) ∈ PAP 0T (R, Lp(P, L0

2),q) for each s ∈ R by Remark 2.1, hence

12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∫ θ

−∞T (θ − s)f2(s)dW (s)

wwwwp

dt → 0 as r →∞

for all s ∈ R. Using the Lebesgue’s dominated convergence theorem, we haveF2 ∈ PAP 0

T (R, Lp(P, H), q). This completes the proof.Lemma 3.4. Assume that (H1) holds. If γi ∈ PAP (Z, Lp(P, H)), i ∈ Z and ifγi is the function defined by

Ri(t) :=∑ti<t

T (t− ti)γi

for each t ∈ R, then Ri ∈ PAPT (R, Lp(P, H), q).Proof. Since γi ∈ PAP (Z,Lp(P, H)), there exist γ1,i ∈ AP (Z, Lp(P, H)) andγ2,i ∈ PAP0(Z, Lp(P, H)), such that γi = γ1,i + γ2,i. Hence,

Ri(t) =∑ti<t

T (t− ti)γ1,i +∑ti<t

T (t− ti)γ2,i =: Π1,i(t) + Π2,i(t).

Next we show that Π1,i(t) ∈ APT (R, Lp(P, H)) and Π2,i(t) ∈ PAP 0T (R, Lp(P,

H), q). Thus, the following verification procedure is divided into three steps.Step 1. Π1,i ∈ UPC(R, Lp(P, H)).Let t′, t′′ ∈ (ti, ti+1), i ∈ Z, t′′ < t′. By (H4), for any ε > 0, we have

‖ T (t′ − t′′)− I ‖p≤ (1− e−δγ)pε

γ1,

where γ1 = Mp ‖ γ1,i ‖p∞ . Using Holder’s inequality, we have

E ‖ Π1,i(t′)−Π1,i(t′′) ‖p

= E

wwww∑

ti<t′T (t′ − ti)γ1,i −

∑

ti<t′′T (t′′ − ti)γ1,i

wwwwp

= E

wwww∑

ti<t′′T (t′′ − ti)[T (t′ − t′′)− I]γ1,i

wwwwp

≤ Mp ‖ T (t′ − t′′)− I ‖p

( ∑

ti<t′′e−δ(t′′−ti)

)p−1

×( ∑

ti<t′′e−δ(t′′−ti)E ‖ γ1,i ‖p

)

≤ Mp ‖ T (t′ − t′′)− I ‖p

( ∑


)p

supi∈Z

E ‖ γ1,i ‖p

23


1365 YAN ET AL 1343-1386

≤ Mp (1− e−δγ)pε

γ1

( ∑


)p

‖ γ1,i ‖p∞

< ε.

Consequently, Π1,i ∈ UPC(R, Lp(P, H)).Step 2. Π1,i ∈ APT (R, Lp(P, H)).Let ti < t ≤ ti+1. For ε > 0, let Ωε be a relatively dense set of R formed

by ε-periods of F. For τ ∈ Ωε and 0 < η < minε, γ/2. By Lemma 2.1, thereexists relative dense sets of real numbers Ωε and integers Qε, for every τ ∈ Ωε,there exists at least one number q ∈ Qε such that |tq − τ | < ε, i ∈ Z andE ‖ γ1,i+q − γ1,i ‖p< ε, q ∈ Qε, i ∈ Z. Then,

E ‖ Π1,i(t + τ)−Π1,i(t) ‖p

= E

wwww∑

ti<t+τ

T (t + τ − ti)γ1,i −∑ti<t

T (t− ti)γ1,i

wwwwp

≤ E

[ ∑ti<t

‖ T (t− ti) ‖‖ γ1,i+q − γ1,i ‖]p

≤ MpE

[( ∑ti<t

e−δ(t−ti)

)p−1( ∑ti<t

e−δ(t−ti) ‖ γ1,i+q − γ1,i ‖p

)]

≤ Mp

( ∑ti<t

e−δ(t−ti)

)p

E ‖ γ1,i+q − γ1,i ‖p

≤ Mpε

(1− e−δγ)p.

Hence, Π1,i ∈ APT (R, Lp(P, H)).Step 3. Π2,i ∈ PAP 0

T (R, Lp(P, H), q).For a given i ∈ Z, define the function v(t) by v(t) = T (t−ti)γ2,i, ti < t ≤ ti+1,

then

limt→∞

supθ∈[t−q,t]

E ‖ v(θ) ‖p

= limt→∞

supθ∈[t−q,t]

E ‖ T (θ − ti)γ2,i ‖p

≤ limt→∞

Mpe−pδ(t−ti) supi∈Z

E ‖ γ2,i ‖p= 0.

Thus v ∈ PC0T (R, Lp(P, H), q) ⊂ PAP 0

T (R, Lp(P, H), q). Define vj : R →Lp(P, H) by

vj(t) = T (t− ti−j)γ2,i−j , ti < t ≤ ti+1, j ∈ N.

So vj ∈ PAP 0T (R, Lp(P, H), q). Moreover,

supθ∈[t−q,t]

E ‖ vj(θ) ‖p

= supθ∈[t−q,t]

E ‖ T (θ − ti−j)γ2,i−j ‖p

24


1366 YAN ET AL 1343-1386

≤ Mpe−pδ(t−ti−j) supi∈Z

E ‖ γ2,i ‖p

≤ Mpe−pδ(t−ti)e−pδγj supi∈Z

E ‖ γ2,i ‖p .

Therefore, the series∑∞

j=0 vj is uniformly convergent on R. By Lemma 2.3, onehas ∑

ti<t

T (t− ti)γ2,i =∞∑

j=0

vj(t) ∈ PAP 0T (R, Lp(P, H), q),

that is

12r

∫ r

−r

supθ∈[t−q,t]

E

wwww∑ti<t

T (θ − ti)γ2,i

wwwwp

dt → 0 as r →∞.

Using the Lebesgue’s dominated convergence theorem, we have Π2,i ∈ PAP 0T (R,

Lp(P, H), q) This completes the proof.Lemma 3.5. If x ∈ PAPT (R, Lp(P, H), q), then t → xt belongs to PAPT (R,D,q).

One can refer to Lemma 3.3 in [4] for the proof of Lemma 3.5.Now, we establish the existence theorem of p-mean piecewise pseudo almost

periodic mild solutions to partial impulsive stochastic differential equation (1)-(2).Theorem 3.1. Assume that assumptions (H1)-(H6) are satisfied. Then system(1)-(2) has a mild solution x ∈ PAPT (R, Lp(P, H), q).Proof. Let Y = PAPT (R, Lp(P, H), q) ∩ UPC(R, Lp(P, H)). Consider theoperator Ψ : Y → PC(R, Lp(P, H)) defined by

(Ψx)(t) =[h(t, xt) +

∫ t

−∞AT (t− s)h(s, xs)ds

]

+[ ∫ t

−∞T (t− s)g(s, xs)ds +

∫ t

−∞T (t− s)f(s, xs)dW (s)

+∑ti<t

T (t− ti)Ii(x(ti))]

=: (Ψ1x)(t) + (Ψ2x)(t), t ∈ R.

Obviously, the operator Ψ1 + Ψ2 has a fixed point if and only if operator Ψ hasa fixed point in Y. To prove which we shall employ Lemma 2.8, we divide theproof into several steps.

Step 1. For every x ∈ Y, Ψx ∈ Y.Let x(·) ∈ Y, by (H2), (H3), (H5) and Lemmas 2.4, 2.5, we deduce that

h(·, x·), g(·, x·), f(·, x·) ∈ PAPT (R, Lp(P, H), q) and Ii(x(ti)) ∈ PAP (Z, Lp(P,H)). Similarly as the proof of Lemmas 3.1-3.5, one has Ψx ∈ Y.

Step 2. For a closed bounded convex subset Br∗ of Y, Ψ1x + Ψ2y ∈ Br∗ ,when x, y ∈ Br∗ .

25


1367 YAN ET AL 1343-1386

Let ρ1, ρ2 > 0 be fixed. By (H4) and (H6) it follows that there exist a positiveconstant µ such that, for all t ∈ R and ψ, x ∈ Y with E ‖ ψ ‖p

D> µ, E ‖ x ‖p> µ,

E ‖ g(t, ψ) ‖p +E ‖ f(t, ψ) ‖pL0

2≤ ρ1Θ(E ‖ ψ ‖p

D),

E ‖ Ii(x) ‖p≤ ρ2Θi(E ‖ x ‖p), i ∈ Z.

Letν = sup

t∈RE ‖ g(t, ψ) ‖p, E ‖ f(t, ψ) ‖p

L02: E ‖ ψ ‖p

D≤ µ,

ν1 = supt∈R,i∈Z

E ‖ Ii(x) ‖p: E ‖ x ‖p≤ µ.

Thus, we have for all t ∈ R,

E ‖ g(t, ψ) ‖p +E ‖ f(t, ψ) ‖pL0

2≤ ρ1Θ(E ‖ ψ ‖p) + ν, ψ ∈ D, (5)

E ‖ Ii(x) ‖p≤ ρ2Θi(E ‖ x ‖p) + ν1, x ∈ Lp(P, H), i ∈ Z. (6)

By (H2), (5), (6), Holder’s inequality and the Ito integral, we have for p > 2,

E ‖ (Ψ1x)(t) + (Ψ2y)(t) ‖p

≤ 5p−1E ‖ h(t, xt) ‖p +5p−1E

wwww∫ t


wwwwp

+5p−1E

wwww∫ t

−∞T (t− s)g(s, ys)ds

wwwwp

+5p−1E

wwww∫ t

−∞T (t− s)f(s, ys)dW (s)

wwwwp

+5p−1E

wwww∑ti<t

T (t− ti)Ii(y(ti))wwww

p

≤ 5p−1 ‖ A−β ‖ E ‖ Aβh(s, xs) ‖p

+5p−1Mp1−β

(∫ t

−∞(t− s)−


)p−1

×(∫ t

−∞e−δ(t−s)E ‖ Aβh(s, xs) ‖p ds

)

+5p−1Mp

(∫ t


)p−1

×(∫ t

−∞e−δ(t−s)E ‖ g(s, ys) ‖p ds

)

+5p−1CpMpE

(∫ t

−∞e−2δ(t−s) ‖ f(s, ys) ‖2L0

2ds

)p/2

+5p−1MpE

[( ∑ti<t

e−δ(t−ti)

)p−1( ∑ti<t

e−δ(t−ti) ‖ Ii(y(ti)) ‖p

)]

26


1368 YAN ET AL 1343-1386

≤ 5p−1 ‖ A−β ‖ L(‖ xt ‖pD +1)

+5p−1Mp1−β

(Γ(1− p(1− β)

p− 1)δ

p(1−β)p−1 −1

)p−1

×(∫ t

−∞e−δ(t−s)L(‖ xs ‖p

D +1)ds

)

+5p−1Mp 1δp−1

(∫ t

−∞e−δ(t−s)[ρ1Θ(E ‖ ys ‖p

D) + ν]ds

)

+5p−1MpCp

(∫ t

−∞e−

pp−2 δ(t−s)ds

) p−2p

×(∫ t

−∞e−

p2 δ(t−s)[ρ1Θ(E ‖ ys ‖p

D) + ν]ds

)

+5p−1Mp 1(1− e−δγ)p−1

( ∑ti<t

e−δ(t−ti)[ρ2Θi(E ‖ y(ti) ‖p) + ν1])

≤ 5p−1 ‖ A−β ‖ L(‖ x ‖p∞ +1)

+5p−1Mp1−β

(Γ(1− p(1− β)

p− 1))p−1

δpβL(‖ x ‖p∞ +1)

+5p−1Mp 1δp

[ρ1Θ(‖ y ‖p∞) + ν]

+5p−1MpCp

(p− 2pδ

) p−2p 2

pδ[ρ1Θ(‖ y ‖p

∞) + ν]

+5p−1Mp 1(1− e−δγ)p

[ρ2 supi∈Z

Θi(‖ y ‖p∞) + ν1].

For p = 2, we have

E ‖ (Ψ1x)(t) + (Ψ2y)(t) ‖2≤ 5 ‖ A−β ‖ L(‖ x ‖p

∞ +1)+5M2

1−β(Γ(1− 2(1− β))δ2βL(‖ x ‖p∞ +1)

+5M2 1δ2

[βΘ(‖ y ‖p∞) + ν] + 5M2 1

2δ[βΘ(‖ y ‖p

∞) + ν]

+5M2 1(1− e−δα)2

[β supi∈Z

Θi(‖ y ‖p∞) + ν1].

Note that, for ρ1, ρ2 sufficiently small, we can choose r∗ > 0 such that for p > 2,

5p−1 ‖ A−β ‖ L(r∗ + 1)

+5p−1Mp1−β

(Γ(1− p(1− β)

p− 1))p−1

δpβL(r∗ + 1)

+5p−1Mp 1δp

[ρ1Θ(r∗) + ν]

27


1369 YAN ET AL 1343-1386

+5p−1MpCp

(p− 2pδ

) p−2p 2

pδ[ρ1Θ(r∗) + ν]

+5p−1Mp 1(1− e−δγ)p

[ρ2 supi∈Z

Θi(r∗) + ν1] ≤ r∗, (7)

and for p = 2, we have

5 ‖ A−β ‖ L(r∗ + 1)+5M2

1−β(Γ(1− 2(1− β))δ2βL(r∗ + 1)

+5M2 1δ2

[βΘ(r∗) + ν] + 5M2 12δ

[βΘ(r∗) + ν]

+5M2 1(1− e−δα)2

[ρ2 supi∈Z

Θi(r∗) + ν1] ≤ r∗. (8)

Let Br∗ = x ∈ Y :‖ x ‖p∞≤ r∗ for r∗ > 0. It is easy to see that Br∗ is a closed

bounded convex subset of Y. Moreover, for all x, y ∈ Br∗ ,

E ‖ (Ψ1x)(t) + (Ψ2y)(t) ‖p≤ r∗.

Therefore, Ψ1x + Ψ2y ∈ Br∗ , when x, y ∈ Br∗ .Step 3. Ψ1 is a contraction.For t ∈ R, and x∗, x∗∗ ∈ Br. From (H2) and Lemma 2.6, we have

E ‖ (Ψ1x∗)(t)− (Ψ1x

∗∗)(t) ‖p

≤ 2p−1E ‖ h(t, x∗t )− h(t, x∗∗t ) ‖p

+2p−1E

wwww∫ t

−∞AT (t− s)[h(s, x∗s)− h(s, x∗∗s )]ds

wwwwp

≤ 2p−1 ‖ A−β ‖ E ‖ Aβh(t, x∗t )−Aβh(t, x∗∗t ) ‖p

+2p−1Mp1−β

(∫ t

−∞(t− s)−


)p−1

×(∫ t

−∞e−δ(t−s)E ‖ Aβh(s, x∗s)−Aβh(s, x∗∗s ) ‖p ds

)

≤ 2p−1 ‖ A−β ‖ L ‖ x∗t − x∗∗t ‖pD

+2p−1Mp1−β

(Γ(1− p(1− β)

p− 1)δ

p(1−β)p−1 −1

)p−1

×(∫ t

−∞e−δ(t−s)L ‖ x∗s − x∗∗s ‖p

D ds

)

≤ L0 ‖ x∗ − x∗∗ ‖p∞ .

Taking supremum over t,

‖ Ψ1x∗ −Ψ1x

∗∗ ‖p∞≤ L0 ‖ x∗ − x∗∗ ‖p

∞ .

where L0 = 2p−1[‖ A−β ‖ +Mp1−β(Γ(1− p(1−β)

p−1 ))p−1δpβ ]L. By (7), we see thatL0 < 1. Hence, Ψ1 is a contractive operator with constant L0.

28


1370 YAN ET AL 1343-1386

Step 4. Ψ2 maps Br∗ into an equicontinuous family.Let τ1, τ2 ∈ (ti, ti+1), i ∈ Z, τ1 < τ2, and x ∈ Br∗ . Then, by (H1), (5), (6),

Holder’s inequality and the Ito integral, we have for p > 2,

E ‖ (Ψ2x)(τ2)− (Ψ2x)(τ1) ‖p

≤ 6p−1E

wwww∫ τ1

−∞T (τ1 − s)[T (τ2 − τ1)− I]g(s, xs)ds

wwwwp

+6p−1E

wwww∫ τ2

τ1

T (τ2 − s)g(s, xs)ds

wwwwp

+6p−1E

wwww∫ τ1

−∞T (τ1 − s)[T (τ2 − τ1)− I]f(s, xs)dW (s)

wwwwp

+6p−1E

wwww∫ τ2

τ1

T (τ2 − s)f(s, xs)dW (s)wwww

p

+3p−1E

wwww∑

ti<τ1

T (τ1 − ti)[T (τ2 − τ1)− I]Ii(x(ti))wwww

p

≤ 6p−1Mp ‖ T (τ2 − τ1)− I ‖p

(∫ τ1

−∞e−δ(τ1−s)ds

)p−1

×(∫ τ1

−∞e−δ(τ1−s)E ‖ g(s, xs) ‖p ds

)

+6p−1Mp

(∫ τ2

τ1

e−δ(τ2−s)ds

)p−1

×(∫ τ2

τ1

e−δ(τ2−s)E ‖ g(s, xs) ‖p ds

)

+6p−1MpCpE

[ ∫ τ1

−∞e−2δ(τ1−s) ‖ T (τ2 − τ1)− I ‖2

× ‖ f(s, xs) ‖2L02

ds

]p/2

+6p−1MpCpE

[ ∫ τ2

τ1

e−2δ(τ2−s) ‖ f(s, xs) ‖2L02

ds

]p/2

+3p−1Mp ‖ T (τ2 − τ1)− I ‖p

( ∑ti<τ1

e−δ(τ1−ti)

)p−1

×( ∑

ti<τ1

e−δ(τ1−ti)E ‖ Ii(x(ti)) ‖p

)

≤ 6p−1Mp ‖ T (τ2 − τ1)− I ‖p

(∫ τ1

−∞e−δ(τ1−s)ds

)p−1

×(∫ τ1

−∞e−δ(τ1−s)[ρ1Θ(E ‖ xs ‖p

D) + ν]ds

)

29


1371 YAN ET AL 1343-1386

+6p−1Mp

(∫ τ2

τ1

e−δ(τ2−s)ds

)p−1

×(∫ τ2

τ1

e−δ(τ2−s)[ρ1Θ(E ‖ xs ‖pD) + ν]ds

)

+6p−1MpCp ‖ T (τ2 − τ1)− I ‖p

(∫ τ1

−∞e−

pp−2 δ(τ1−s)ds

) p−2p

×(∫ τ1

−∞e−

p2 δ(τ1−s)[ρ1Θ(E ‖ xs ‖p

D) + ν]ds

)

+6p−1MpCp

(∫ τ2

τ1

e−p

p−2 δ(τ2−s)ds

) p−2p

×(∫ τ2

τ1

e−p2 δ(τ2−s)[ρ1Θ(E ‖ xs ‖p

D) + ν]ds

)

+3p−1Mp ‖ T (τ2 − τ1)− I ‖p

( ∑ti<τ1

e−δ(τ1−ti)

)p−1

×( ∑

ti<τ1

e−δ(τ1−ti)[ρ2Θi(E ‖ x(ti) ‖p) + ν1])

≤ 6p−1Mp ‖ T (τ2 − τ1)− I ‖p 1δp

[ρ1Θ(r∗) + ν]

+6p−1Mp

(∫ τ2

τ1

e−δ(τ2−s)ds

)p

[ρ1Θ(r∗) + ν]

+6p−1MpCp ‖ T (τ2 − τ1)− I ‖p

(p− 2pδ

) p−2p 2

pδ[ρ1Θ(r∗) + ν]

+6p−1MpCp

(∫ τ2

τ1

e−p

p−2 δ(τ2−s)ds

) p−2p

×(∫ τ2

τ1

e−p2 δ(τ2−s)ds

)[ρ1Θ(r∗) + ν]

+3p−1Mp ‖ T (τ2 − τ1)− I ‖p 1(1− e−δγ)p

[ρ2Θi(r∗) + ν1].

For p = 2, we have

E ‖ (Ψx)(τ2)− (Ψx)(τ1)) ‖2

≤ 6M2 ‖ T (τ2 − τ1)− I ‖2 1δ2

[ρ1Θ(r∗) + ν]

+6M2

(∫ τ2

τ1

e−δ(τ2−s)ds

)2

[ρ1Θ(r∗) + ν]

+6M2 ‖ T (τ2 − τ1)− I ‖2 2δ[ρ1Θ(r∗) + ν]

30


1372 YAN ET AL 1343-1386

+6M2

(∫ τ2

τ1

e−2δ(τ2−s)ds

)[ρ1Θ(r∗) + ν]

+3M2 ‖ T (τ2 − τ1)− I ‖2 1(1− e−δγ)2

[ρ2Θi(r∗) + ν1].

The right-hand side of the above inequality is independent of x ∈ Br∗ andtends to zero as τ2 → τ1, since the compactness of T (t) for t > 0 implies implythe continuity in the uniform operator topology. Thus, Ψ maps Br∗ into anequicontinuous family of functions.

Step 5. Ψ2Br∗ is precompact.For each t ∈ R, and let ε be a real number satisfying 0 < ε < 1. For x ∈ Br∗ ,

we define

(Ψ2,εx)(t)

= T (ε)[ ∫ t−ε

−∞T (t− ε− s)g(s, xs)ds

+∫ t−ε

−∞T (t− ε− s)f(s, xs)dW (s)

+∑

ti<t−ε

T (t− ε− ti)Ii(x(ti))]

= T (ε)[(Ψ2x)(t− ε)].

Since T (t)(t > 0) is compact, then the set Vε(t) = (Ψ2,εx)(t) : x ∈ Br∗ isrelatively compact in Lp(P, H) for each t ∈ R. Moreover, for every x ∈ Br∗ , wehave for p > 2,

E ‖ (Ψ2x)(t)− (Ψ2,εx)(t) ‖p

≤ 3p−1E

wwww∫ t

t−ε

T (t− s)g(s, xs)ds

wwwwp

+3p−1E

wwww∫ t

t−ε

T (t− s)f(s, xs)dW (s)wwww

p

+3p−1E

wwww∑

t−ε<ti<t

T (t− ti)Ii(x(ti))wwww

p

≤ 3p−1Mp

(∫ t

t−ε

e−δ(t−s)ds

)p−1(∫ t

t−ε

e−δ(t−s)E ‖ g(s, xs) ‖p ds

)

+3p−1CpMpE

(∫ t

t−ε

e−2δ(t−s) ‖ f(s, xs) ‖2L02

ds

)p/2

+3p−1MpE

[( ∑t−ε<ti<t

e−δ(t−ti)

)p−1

×( ∑

t−ε<ti<t

e−δ(t−ti) ‖ Ii(x(ti)) ‖p

)]

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1373 YAN ET AL 1343-1386

≤ 3p−1Mp

(∫ t

t−ε

e−δ(t−s)ds

)p−1

×(∫ t

t−ε

e−δ(t−s)[ρ1Θ(E ‖ xs ‖pD) + ν]ds

)

+3p−1CpMp

(∫ t

t−ε

e−p

p−2 δ(t−s)ds

) p−2p

×(∫ t

t−ε

e−p2 δ(t−s)[ρ1Θ(E ‖ xs ‖p

D) + ν]ds

)

+3p−1Mp

( ∑t−ε<ti<t

e−δ(t−ti)

)p−1

×( ∑

t−ε<ti<t

e−δ(t−ti)[ρ2Θi(E ‖ xi(ti) ‖p) + ν1])

≤ 3p−1Mp

(∫ t

t−ε

e−δ(t−s)ds

)p

[ρ1Θ(r∗) + ν]

+3p−1CpMp

(∫ t

t−ε

e−p

p−2 δ(t−s)ds

) p−2p

×(∫ t

t−ε

e−p2 δ(t−s)ds

)[ρ1Θ(r∗) + ν]

+3p−1Mp

( ∑t−ε<ti<t

e−δ(t−ti)

)p

[ρ2 supi∈Z

Θi(r∗) + ν1].

For p = 2, we have

E ‖ (Ψ2x)(t)− (Ψ2,εx)(t) ‖2

≤ 3M2

(∫ t

t−ε

e−δ(t−s)ds

)2

[ρ1Θ(r∗) + ν]

+3M2

(∫ t

t−ε

e−2δ(t−s)ds

)[ρ1Θ(r∗) + ν]

+3M2

( ∑t−ε<ti<t

e−δ(t−ti)

)2

[ρ2 supi∈Z

Θi(r∗) + ν1].

Therefore, letting ε → 0, it follows that there are relatively compact sets Vε(t)arbitrarily close to V (t) = (Ψ2x)(t) : x ∈ Br∗, and hence V (t) is alsorelatively compact in Lp(P, H) for each t ∈ R. Since Ψ2x : x ∈ Br∗ ⊂PC0

h(R, Lp(P, H), q), then Ψ2x : x ∈ Br∗ is a relatively compact set byLemma 2.7, then Ψ2 is a compact operator.

Step 6. Ψ2 is continuous.Let x(n) ⊆ Br∗ with x(n) → x(n →∞) in Y, then there exists a bounded

subset K ⊆ Lp(P, K) such that R(x) ⊆ K, R(xn) ⊆ K, n ∈ N. By the assump-tion (H2) and (H4), for any ε > 0, there exists ξ > 0 such that x, y ∈ K and

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1374 YAN ET AL 1343-1386

‖ x− y ‖p∞< ξ implies that

E ‖ g(s, xs)− g(s, ys) ‖p< ε for all t ∈ R,

E ‖ f(s, xs)− f(s, ys) ‖pL0

2< ε for all t ∈ R,

andE ‖ Ii(x)− Ii(y) ‖p< ε for all i ∈ Z.

For the above ξ there exists n0 such that ‖ x(n) − x ‖p∞< ε for n > n0, then for

n > n0, we have

E ‖ g(s, x(n)s )− g(s, xs) ‖p< ε for all t ∈ R,

E ‖ f(s, x(n)s )− f(s, xs) ‖p

L02< ε for all t ∈ R,

andE ‖ Ii(x(n))− Ii(x) ‖p< ε for all i ∈ Z.

Then, by Holder’s inequality, we have that for p > 2,

E ‖ (Ψ2x(n))(t)− (Ψ2x)(t) ‖p

≤ 3p−1E

wwww∫ t

−∞T (t− s)[g(s, x(n)

s )− g(s, xs)]ds

wwwwp

+3p−1E

wwww∫ t

−∞T (t− s)[f(s, x(n)

s )− f(s, xs)]dW (s)wwww

p

+3p−1E

wwww∑ti<t

T (t− ti)[Ii(x(n)(ti))− Ii(x(ti))]wwww

p

≤ 3p−1Mp

(∫ t


)p−1

×(∫ t

−∞e−δ(t−s)E ‖ g(s, x(n)

s )− g(s, xs) ‖p ds

)

+3p−1CpMp

(∫ t

−∞e−2δ(t−s)E ‖ f(s, x(n)

s )− f(s, xs) ‖2L02

ds

)p/2

+3p−1MpE

[( ∑ti<t

e−δ(t−ti)

)p−1

×( ∑

ti<t

e−δ(t−ti) ‖ Ii(x(n)(ti))− Ii(x(ti)) ‖p

)]

≤ 3p−1Mp

(∫ t


)p

ε

+3p−1CpMp

(∫ t

−∞e−

pp−2 δ(t−s)ds

) p−2p

(∫ t

−∞e−

p2 δ(t−s)ds

)ε

33


1375 YAN ET AL 1343-1386

+3p−1Mp 1(1− e−δγ)p−1

( ∑ti<t

e−δ(t−ti)

)ε

≤ 3p−1Mp

[1δp

+ Cp

(p− 2pδ

) p−22 2

pδ+

1(1− e−δγ)p

]ε.

For p = 2, we have

E ‖ (Ψ2x(n))(t)− (Ψ2x)(t) ‖2

≤ 3M2

[1δ2

+12δ

+1

(1− e−δγ)2

]ε.

Thus Ψ2 is continuous on Br∗ and Ψ2 is completely continuous.Therefore, all the conditions of Lemma 2.8 are satisfied and thus operator

Ψ has a fixed point x in Br∗ , which is in turn a mild solution of the system(1)-(2), that is

x(t) = h(t, xt) +∫ t


+∫ t

−∞T (t− s)g(s, xs)ds +

∫ t


+∑ti<t

T (t− ti)Ii(x(ti)), t ∈ R.

Finally, to prove that x satisfies (4) for all t ≥ s, all s ∈ R. Fix σ, σ 6= ti, i ∈Z, we have for t ∈ [σ, σ + b), b > 0,

x(σ) = h(σ, xσ) +∫ σ

−∞AT (σ − s)h(s, xs)ds

+∫ σ

−∞T (σ − s)g(s, xs)ds +

∫ σ

−∞T (σ − s)f(s, xs)dW (s)

+∑ti<σ

T (σ − ti)Ii(x(ti)) = ϕ(σ).

Since T (t) : t ≥ 0 is an analytic semigroup, we have for all t ∈ [σ, σ + b),

x(t)

= h(t, xt) +∫ σ

−∞AT (t− s)h(s, xs)ds +

∫ σ

−∞T (t− s)g(s, xs)ds

+∫ σ

−∞T (t− s)f(s, xs)dW (s) +

∑ti<σ

T (t− ti)Ii(x(ti))

+∫ t

σ

AT (t− s)h(s, xs)ds +∫ t

σ


+∫ t

σ

T (t− s)f(s, xs)dW (s) +∑

σ<ti<t

T (t− ti)Ii(x(ti))

34


1376 YAN ET AL 1343-1386

= T (t− σ)[ϕ(σ)− h(σ, ϕ)] + h(t, xt) +∫ t

σ


+∫ t

σ

T (t− s)f(s, xs)dW (s) +∑

σ<ti<t

T (t− ti)Ii(x(ti)).

Hence x ∈ PAPT (R, Lp(P, H), q) is an p-mean piecewise pseudo almost periodicmild solution to system (1)-(2). This completes the proof.

4 Global attractiveness

In this section, we present the global attractiveness of a piecewise pseudo almostperiodic solution of (1)- (2). To do this, we also need the following assumptions:

(B1) There exist constants 0 < β < 1, lj > 0, j = 1, 2,such that

E ‖ Aβh(t, ψ) ‖p≤ l1 ‖ ψ ‖pD, t ∈ R, ψ ∈ D,

E ‖ g(t, ψ) ‖p +E ‖ f(t, ψ) ‖pL0

2≤ l2 ‖ ψ ‖p

D, t ∈ R, ψ ∈ D.

(B2) There exist constant ci > 0, i ∈ Z, such that

E ‖ Ii(x) ‖p≤ ciE ‖ x ‖p, x ∈ Lp(P, K).

Theorem 4.1. Assume that assumptions of Theorem 3.1 hold and, in addition,hypotheses (B1), (B2) are satisfied. Then the piecewise pseudo almost periodicmild solution of (1)-(2) is globally exponentially stable.Proof. Let x(·) be a fixed point of Ψ in Y. By Theorem 3.1, any fixed pointof Ψ is a mild solution of the system (1)-(2). We now can choose a positiveconstant β such that 0 < β < pδ

2 ,

6p−1Mp 1(1− e−δγ)p−1(1− e−(δ−β)γ)

supi∈Z

ci < 1,

and

eβtE ‖ x(t) ‖p

≤ 5p−1eβtE ‖ h(t, xt) ‖p +5p−1eβtE

wwww∫ t


wwwwp

+5p−1eβtE

wwww∫ t

−∞T (t− s)g(s, xs)ds

wwwwp

+5p−1eβtE

wwww∫ t


wwwwp

+5p−1eβtE

wwww∑ti<t

T (t− ti)Ii(x(ti))wwww

p

=5∑

j=1

νj .

35


1377 YAN ET AL 1343-1386

Now, we estimate the terms on the right-hand side of the above inequality. By(B1) and ν1, we have

ν1 ≤ 5p−1 ‖ A−β ‖ l1eβt ‖ xt ‖p

D .

For any x(t) ∈ Lp(P, H) and any ε > 0, there exists a t1 > 0 such that eβtE ‖x(t− q) ‖p< ε for t > t1. Thus, we obtain

ν1 ≤ 5p−1 ‖ A−β ‖ l1ε,

which implies ν1 → 0 as t →∞. As to ν2, for any x(t) ∈ Lp(P, H), t ∈ [−q,∞),we have

ν2

≤ 5p−1eβtMp1−β

(∫ t

−∞(t− s)−


)p−1

×(∫ t

−∞e−δ(t−s)E ‖ Aβh(s, xs) ‖p ds

)

≤ 5p−1Mp1−β

(Γ(1− p(1− β)

p− 1)δ

p(1−β)p−1 −1

)p−1

×(∫ t

−∞e−(δ−β)(t−s)l1e

βsE ‖ xs ‖pD ds

).

For any x(t) ∈ Lp(P, H) and any ε > 0, there exists a t2 > 0 such that eβtE ‖x(s− θ) ‖p< ε for s > t2. Thus, we obtain

ν2

≤ 5p−1Mp1−β

(Γ(1− p(1− β)

p− 1)δ

p(1−β)p−1 −1

)p−1

×[ ∫ t

t2

e−(δ−β)(t−s)l1eβsE ‖ xs ‖p

D ds

)

+e−(δ−β)t

∫ t2

−∞e(δ−β)sl1e


]

≤ 5p−1Mp1−β

(Γ(1− p(1− β)

p− 1)δ

p(1−β)p−1 −1

)p−1

×[

1δ − β

l1ε + e−(δ−β)t

∫ t2



]

Since e−(δ−β)t → 0 as t →∞, then there exists t3 ≥ t2 such that for any t ≥ t3,

5p−1Mp1−β

(Γ(1− p(1− β)

p− 1)δ

p(1−β)p−1 −1

)p−1

36


1378 YAN ET AL 1343-1386

×e−(δ−β)t

∫ t2



≤ ε− 5p−1Mp1−β

(Γ(1− p(1− β)

p− 1)δ

p(1−β)p−1 −1

)p−1 1δ − β

l1ε.

Thus, for any t ≥ t3, we obtain ν2 ≤ ε, which implies ν2 → 0 as t → ∞. As toν3, for any x(t) ∈ Lp(P, H), t ∈ [−q,∞), we have

ν3

≤ 5p−1Mpeβt

(∫ t


)p−1

×(∫ t

−∞e−δ(t−s)E ‖ g(s, xs) ‖p ds

)

≤ 5p−1Mp 1δp−1

(∫ t

−∞e−(δ−β)(t−s)eβsl2 ‖ xs ‖p ds

).

Similar to the discussion of ν2, we obtain ν2 → 0 as t → ∞. As to ν4, for anyx(t) ∈ Lp(P, H), t ∈ [−q,∞), we have for p > 2,

ν4

≤ 5p−1CpMpeβtE

(∫ t

−∞e−2δ(t−s) ‖ f(s, xs) ‖2L0

2ds

)p/2

≤ 5p−1CpMpeβt

(∫ t

−∞e−

pp−2 δ(t−s)ds

) p−2p

×(∫ t

−∞e−

p2 δ(t−s)l2E ‖ x(s) ‖p ds

)

≤ 5p−1CpMp

(p− 2pδ

) p−2p

×(∫ t

−∞e−( pδ

2 −β)(t−s)l2eβsE ‖ xs ‖p ds

).

Similar to the discussion of ν2, we obtain ν4 → 0 as t →∞. For p = 2, we have

ν4 ≤ 5M2

(∫ t

−∞e−(2δ−β)(t−s)l2e

βsE ‖ xs ‖2 ds

).

Similarly, we obtain ν4 ≤ ε. By (B2) and Holder’s inequality, we have

ν5

≤ 6p−1MpeβtE

[( ∑ti<t

e−δ(t−ti)

)p−1( ∑ti<t

e−δ(t−ti) ‖ Ii(x(ti)) ‖p

)]

≤ 6p−1Mp 1(1− e−δγ)p−1

eβt

( ∑ti<t

e−δ(t−ti)[ciE ‖ x(ti) ‖p])

37


1379 YAN ET AL 1343-1386

≤ 6p−1Mp 1(1− e−δγ)p

( ∑ti<t

e−(δ−β)(t−ti)[cieβtiE ‖ x(ti) ‖p]

)

≤ 6p−1Mp 1(1− e−δγ)p−1(1− e−(δ−β)γ)

[supi∈Z

cieβtiE ‖ x(ti) ‖p].

Then there exists a θ ∈ [−q,∞), such that for any t ≥ θ, we have

eβtE ‖ x(t) ‖p

≤ L∗ supt∈R

eβtE ‖ x(t) ‖p +(5p−1 ‖ A−β ‖p l1 + 3)ε,

where L∗ = 6p−1Mp 1

(1−e−δγ)p−1(1−e−(δ−β)γ)supi∈Z ci < 1. Thus we get that

supt∈R

eβtE ‖ x(t) ‖p≤ (5p−1 ‖ A−β ‖p l1 + 3)1− L∗

ε.

It follows that eβtE ‖ x(t) ‖p≤ (5p−1‖A−β‖pl1+3)1−L∗ ε, which is implies that eβtE ‖

x(t) ‖p→ 0 as t →∞. So we conclude that the piecewise pseudo almost periodicmild solution of (1)-(2) is globally exponentially stable. The proof is completed.

5 An example

Consider following partial stochastic differential equations of the form

d[z(t, x)−$1(t, z(t− q, x))] =∂2

∂x2z(t, x)dt + $2(t, z(t− q, x))dt

+$3(t, z(t− q, x))dW (t), t ∈ R, t 6= ti, i ∈ Z, x ∈ [0, π], (9)

∆z(ti, x) = βiz(ti, x), i ∈ Z, x ∈ [0, π], (10)

z(t, 0) = z(t, π) = 0, t ∈ R, (11)

where W (t) is a two-sided standard one-dimensional Brownian motion definedon the filtered probability space (Ω,F , P,Ft). In this system, βi ∈ PAP (Z, R),ti = i + 1

4 | sin i + sin√

2i|, tji, i ∈ Z, j ∈ Z are equipotentially almost periodicand γ = infi∈Z(ti+1 − ti) > 0, one can see [22] for more details.

Let H = L2([0, 1]) with the norm ‖ · ‖ and define the operators A : H → Hby Aω = ω′′ with the domain

D(A) := ω ∈ H : ω, ω′ are absolutely continuous, ω′′ ∈ H, ω(0) = ω(π) = 0.Then

Aω = −∞∑

n=1

n2〈ω, zn〉zn, ω ∈ D(A),

where zn(x) =√

2π sin(nx), n = 1, 2, 3, ..., is an orthogonal set of eigenvector of

A.

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1380 YAN ET AL 1343-1386

The bounded linear operator (−A)34 is given by

(−A)34 ω =

∞∑n=1

n32 〈ω, zn〉zn

on the space

D((−A)34 ) =

ω(·) ∈ H :

∞∑n=1

n32 〈ω, zn〉zn ∈ H

,

and (−A)−34 ω =

∑∞n=1

1

n32〈ω, zn〉zn for every ω ∈ H and ‖ (−A)−

34 ‖ is

bounded. It is well known that A is the infinitesimal generator of an analyticsemigroup T (t)(t ≥ 0) in H, and is given (See [40]) by

T (t)ω =∞∑

n=1

exp(−n2t)〈ω, ωn〉ωn, ω ∈ H,

that satisfies ‖ T (t) ‖≤ exp(−π2t), t ≥ 0 and satisfies (H1).Let Let y(t) = z(t, x), t ∈ [−q,∞), x ∈ [0, π]. Taking

A34 h(t, yt)(x) = $1(t, z(t− q, x)),

g(t, yt)(x) = $2(t, z(t− q, x)),

f(t, yt)(x) = $3(t, z(t− q, x)),

andIi(y)(x) = βiz(ti, x), i ∈ Z.

Then, the above equation (9)-(11) can be written in the abstract form as thesystem (1)-(2).

From Theorem 3.1, it follows that the following proposition holds.Proposition 5.1. Let $1, $2, $3 satisfy (H2)-(H6), then system (9)-(11) hasan p-mean piecewise pseudo almost periodic mild solution on R.

In the above example, we can take

$1(t, z(t− q, ·)) = k1[sin t + sin√

2t + l(t)] sin z(t− q, ·),

$2(t, z(t− q, ·)) = k2[sin t + sin√

2t + l(t)] sin z(t− q, ·),$3(t, z(t− q, ·)) = k3[sin t + sin

√2t + l(t)] sin z(t− q, ·),

andβiz(ti, ·) = ci[sin i + sin

√2i + l(i)] sin z(ti, ·), i ∈ Z,

where kj > 0, j = 1, 2, 3 and ci > 0, i ∈ Z, l ∈ UPC(R, R) defined by

l(t) =

0, t ≤ 0,e−t, t ≥ 0.

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1381 YAN ET AL 1343-1386

From [3], sin t + sin√

2t is almost periodic. On the other hand,

12r

∫ r

−r

|l(t)|pdt

=12r

∫ r

0

|l(t)|pdt =12r

∫ r

0

e−ptdt =12r

1− e−pr

p.

Consequently

limr→∞

12r

∫ r

−r

|l(t)|pdt = 0.

TakingA

34 h(t, yt)(x) = k1[sin t + sin

√2t + l(t)] sin z(t− q, x),

g(t, yt)(x) = k2[sin t + sin√

2t + l(t)] sin z(t− q, x),

f(t, yt)(x) = k3[sin t + sin√

2t + l(t)] sin z(t− q, x),

andIi(y)(x) = ci[sin i + sin

√2i + l(i)] sin z(ti, x), i ∈ Z.

Thus, one has

E ‖ A34 h(t, ψ)−A

34 h(t1, ψ1) ‖p

≤ ((2 +√

2)k1)p[|t− t1|+ ‖ ψ − ψ1 ‖pD],

E ‖ g(t, ψ)− g(t, ψ1) ‖p≤ (3k2)p ‖ ψ − ψ1 ‖pD,

E ‖ f(t, ψ)− f(t, ψ1) ‖p≤ (3k3)p ‖ ψ − ψ1 ‖pD,

and E ‖ A34 h(t, ψ) ‖p≤ (3k1)p ‖ ψ ‖p

D, E ‖ g(t, ψ) ‖p≤ (3k2)p ‖ ψ ‖pD,E ‖

f(t, ψ) ‖p≤ (3k3)p ‖ ψ ‖pD, for all (t, ψ), (t1, ψ1) ∈ R×D. Further,

E ‖ I(y)− I(y1) ‖p≤ (3ci)p ‖ y − y1 ‖p,

and E ‖ I(y) ‖p≤ (3ci)p ‖ y ‖p, for all (t, y), (t, y1) ∈ Lp(P, H)). Then, all con-ditions in Theorem 4.1 are satisfied. Hence, the system (9)-(11) has an p-meanpiecewise globally exponentially stable pseudo almost periodic mild solution.

Acknowledgments

This work is supported by the National Natural Science Foundation of China(11461019), the President Fund of Scientific Research Innovation and Appli-cation of Hexi University (xz2013-10), the Scientific Research Fund of YoungTeacher of Hexi University (QN2015-01).

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1382 YAN ET AL 1343-1386

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[31] R. Sakthivel, J. Luo, Asymptotic stability of impulsive stochastic partialdifferential equations with infinite delays, J. Math. Anal. Appl. 356 (2009),1-6.

[32] L. Hu, Y. Ren, Existence results for impulsive neutral stochastic functionalintegro-differential equations with infinite delays, Acta Appl. Math. 111(2010), 303-317

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[34] R.J. Zhang, N. Ding, L.S. Wang, Mean square almost periodic solutionsfor impulsive stochastic differential equations with delays, J. Appl. Math.2012 (2012), Article ID 414320, 1-14.

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43


1385 YAN ET AL 1343-1386

Zuomao Yan and Fangxia LuDepartment of Mathematics,Hexi University,Zhangye, Gansu734000, P.R. China∗ Corresponding author.E-mail address: [email protected]

44


1386 YAN ET AL 1343-1386

FOURIER SERIES OF FUNCTIONS ASSOCIATED WITH

POLY-GENOCCHI POLYNOMIALS

TAEKYUN KIM1,2, DAE SAN KIM3, DMITRY V. DOLGY4, AND JIN-WOO PARK5,∗

Abstract. In this paper, we will consider three types functions associated

with poly-Genocchi polynomials and find their Fourier series expansions. In

addition, we will express each of them in terms of Bernoulli functions.

1. Introduction

Let r be any integer. Then we recall that

Lir(x) =∞∑m=1

xm

mr, (see [1,6]),

is the rth polylogarithm function for r ≥ 1, and a rational function for r ≤ 0. Herewe note

d

dxLir+1(x) =

1

xLir(x).

The poly-Bernoulli polynomials B(r)m (x) of index r are given by

Lir(1− e−t)et − 1

ext =

∞∑m=0

B(r)m (x)

tm

m!. (1.1)

For x = 0, B(r)m = B(r)

m (0) are called poly-Bernoulli numbers of index r. Note here

that B(1)m (x) = Bm(x) are the Bernoulli polynomials given by

t

et − 1ext =

∞∑m=0

Bmtm

m!.

Here we mention in passing that our definition in (1.1) of poly-Bernoulli poly-nomials was introduced in [4]. Analogously to the construction of poly-Bernoulli

polynomials, the poly-Genocchi polynomials G(r)m (x) of index r are defined by

2Lir(1− e−t)et + 1

ext =

∞∑m=0

G(r)m (x)

tm

m!. (1.2)

When x = 0, G(r)m = G(r)

m (0) are called poly-Genocchi numbers of index r. Observe

here that G(1)m (x) = Gm(x) are the Genocchi polynomials given by

2t

et + 1ext =

∞∑m=0

Gm(x)tm

m!.

2010 Mathematics Subject Classification. 11B68, 11B83, 42A16.Key words and phrases. Fourier series, Bernoulli functions, poly-Genocchi polynomials, poly-

Bernoulli polynomials.∗ Corresponding author.

1


1387 T. KIM ET AL 1387-1400

2 Fourier series of functions associated with poly-Genocchi polynomials

The poly-Genocchi polynomials G(r)m (x) were first introduced in [3], where they

were called poly-Euler polynomials and denoted by E(r)m (x). However, for the ob-

vious reason it seems more appropriate to call them poly-Genocchi polynomialsrather than poly-Euler polynomials. For the definitions of Genocch numbers andpolynomials, the reader refers to [2, 7, 8] and [3], respectively.

For poly-Bernoulli polynomials and poly-Genocchi polynomials, we will need thefollowing facts.

d

dxB(r)m (x) = mB(r)

m−1(x), (m ≥ 1), B(r)0 (x) = 1

B(0)m (x) = xm, B(0)

m = δm,0,

d

dxG(r)m (x) = mG(r)

m−1(x), (m ≥ 1),

G(r+1)m (1) + G(r+1)

m = 2B(r)m−1, (m ≥ 1), (1.3)

G(r)0 (x) = 0, G(r)

1 (x) = 1, degG(r)m (x) = m− 1, (m ≥ 1). (1.4)

Here we obtain (1.3) by differentiating with respect to t both sides of the identity∞∑m=0

(G(r+1)m (1) + G(r+1)

m

) tmm!

= 2Lir+1(1− e−t),

which follows immediately from (1.2).Let Em(x) be the Euler polynomials defined by

2

et + 1ext =

∞∑m=0

Em(x)tm

m!,

and let

Lir(1− e−t) =∞∑n=1

antn

n!= t+

∞∑n=2

antn

n!.

Then in view of (1.2) we have

∞∑m=0

G(r)m (x)

tm

m!=

∞∑m=1

(m−1∑l=0

(m

l

)am−lEl(x)

)tm

m!,

from which (1.4) follows.

In [5], G(r+1)m (x) are expressed as linear combinations of Euler polynomials and

of Genocchi polynomials as follows.

G(r+1)m (x) =

m−1∑j=0

(m

j

)B(r)m−j−1Ej(x)

=1

m+ 1

m∑j=1

(m+ 1

j

)B(r)m−jGj(x), (m ≥ 1).

We will need the following facts about Bernoulli functions Bm(〈x〉), where forany real number x, 〈x〉 = x− bxc ∈ [0, 1) denotes the fractional part of x :

(a) for m ≥ 2,

Bm(〈x〉) = −m!

∞∑n=−∞n6=0

e2πinx

(2πin)m,


1388 T. KIM ET AL 1387-1400

T. Kim, D. S. Kim, D. V. Dolgy, J.-W. Park 3

(b) for m = 1,

−∞∑

n=−∞n6=0

e2πinx

2πin=

B1(〈x〉), for x /∈ Z,

0, for x ∈ Z.

In this paper, we will consider the following three types of functions αm(〈x〉),βm(〈x〉), and γm(〈x〉) associated with poly-Genocchi polynomials and find theirFourier series expansions. In addition, we will express each of them in terms ofBernoulli functions.

(a) αm(〈x〉) =∑mk=1 G

(r+1)k (x)xm−k, (m ≥ 2);

(b) βm(〈x〉) =∑mk=1

1k!(m−k)!G

(r+1)k (x)xm−k, (m ≥ 2);

(c) γm(〈x〉) =∑m−1k=1

1k(m−k)G

(r+1)k (x)xm−k, (m ≥ 2).

2. Fourier series of functions of the first type

Let

αm(x) =m∑k=1

G(r+1)k (x)xm−k, (m ≥ 2).

Then we will consider the function

αm(〈x〉) =m∑k=1

G(r+1)k (〈x〉) 〈x〉m−k , (m ≥ 2),

defined on R, which is periodic with period 1. The Fourier series of αm(〈x〉) is

∞∑n=−∞

A(m)n e2πinx,

where

A(m)n =

∫ 1

0

αm(〈x〉)e−2πinxdx =

∫ 1

0

α,(x)e−2πinxdx.

Before proceeding further, we need to observe the following.

α′m(x) =m∑k=1

(kG(r+1)

k−1 (x)xm−k + (m− k)G(r+1)k (x)xm−k−1

)=

m∑k=2

kG(r+1)k−1 (x)xm−k +

m−1∑k=1

(m− k)G(r+1)k (x)xm−k−1

=m−1∑k=1

(k + 1)G(r+1)k (x)xm−k−1 +

m−1∑k=1

(m− k)G(r+1)k (x)xm−k−1

=(m+ 1)αm−1(x).

From this, we obtain (αm+1(x)

m+ 2

)′= αm(x),

and ∫ 1

0

αm(x)dx =1

m+ 2(αm+1(1)− αm+1(0)) .


1389 T. KIM ET AL 1387-1400


For each integer m ≥ 2, we let

∆m =αm(1)− αm(0)

=m∑k=1

(G(r+1)k (1)−G(r+1)

k δm,k

)=

m∑k=1

(−G(r+1)

k + 2B(r)k−1 −G(r+1)

k δm,k

)=2

m∑k=1

B(r)k−1 −

m∑k=1

G(r+1)k −G(r)

m .

Then

αm(0) = αm(1)⇐⇒ ∆m = 0

⇐⇒ 2m∑k=1

B(r)k−1 =

m∑k=1

G(r+1)k + G(r+1)

m ,

and ∫ 1

0

αm(x)dx =1

m+ 2∆m+1.

We are now going to determine the Fourier coefficients A(m)n .

Case 1 : n 6= 0.

A(m)n =

∫ 1

0

αm(x)e−2πinxdx

=− 1

2πin

[αm(x)e−2πinx

]10

+1

2πin

∫ 1

0

α′m(x)e−2πinxdx

=− 1

2πin(αm(1)− αm(0)) +

m+ 1

2πin

∫ 1

0

αm−1(x)e−2πinxdx

=m+ 1

2πinA(m−1)n − 1

2πin∆m,

from which by induction on m we can deduce tat

A(m)n = − 1

m+ 2

m−1∑j=1

(m+ 2)j(2πin)j

∆m−j+1.

Case 2 : n = 0.

A(m)0 =

∫ 1

0

αm(x)dx =1

m+ 2∆m+1.

αm(〈x〉), (m ≥ 2) is piecewise C∞. Moreover, αm(〈x〉) is continuous for thoseintegers m ≥ 2 with ∆m = 0, and discontinuous with jump discontinuities for thoseintegers m ≥ 2 with ∆m 6= 0.

Assume first that m is an integer ≥ 2 with ∆m = 0. Then αm(0) = αm(1).Hence αm(〈x〉) is piecewise C∞, and continuous. Thus the Fourier series of αm(〈x〉)


1390 T. KIM ET AL 1387-1400


converges uniformly to αm(〈x〉), and

αm(〈x〉) =1

m+ 2∆m+1 +

∞∑n=−∞n6=0

− 1

m+ 2

m−1∑j=1

(m+ 2)j(2πin)j

∆m−j+1

e2πinx

=1

m+ 2∆m+1 +

1

m+ 2

m−1∑j=1

(m+ 2

j

)∆m−j+1

−j! ∞∑n=−∞n6=0

e2πinx

(2πin)j

=

1

m+ 2∆m+1 +

1

m+ 2

m−1∑j=2

(m+ 2

j

)∆m−j+1Bj(〈x〉)

+ ∆m ×B1(〈x〉), for x /∈ Z,

0, for x ∈ Z.Now, we can state our first theorem.

Theorem 2.1. For each integer l ≥ 2, we let

∆l = 2

l∑k=1

B(r)k−1 −

l∑k=1

G(r+1)k −G(r+1)

l .

Assume that ∆m = 0, for an integer m ≥ 2. Then we have the following.

(a)∑mk=1 G

(r+1)k (〈x〉) 〈x〉m−k has the Fourier series expansion

m∑k=1

G(r+1)k (〈x〉) 〈x〉m−k

=1

m+ 2∆m+1 +

∞∑n=−∞n6=0

− 1

m+ 2

m−1∑j=1

(m+ 2)j(2πin)j

∆m−j+1

e2πinx,

for all x ∈ R, where the convergence is uniform.(b)

m∑k=1

G(r+1)k (〈x〉) 〈x〉m−k =

1

m+ 2∆m+1 +

1

m+ 2

m−1∑j=2

(m+ 2

j

)∆m−j+1Bj(〈x〉),

for all x ∈ R.

Assume next that ∆m 6= 0, for an integer m ≥ 2. Then αm(0) 6= αm(1). Henceαm(〈x〉) is piecewise C∞, and discontinuous with jump discontinuities at integers.Then the Fourier series of αm(〈x〉) converges pointwise to αm(〈x〉), for x /∈ Z, andconverges to

1

2(αm(0) + αm(1)) = αm(0) +

1

2∆m.

We can now state our second theorem.


∆l = 2l∑

k=1

B(r)k−1 −

l∑k=1

G(r+1)k −G(r+1)

l .

Assume that ∆m 6= 0, for an integer m ≥ 2. Then we have the following.


1391 T. KIM ET AL 1387-1400


(a)

1

m+ 2∆m+1 +

∞∑n=−∞n6=0

− 1

m+ 2

m−1∑j=1

(m+ 2)j(2πin)j

∆m−j+1

e2πinx

=

∑mk=1 G

(r+1)k (〈x〉) 〈x〉m−k , for x /∈ Z,

G(r+1)m + 1

2∆m, for x ∈ Z.

(b)

1

m+ 2

m−1∑j=0

(m+ 2

j

)∆m−j+1Bj(〈x〉) =

m∑k=1

G(r+1)k (〈x〉) 〈x〉m−k , for x /∈ Z;

1

m+ 2

m−1∑j=0j 6=1

(m+ 2

j

)∆m−j+1Bj(〈x〉) = G(r+1)

m +1

2∆m, for x ∈ Z.

3. Fourier series of functions of the second type

Let

βm(x) =

m∑k=1

1

k!(m− k)!G(r+1)k (x)xm−k, (m ≥ 2).


βm(〈x〉) =

m∑k=1

1

k!(m− k)!G(r+1)k (〈x〉) 〈x〉m−k , (m ≥ 2),

defined on R, which is periodic with period 1.The Fourier series of βm(〈x〉) is

∞∑n=−∞

B(m)n e2πinx,

where

B(m)n =

∫ 1

0

βm(〈x〉)e−2πinxdx =

∫ 1

0

βm(x)e−2πinxdx.

To continue further, we need to observe the following.

β′m(x) =

m∑k=1

(k

k!(m− k)!G(r+1)k−1 (x)xm−k +

m− kk!(m− k)!

G(r+1)k (x)xm−k−1

)

=

m∑k=2

1

(k − 1)!(m− k)!G(r+1)k−1 (x)xm−k +

m−1∑k=1

1

k!(m− k − 1)!G(r+1)k (x)xm−k−1

=

m−1∑k=1

1

k!(m− 1− k)!G(r+1)k (x)xm−1−k +

m−1∑k=1

1

k!(m− 1− k)!G(r+1)k (x)xm−1−k

=2βm−1(x).

From this, we have (βm+1(x)

2

)′= βm(x),


1392 T. KIM ET AL 1387-1400


and ∫ 1

0

βm(x)dx =1

2(βm+1(1)− βm+1(0)) .

For each integer m ≥ 2, we put

Ωm =βm(1)− βm(0)

=m∑k=1

1

k!(m− k)!

(G(r+1)k (1)−G(r+1)

k δm,k

)=

m∑k=1

1

k!(m− k)!

(−G(r+1)

k + 2B(r)k−1 −G(r+1)

k δm,k

)=2

m∑k=1

B(r)k−1

k!(m− k)!−

m∑k=1

G(r+1)k

k!(m− k)!− G(r+1)

m

m!.

Then

βm(0) = βm(1)⇐⇒ Ωm = 0

⇐⇒2m∑k=1

B(r)k−1

k!(m− k)!=

m∑k=1

G(r+1)k

k!(m− k)!+

G(r+1)m

m!,

and ∫ 1

0

βm(x)dx =1

2Ωm+1.

Now, we would like to determine the Fourier coefficients B(m)n .

Case 1 : n 6= 0.

B(m)n =

∫ 1

0

βm(x)e−2πinxdx

=− 1

2πin

[βm(x)e−2πinx

]10

+1

2πin

∫ 1

0

β′m(x)e−2πinxdx

=− 1

2πin(βm(1)− βm(0)) +

2

2πin

∫ 1

0

βm−1(x)e−2πinxdx

=2

2πinB(m−1)n − 1

2πinΩm,

from which by induction on m we can deduce that

B(m)n = −

m−1∑j=1

2j−1

(2πin)jΩm−j+1.

Case 2 : n = 0.

B(m)0 =

∫ 1

0

βm(x)dx =1

2Ωm+1.

βm(〈x〉), (m ≥ 2) is piecewise C∞. Moreover, βm(〈x〉) is continuous for thoseintegers m ≥ 2 with Ωm = 0, and discontinuous with jump discontinuities atintegers for those integers m ≥ 2 with Ωm 6= 0.

Assume first that m is an integer ≥ 2 with Ωm = 0. Then βm(0) = βm(1).βm(〈x〉) is piecewise C∞, and continuous. Hence the Fourier series of βm(〈x〉)


1393 T. KIM ET AL 1387-1400


converges uniformly to βm(〈x〉), and

βm(〈x〉) =1

2Ωm+1 +

∞∑n=−∞n6=0

−m−1∑j=1

2j−1

(2πin)jΩm−j+1

e2πinx

=1

2Ωm+1 +

m−1∑j=1

2j−1

j!Ωm−j+1

−j! ∞∑n=−∞n6=0

e2πinx

(2πin)j

=

1

2Ωm+1 +

m−1∑j=2

2j−1

j!Ωm−j+1Bj(〈x〉) + Ωm ×

B1(〈x〉), for x /∈ Z,

0, for x ∈ Z.

Now, we are ready to state our first theorem.


Ωl = 2l∑

k=1

B(r)k−1

k!(l − k)!−

l∑k=1

G(r+1)k

k!(l − k)!−

G(r+1)l

l!.

Assume that Ωm = 0, for an integer m ≥ 2. Then we have the following

(a)∑mk=1

1k!(m−k)!G


m∑k=1

1

k!(m− k)!G(r+1)k (〈x〉) 〈x〉m−k =

1

2Ωm+1+

∞∑n=−∞n6=0

−m−1∑j=1

2j−1

(2πin)jΩm−j+1

e2πinx,

for all x ∈ R, where the convergence is uniform.(b)

m∑k=1

1

k!(m− k)!G(r+1)k (〈x〉) 〈x〉m−k =

1

2Ωm+1 +

m−1∑j=2

2j−1

j!Ωm−j+1Bj(〈x〉),

for all x ∈ R.

Next, we assume that m is an integer ≥ 2 with Ωm 6= 0. Then βm(0) 6= βm(1).Hence βm(〈x〉) is piecewise C∞, and discontinuous with jump discontinuities atintegers. Thus the Fourier series of βm(〈x〉) converges pointwise to βm(〈x〉), forx /∈ Z, and converges to

1

2(βm(0) + βm(1)) = βm(0) +

1

2Ωm,

for x ∈ Z.We are now ready to state our second theorem.


Ωl = 2l∑

k=1

B(r)k−1

k!(l − k)!−

l∑k=1

G(r+1)k

k!(l − k)!−

G(r+1)l

l!.

Assume that Ωm 6= 0, for an integer m ≥ 2. Then we have the following.


1394 T. KIM ET AL 1387-1400


(a)

1

2Ωm+1 +

∞∑n=−∞n6=0

−m−1∑j=1

2j−1

(2πin)jΩm−j+1

e2πinx

=

∑mk=1

1k!(m−k)!G

(r+1)k (〈x〉) 〈x〉m−k , for x /∈ Z,

1m!G

(r+1)m + 1

2Ωm, for x ∈ Z.

(b)

m−1∑j=0

2j−1

j!Ωm−j+1Bj(〈x〉) =

m∑k=1

1

k!(m− k)!G(r+1)k (〈x〉) 〈x〉m−k , for x /∈ Z;

m−1∑j=0j 6=1

2j−1

j!Ωm−j+1Bj(〈x〉) =

1

m!G(r+1)m +

1

2Ωm, for x ∈ Z.

4. Fourier series of functions of the third type

Let

γm(x) =m−1∑k=1

1

k(m− k)G(r+1)k (x)xm−k, (m ≥ 2).


γm(〈x〉) =m−1∑k=1

1

k(m− k)G(r+1)k (〈x〉) 〈x〉m−k , (m ≥ 2),

defined on R, which is periodic with period 1.The Fourier series of γm(〈x〉) is

∞∑n=−∞

C(m)n e2πinx,

where

C(m)n =

∫ 1

0

γm(〈x〉)e−2πinxdx =

∫ 1

0

γm(x)e−2πinxdx.

To proceed further, we need to observe the following.

γ′m(x) =

m−1∑k=1

1

k(m− k)

(kG(r+1)

k−1 (x)xm−k + (m− k)G(r+1)k (x)xm−k−1

)=m−1∑k=2

1

m− kG(r+1)k−1 (x)xm−k +

m−1∑k=1

1

kG(r+1)k (x)xm−k−1

=m−2∑k=1

1

m− 1− kG(r+1)k (x)xm−1−k +

m−1∑k=1

1

kG(r+1)k (x)xm−1−k

=(m− 1)m−2∑k=1

1

k(m− 1− k)G(r+1)k (x)xm−1−k +

1

m− 1Gr+1m−1(x)

=(m− 1)γm−1(x) +1

m− 1G(r+1)m−1 (x).


1395 T. KIM ET AL 1387-1400


From this, we have(1

m

(γm+1(x)− 1

m(m+ 1)G(r+1)m+1 (x)

))′= γm(x),

and ∫ 1

0

γm(x)dx =1

m

[γm+1(x)− 1

m(m+ 1)G(r+1)m+1 (x)

]10

=1

m

(γm+1(1)− γm+1(0)− 1

m(m+ 1)

(G(r+1)m+1 (1)−G(r+1)

m+1 (0)))

=1

m

(γm+1(1)− γm+1(0) +

2

m(m+ 1)

(G(r+1)m+1 − B(r)

m

)).

For each integer m ≥ 2, we let

Λm =γm(1)− γm(0)

=m−1∑k=1

1

k(m− k)

(G(r+1)k (1)−G(r+1)

k δm,k

)=

m−1∑k=1

1

k(m− k)

(−G(r+1)

k + 2B(r)k−1 −G(r+1)

k δm,k

)=2

m−1∑k=1

B(r)k−1

k(m− k)−m−1∑k=1

G(r+1)k

k(m− k).

Then

γm(0) = γm(1)⇐⇒Λm = 0

⇐⇒2m−1∑k=1

B(r)k−1

k(m− k)=m−1∑k=1

G(r+1)k

k(m− k),

∫ 1

0

γm(x)dx =1

m

(Λm+1 +

2

m(m+ 1)

(G(r+1)m+1 − B(r)

m

)).

Now, we are going to determine the Fourier coefficients C(m)n .

Case 1 : n 6= 0.Observe first that∫ 1

0

G(r+1)m (x)e−2πinxdx =

2∑m−1k=1

(m)k−1

(2πin)k

(G(r+1)m−k+1 − B(r)

m−k

), for n 6= 0,

2m+1

(B(r)m −G(r+1)

m+1

), for n = 0.

Then we have

C(m)n = − 1

2πin

[γm(x)e−2πinx

]10

+1

2πin

∫ 1

0

γ′m(x)e−2πinxdx

=− 1

2πin(γm(1)− γm(0)) +

1

2πin

∫ 1

0

((m− 1)γm−1(x) +

1

m− 1G(r+1)m−1 (x)

)e−2πinxdx

=− 1

2πinΛm +

m− 1

2πinC(m−1)n +

1

2πin(m− 1)

∫ 1

0

G(r+1)m−1 (x)e−2πinxdx

=m− 1

2πinC(m−1)n − 1

2πinΛm +

2

2πin(m− 1)Φm,


1396 T. KIM ET AL 1387-1400


where

Φm =m−2∑k=1

(m− 1)k−1(2πin)k

(G(r+1)m−k − B(r)

m−k−1

), (4.1)

From (4.1), by induction on m we can show that

C(m)n = − 1

m

m−1∑j=1

(m)j(2πin)j

Λm−j+1 +1

m

m−2∑j=1

2(m)j(2πin)j(m− j)

Φm−j+1.

We now observe that

m−2∑j=1


Φm−j+1

=m−2∑j=1


m−j−1∑k=1

(m− j)k−1(2πin)k

(G(r+1)m−j−k+1 − B(r)

m−j−k

)

=m−2∑j=1

m−j−1∑k=1

2(m)j+k−1(2πin)j+k(m− j)

(G(r+1)m−j−k+1 − B(r)

m−j−k

)

=2m−2∑j=1

1

m− j

m−1∑s=j+1

(m)s−1(2πin)s

(G(r+1)m−s+1 − B(r)

m−s

)

=2

m−1∑s=2

(m)s−1(2πin)s

(G(r+1)m−s+1 − B(r)

m−s

) s−1∑j=1

1

m− j

=2

m−1∑s=2

(m)s−1(2πin)s

(G(r+1)m−s+1 − B(r)

m−s

)(Hm−1 −Hm−s)

=2m−1∑s=1

(m)s−1(2πin)s

(G(r+1)m−s+1 − B(r)

m−s

)(Hm−1 −Hm−s)

=2m−1∑s=1

(m)s(2πin)s

G(r+1)m−s+1 − B(r)

m−sm− s+ 1

(Hm−1 −Hm−s).

Putting everything altogether,

C(m)n = − 1

m

m−1∑s=1

(m)s(2πin)s

(Λm−s+1 −

2(G(r+1)m−s+1 − B(r)

m−s)

m− s+ 1(Hm−1 −Hm−s)

).

Case 2 : n = 0.

C(m)0 =

∫ 1

0

γm(x)dx

=1

m

(Λm+1 +

2

m(m+ 1)

(G(r+1)m+1 − B(r)

m

)).

γm(〈x〉), (m ≥ 2) is piecewise C∞. In addition, γm(〈x〉) is continuous for thoseintegers m ≥ 2 with Λm = 0, and discontinuous with jump discontinuities atintegers for those integers m ≥ 2 with Λm 6= 0.


1397 T. KIM ET AL 1387-1400


Assume first that Λm = 0. Then γm(0) = γm(1). Hence γm(〈x〉) is piecewise C∞,and continuous. Thus Fourier series of γm(〈x〉) converges uniformly to γm(〈x〉), and

γm(〈x〉) =1

m

(Λm+1 +

2

m(m+ 1)

(G(r+1)m+1 − B(r)

m

))

− 1

m

∞∑n=−∞n6=0

m−1∑s=1

(m)s(2πin)s

Λm−s+1 −2(G(r+1)m−s+1 − B(r)

m−s

)m− s+ 1

(Hm−1 −Hm−s)

e2πinx

=1

m

(Λm+1 +

2

m(m+ 1)

(G(r+1)m+1 − B(r)

m

))

+1

m

m−1∑s=1

(m

s

)Λm−s+1 −2(G(r+1)m−s+1 − B(r)

m−s

)m− s+ 1

(Hm−1 −Hm−s)

×

−s! ∞∑n=−∞n6=0

e2πinx

(2πin)s

=

1

m

(Λm+1 +

2

m(m+ 1)

(G(r+1)m+1 − B(r)

m

))

+1

m

m−1∑s=2

(m

s

)Λm−s+1 −2(G(r+1)m−s+1 − B(r)

m−s

)m− s+ 1

(Hm−1 −Hm−s)

Bs(〈x〉)

+ Λm ×B1(〈x〉), for x /∈ Z,

0, for x ∈ Z.

Now, we can state our first theorem.


Λl = 2l−1∑k=1

B(r)k−1

k(l − k)−

l−1∑k=1

G(r+1)k

k(l − k).

Assume that Ωm = 0, for an integer m ≥ 2. Then we have the following.

(a)∑m−1k=1

1k(m−k)G


m−1∑k=1

1

k(m− k)G(r+1)k (〈x〉) 〈x〉m−k

=1

m

(Λm+1 +

2

m(m+ 1)

(G(r+1)m+1 − B(r)

m

))

− 1

m

∞∑n=−∞n6=0

m−1∑s=1

(m)s(2πin)s

Λm−s+1 −2(G(r+1)m−s+1 − B(r)

m−s

)m− s+ 1

(Hm−1 −Hm−s)

e2πinx,

for all x ∈ R, where the convergence is uniform.


1398 T. KIM ET AL 1387-1400


(b)

m−1∑k=1

1

k(m− k)G(r+1)k (〈x〉) 〈x〉m−k

=1

m

m−1∑s=0s6=1

(m

s

)Λm−s+1 −2(G(r+1)m−s+1 − B(r)

m−s

)m− s+ 1

(Hm−1 −Hm−s)

Bs(〈x〉),

for all x ∈ R.

Assume next that m is an integer ≥ 2 with Λm 6= 0. Then γm(0) 6= γm(1). Henceγm(〈x〉) is piecewise C∞, and discontinuous with jump discontinuities at integers.Thus the Fourier series of γm(〈x〉) converges pointwise to γm(〈x〉), for x /∈ Z, andconverges to

1

2(γm(0) + γm(1)) =

1

2Λm,

for x ∈ Z.We can now state our second theorem.


Λl = 2l−1∑k=1

B(r)k−1

k(l − k)−

l−1∑k=1

G(r+1)k

k(l − k).

Assume that Λm 6= 0, for an integer m ≥ 2. Then we have the following.

(a)

1

m

(Λm+1 +

2

m(m+ 1)

(G(r+1)m+1 − B(r)

m

))

− 1

m

∞∑n=−∞n6=0

m−1∑s=1

(m)s(2πin)s

Λm−s+1 −2(G(r+1)m−s+1 − B(r)

m−s

)m− s+ 1

(Hm−1 −Hm−s)

e2πinx

=

∑m−1k=1

1k(m−k)G

(r+1)k (〈x〉) 〈x〉m−k , for x /∈ Z,

12Λm, for x ∈ Z.

(b)

1

m

m−1∑s=0

(m

s

)Λm−s+1 −2(G(r+1)m−s+1 − B(r)

m−s

)m− s+ 1

(Hm−1 −Hm−s)

Bs(〈x〉)

=m−1∑k=1

1

k(m− k)G(r+1)k (〈x〉) 〈x〉m−k , for x /∈ Z;

1

m

m−1∑s=0s6=1

(m

s

)Λm−s+1 −2(G(r+1)m−s+1 − B(r)

m−s

)m− s+ 1

(Hm−1 −Hm−s)

Bs(〈x〉)

=1

2Λm, for x ∈ Z.


1399 T. KIM ET AL 1387-1400


5. Acknowledgements

This research was supported by the Daegu University Research Grant, 2017.

References

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY,

1966.

[2] Y. He, S. Araci, H. M. Srivastava, M. Acikgoz, Some new identities for the Apostol-Bernoullipolynomials and the Apostol-Genocchi polynomials, Appl. Math. Comput., 262 (2015), 31-41.

[3] H. Jolany, M. Aliabadi, R. B. Corcino and M. R. Darafsheh, A note on multi poly-Euler

numbers and Bernoulli polynomials, Gen. Math., 20 (2012), no. 2 − 3, 122-134.[4] D. S. Kim, T. Kim, A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials,

Russ. J. Math. Phys. 22 (2015), no. 1, 26–33.

[5] T. Kim, D. S. Kim, G.-W. Jang, J. Kwon, Fourier series of sums of products of Genocchifunctions and their applications, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1683–1694.

[6] W. F. Pickard, On polylogarithms, Publ. Math. Debrecen, 15 (1968), 33–43.[7] C. S. Ryoo, Calculating zeros of the twisted Genocchi polynomials, Adv. Stud. Contemp.

Math. (Kyungshang) 17 (2008), no. 2, 147–159.

[8] Y. Simsek, Identities on the Changhee numbers and Apostol-type Daehee polynomials,Adv.Stud. Contemp. Math. (Kyungshang) 27 (2017), no. 2, 199–212.

1 Department of Mathematics, College of Science, Tianjin Polytechnic University,

Tianjin, 300160, China

2 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of

Korea.

E-mail address: [email protected]

3 Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea.


4 Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea.


5 Department of Mathematics Education, Daegu University, Gyeongsan-si, Gyeongsangbuk-

do, 712-714, Republic of Korea.



1400 T. KIM ET AL 1387-1400

Tracy-Singh Products and Classes of Operators

Arnon Ploymukda1, Pattrawut Chansangiam1∗, Wicharn Lewkeeratiyutkul21Department of Mathematics, Faculty of Science,

King Mongkut’s Institute of Technology Ladkrabang,Bangkok 10520, Thailand.

2Department of Mathematics and Computer Science,Faculty of Science, Chulalongkorn University,

Bangkok 10330, Thailand.

Abstract

We investigate relationship between Tracy-Singh products and certainclasses of Hilbert space operators. We show that the normality, hyponor-mality, paranormality of operators are preserved by Tracy-Singh products.Operators of class-A type are also preserved under Tracy-Singh products.Moreover, we obtain necessary and sufficient conditions for the Tracy-Singh product of two operators to be normal, quasinormal, (co)isometry,and unitary.

Keywords: Tracy-Singh product, tensor product, normality, class A operatorMathematics Subject Classifications 2010: 47A05, 47A80, 47B20, 47B47.

1 Introduction

Tensor product of bounded linear operators plays a crucial role in functionalanalysis and operator theory. Many algebraic-order-analytic properties of oper-ators are preserved under taking tensor products, but by no means all of them.Importance results on tensor product involving certain classes of operators (e.g.positive, unitary, normal, compact) have been noticed by many mathematiciansfrom the beginning of the theory to nowadays (e.g. [22]). In the last two decades,the concepts of normality, hyponormality, and paranormality have been intro-duced and investigated by many authors, see e.g., [5, 13, 21]. Relations betweentensor products and class-A type operators also have received much attention,e.g., [10, 11, 12, 19, 20]. See more information about classes of operators in themonograph [7].

Recently, the notion of tensor product was extended to the Tracy-Singhproduct for Hilbert space operators in [15]. It was shown that compactness,

∗Corresponding author. Email: [email protected]

1


1401 Arnon Ploymukda ET AL 1401-1413


positivity and strict-positivity of operators are preserved under Tracy-Singhproducts [15, 16].

In this paper, we investigate relationship between Tracy-Singh products andcertain classes of operators. We divide such classes into three categories. Thefirst category consists of nilpotent, (skew)-Hermitian, (co)isometry, and unitaryoperators. The second one contains operator normality, hyponormality, andparanormality. The last one is the class-A type operators, which includes classA(k), class A, quasi-class (A, k), quasi-class A, ∗-class A, quasi-∗-class A, andquasi-∗-class (A, k) operators. We will show that the mentioned propertiesof operators are preserved under taking Tracy-Singh products. Moreover, weobtain necessary and sufficient conditions for the Tracy-Singh product of twooperators to be normal, quasinormal, (co)isometry, and unitary operators.

The paper is structured as follows. The next section supplies some pre-requisites about the tensor product and the Tracy-Singh product of operators.Next, we discuss relationship between Tracy-Singh products and the normality,hyponormality, and paranormality of operators. Then we consider Tracy-Singhproducts and certain properties of operators–being nilpotent, (skew)-Hermitian,(co)isometry, and unitary. The last section deals with class A type operators.

2 Preliminaries

In what follows, H and K denote complex separable Hilbert spaces. When Xand Y are Hilbert spaces, denote by B(X,Y ) the Banach space of bounded linearoperators from X into Y , equipped with the operator norm ∥·∥ and abbreviateB(X,X) to B(X). For Hermitian operators A and B on the same Hilbert space,we use the notation A > B to mean that A−B is a positive operator.

In order to define the Tracy-Singh product, we have to fix the orthogonaldecompositions of Hilbert spaces, namely,

H =m⊕i=1

Hi, K =n⊕

l=1

Kl

where all Hi’s and Kk’s are Hilbert spaces. Any operator A ∈ B(H) and B ∈B(K) thus can be expressed uniquely as operator matrices

A = [Aij ]m,mi,j=1 and B = [Bkl]

n,nk,l=1

where Aij ∈ B(Hj ,Hi) and Bkl ∈ B(Kl,Kk) for each i, j, k, l. Then the Tracy-Singh product of A and B is defined to be

AB =[[Aij ⊗Bkl]kl

]ij, (1)

which is a bounded linear operator from⊕m,n

i,k=1 Hi ⊗Kk into itself. Note thatwhen m = n = 1, the Tracy-Singh product AB reduces to the tensor productA⊗B.



A. Ploymukda, P. Chansangiam, W. Lewkeeratiyutkul

Lemma 1 ([15]). Algebraic and order properties of the Tracy-Singh product foroperators are listed here (provided that every operation is well-defined):

1. The map (A,B) 7→ AB is bilinear.

2. Compatibility with adjoints: (AB)∗ = A∗ B∗.

3. Compatibility with ordinary products: (AB)(C D) = AC BD.

4. Compatibility with powers: (AB)r = Ar Br for any r ∈ N.

5. Compatibility with inverses: if A and B are invertible, then A B isinvertible with (AB)−1 = A−1 B−1.

6. Positivity: if A > 0 and B > 0, then AB > 0.

7. Monotonicity: if A1 > B1 and A2 > B2, then A1 A2 > B1 B2.

Lemma 2 ([15]). Let A = [Aij ] ∈ B(H) and B ∈ B(K) be operator matrices.Then each (i, j)-block of AB is Aij B.

Analytic properties of the Tracy-Singh product for operators are listed below.

Lemma 3 ([16]). Let A = [Aij ]m,mi,j=1 ∈ B(H) and B = [Bkl]

n,nk,l=1 ∈ B(K). Then

we have

(i)1

mn∥A∥∥B∥ 6 ∥AB∥ 6 mn∥A∥∥B∥.

(ii) |AB| = |A||B|, here the absolute value of A is defined by |A| = (A∗A)12 .

(iii) If A and B are positive operators, then (A B)α = Aα Bα for anynonnegative real α.

Lemma 4. Let A ∈ B(H) and B ∈ B(K).

(i) The condition AB = 0 holds if and only if A = 0 or B = 0.

(ii) If AB = A C and A = 0, then B = C.

(iii) If B A = C A and A = 0, then B = C.

Proof. From the norm estimation in Lemma 3(i), one can deduce property (i).Properties (ii) and (iii) follow from (i) and the bilinearity of Tracy-Singh productin Lemma 1.

Lemma 5 ([21]). Let A,C ∈ B(H) and B,D ∈ B(K) be nonzero operators.Then A⊗B = C ⊗D if and only if there exists α ∈ C \ 0 such that C = αAand D = α−1B.

Proposition 6. Let A = [Aij ]m,mi,j=1, C = [Cij ]

m,mi,j=1 ∈ B(H) and B = [Bkl]

n,nk,l=1, D =

[Dkl]n,nk,l=1 ∈ B(K) be operator matrices such that Aij , Bkl, Cij and Dkl are

nonzero operators for all i, j = 1, . . . ,m and k, l = 1, . . . , n. Then AB = CDif and only if there exists α ∈ C \ 0 such that C = αA and D = α−1B.




Proof. If C = αA and D = α−1B for some α ∈ C \ 0, then by Lemma 1,

C D = (αA) (α−1B) = αα−1(AB) = AB.

Assume that AB = CD. By using Lemma 2, we get AijB = CijD for alli, j = 1, . . . ,m. For any fixed i, j ∈ 1, . . . ,m, we have Aij ⊗ Bkl = Cij ⊗Dkl

for all k, l = 1, . . . , n. For each i, j ∈ 1, . . . ,m and k, l ∈ 1, . . . , n, byapplying Lemma 5, there exists αij,kl ∈ C \ 0 such that Cij = αij,klAij andDkl = α−1

ij,klBkl. For any fixed i, j ∈ 1, . . . ,m, we have Cij = αij,klAij forall k, l = 1, . . . , n. This implies that αij,11 = · · · = αij,nn = αij . For anyfixed k, l ∈ 1, . . . , n, we have Dkl = α−1

ij Bkl for all i, j = 1, . . . ,m. It follows

that α11 = · · · = αmm = α. Thus Cij = αAij and Dkl = α−1Bkl for alli, j = 1, . . . ,m and k, l = 1, . . . , n. Therefore C = αA and D = α−1B.

Recall that the commutator of A and B in B(H) is defined by

[A,B] = AB −BA.

Proposition 7. Let A,C ∈ B(H) and B,D ∈ B(K).

(i) If [A,C] > 0 and [B,D] > 0, then [AB,C D] > 0.

(ii) If [A,C] 6 0 and [B,D] 6 0, then [AB,C D] 6 0.

(iii) If [A,C] = 0 and [B,D] = 0, then [AB,C D] = 0.

Proof. (i) Since AC > CA and BD > DB, we have AC BD > CADB byLemma 1. Then

[AB,C D] = AC BD − CADB > 0.

The assertion (ii) follows from (i) and the fact that −[X,Y ] = [Y,X] for anyoperators X and Y . The assertion (iii) follows from (i) and (ii).

3 Tracy-Singh products and operator normality

In this section, we discuss normality of Tracy-Singh products of operators. Thecontents can be divided into three parts. The first part deals with generalproperties of normality, the second one concerns hyponormality, and the lastone consists of paranormality.

3.1 Normality

Recall the following types of operator normality; see e.g. [7, Chapter 2] and [17]for more details.

Definition 8. An operator T ∈ B(H) is said to be

• normal if [T ∗, T ] = 0 ;




• binormal if [T ∗T, TT ∗] = 0 ;

• quasinormal if [T, T ∗T ] = 0 ;

• posinormal if TT ∗ = T ∗PT for some positive operator P .

Stochel [21] showed that for non-zero A ∈ B(H) and B ∈ B(K), the tensorproduct A⊗B is normal (resp. quasinormal) if and only if A and B are normal(resp. quasinormal). Now, we will extend this result to the case of Tracy-Singhproducts.

Theorem 9. Let A = [Aij ]m,mi,j=1 ∈ B(H) and B = [Bkl]

n,nk,l=1 ∈ B(K) be operator

matrices such that Aij and Bkl are nonzero operators for all i, j = 1, . . . ,m andk, l = 1, . . . , n. Then AB is normal if and only if so are A and B.

Proof. If A and B are normal, then by Lemma 1 and Proposition 7 we have

[(AB)∗, AB] = [A∗ B∗, AB] = [A∗, A] [B∗, B] = 0,

i.e., AB is also normal. Conversely, suppose that AB is normal. Note that

A∗AB∗B = (AB)∗(AB) = (AB)(AB)∗ = AA∗ BB∗.

By Proposition 6, there exists α ∈ C \ 0 such that AA∗ = αA∗A and BB∗ =α−1B∗B. Since AA∗ and A∗A are positive, we have α > 0. Then

∥A∥2 = ∥AA∗∥ = ∥αA∗A∥ = α∥A∥2,∥B∥2 = ∥BB∗∥ = ∥α−1B∗B∥ = α−1∥B∥2.

We arrive at α = 1, meaning that both A and B are normal.



matrices such that Aij and Bkl are nonzero operators for all i, j = 1, . . . ,m andk, l = 1, . . . , n. Then AB is quasinormal if and only if so are A and B.

Proof. Assume thatA andB are quasinormal. Since [A,A∗A] = 0 and [B,B∗B] =0, we have

[AB, (AB)∗(AB)] = [AB,A∗AB∗B] = 0.

Hence, AB is quasinormal. Suppose that AB is quasinormal. Note that

AA∗ABB∗B = (AB)(AB)∗(AB)

= (AB)∗(AB)2

= A∗A2 B∗B2.

Then there exists α ∈ C\0 such thatA∗A2 = αAA∗A andB∗B2 = α−1BB∗B.This in turn implies that

(A2)∗A2 = A∗(A∗A2) = α(A∗A)2,

(B2)∗B2 = B∗(B∗B2) = α−1(B∗B)2.




Since (A2)∗A2 and α(A∗A)2 are positive, we conclude α > 0. We have

α∥A∥4 = α∥(A∗A)2∥2 = ∥(A2)∗A2∥ = ∥A2∥2 6 ∥A∥4

and, similarly, α−1∥B∥4 6 ∥B∥4. This forces α = 1 and, thus, both A and Bare quasinormal.

Proposition 11. Let A ∈ B(H) and B ∈ B(K). If both A and B satisfy oneof the following properties, then the same property holds for A B: binormal,posinormal.

Proof. The assertion for binormality follows from Lemma 1 and Proposition 7.Now, suppose that AA∗ = A∗PA and BB∗ = B∗QB for some positive operatorsP and Q. By Lemma 1, we get

(AB)(AB)∗ = AA∗ BB∗ = A∗PAB∗QB

= (AB)∗(P Q)(AB).

According to Lemma 1, P Q is positive. Therefore AB is posinormal.

3.2 Hyponormality

Recall the following hyponormal structures of operators; see e.g. [1, 4, 13] and[7, Chapter 2] for more information.

Definition 12. Let p > 0 be a constant. An operator T ∈ B(H) is said to be

• hyponormal if [T ∗, T ] is positive ;

• p-hyponormal if (T ∗T )p > (TT ∗)p ;

• quasihyponormal if T ∗[T ∗, T ]T is positive ;

• p-quasihyponormal if T ∗(T ∗T )pT > T ∗(TT ∗)pT ;

• cohyponormal if T ∗ is hyponormal ;

• log-hyponormal if T is invertible and log(T ∗T ) > log(TT ∗).

Definition 13. Let T ∈ B(H) have the polar decomposition T = U |T | where Uis a unitary operator. The Aluthge transformation of T is defined by

T = |T | 12U |T | 12 .

Then T is said to be

• w-hyponormal if |T | > |T | > |T ∗| ;

• iw-hyponormal if T is invertible and |T | >(|T | 12U |T | 12U∗|T | 12

) 12

.




Theorem 14. Let A ∈ B(H), B ∈ B(K), and let p > 0 be a constant. Ifboth A and B satisfy one of the following properties, then the same propertyholds for A B: hyponormal, p-hyponormal, cohyponormal, quasihyponormal,p-quasihyponormal.

Proof. The assertions for hyponormality and cohyponormality follow from Lemma1 and Proposition 7. The assertion for p-hyponormality is done by applyingLemmas 1 and 3. Now, suppose that A and B are quasihyponormal. By Lemma1, we obtain

(AB)∗ [(AB)∗, AB] (AB)

= (A∗ B∗)((A∗ B∗)(AB)− (AB)(A∗ B∗))(AB)

= (A∗ B∗)(A∗ B∗)(AB)(AB)− (A∗ B∗)(AB)(A∗ B∗)(AB)

= A∗A∗AAB∗B∗BB −A∗AA∗AB∗BB∗B.

Since A∗A∗AA − A∗AA∗A = A∗[A∗, A]A > 0 and B∗B∗BB − B∗BB∗B =B∗[B∗, B]B > 0, we have by Lemma 1 that

A∗A∗AAB∗B∗BB −A∗AA∗AB∗BB∗B > 0.

Hence, (A B)∗ [(AB)∗, AB] (A B) > 0. This means that A B isquasihyponormal.

Assume that A and B are p-quasihyponormal. Lemmas 1 and 3 togetherimply that

(AB)∗ ((AB)∗(AB))p(AB)

= (A∗ B∗) (A∗AB∗B)p(AB)

= A∗(A∗A)pAB∗(B∗B)pB

> A∗(AA∗)pAB∗(BB∗)pB

= (AB)∗ (AA∗ BB∗)p(AB)

= (AB)∗ ((AB)(AB)∗)p(AB).

This show that AB is p-quasihyponormal.

Kim [13] investigated the tensor product of log-hyponormal (reps. w-hyponormal,iw-hyponormal) operators. Now, we consider the case of Tracy-Singh products.

Lemma 15 ([6]). Let S and T be positive invertible operators. Then log T >logS if and only if T p >

(T

p2 SpT

p2

) 12 for all p > 0.

Theorem 16. Let A ∈ B(H) and B ∈ B(K) be positive invertible operators. IfA and B are log-hyponormal, then AB is also log-hyponormal.

Proof. Assume that A and B are log-hyponormal operators. Since A and B areinvertible, Lemma 1 implies that A B is invertible. Using Lemmas 1 and 3,




we obtain that for any p > 0,

[(AB)∗(AB)]p

= (A∗AB∗B)p

= (A∗A)p (B∗B)p

> [(A∗A)p2 (AA∗)p(A∗A)

p2 ]

12 [(B∗B)

p2 (BB∗)p(B∗B)

p2 ]

12

= [(A∗A)p2 (AA∗)p(A∗A)

p2 (B∗B)

p2 (BB∗)p(B∗B)

p2 ]

12

= [(A∗AB∗B)p2 (AA∗ BB∗)

p(A∗AB∗B)

p2 ]

12

= [((AB)∗(AB))p2 ((AB)(AB)∗)

p((AB)∗(AB))

p2 ]

12 .

By Lemma 15, we have log(AB)∗(AB) > log(AB)(AB)∗. This meansthat AB is log-hyponormal.

Lemma 17 ([1]). An operator T ∈ B(H) is w-hyponormal if and only if |T | >(|T | 12 |T ∗||T | 12

) 12

and |T ∗| 6(|T ∗| 12 |T ||T ∗| 12

) 12

.

Theorem 18. Let A ∈ B(H) and B ∈ B(K). If A and B are w-hyponormal,then AB is also w-hyponormal.

Proof. Assume that A and B are w-hyponormal. By applying Lemmas 1 and3, we have

|AB| = |A| |B|

>(|A| 12 |A∗||A| 12

) 12

(|B| 12 |B∗||B| 12

) 12

=(|A| 12 |A∗||A| 12 |B| 12 |B∗||B| 12

) 12

=[(

|A| 12 |B| 12)(|A∗| |B∗|)

(|A| 12 |B| 12

)] 12

=(|AB| 12 |(AB)∗||AB| 12

) 12

.

Similarly, we get

|(AB)∗| = |A∗| |B∗|

6(|A∗| 12 |A||A∗| 12

) 12

(|B∗| 12 |B||B∗| 12

) 12

=(|A∗| 12 |A||A∗| 12 |B∗| 12 |B||B∗| 12

) 12

=[(

|A∗| 12 |B∗| 12)(|A| |B|)

(|A∗| 12 |B∗| 12

)] 12

=(|(AB)∗| 12 |AB||(AB)∗| 12

) 12

.

By Lemma 17, the operator AB is w-hyponormal.




Corollary 19. Let A ∈ B(H) and B ∈ B(K) be invertible operators. If A andB are iw-hyponormal, then AB is also iw-hyponormal.

Proof. It follows from Lemma 1, Proposition 18 and the fact that every iw-hyponormal operator is w-hyponormal and every invertible w-hyponormal op-erator is iw-hyponormal ([13]).

3.3 Paranormality

Consider the following paranormality of operators; see [2, 3, 14, 18].

Definition 20. Let M > 1 be a constant. An operator T ∈ B(H) is said to be

• M -paranormal if M2T ∗2T 2 − 2αT ∗T + α2I > 0 for all α > 0 ;

• paranormal if T ∗2T 2 − 2αT ∗T + α2I > 0 for all α > 0 ;

• M∗-paranormal if M2T ∗2T 2 − 2αTT ∗ + α2I > 0 for all α > 0 ;

• ∗-paranormal if T ∗2T 2 − 2αTT ∗ + α2I > 0 for all α > 0.

Recall that an operator T ∈ B(H) is an isometry if T ∗T = I; it is called aninvolution if T 2 = I.

Proposition 21. Let A ∈ B(H), X ∈ B(K) and let M > 1 be a constant. IfX is an isometry and A is M -paranormal (resp. paranormal), then AX andX A are M -paranormal (resp. paranormal).

Proof. Assume that A is M -paranormal and X is an isometry. It follows thatfor any α > 0 we have

M2(AX)∗2(AX)2 − 2α(AX)∗(AX) + α2(I I)

= M2A∗2A2 X∗2X2 − 2αA∗AX∗X + α2I I

= M2A∗2A2 I − 2αA∗A I + α2I I

=(M2A∗2A2 − 2αA∗A+ α2I

) I

> 0.

Thus AX is M -paranormal. Similarly, the operator X A is M -paranormal.The case of paranormality is just the case ofM -paranormality whenM = 1.

Proposition 22. Let A ∈ B(H), X ∈ B(K) and let M > 1 be a constant. If Xis a self-adjoint involution and A is an M∗-paranormal (resp. ∗-paranormal)operator, then AX and X A are M -paranormal (resp. ∗-paranormal).

Proof. The proof is similar to that of Proposition 21.

Ando [2] showed that for any paranormal operator A, the tensor productsA⊗ I and I ⊗A are paranormal. The next result is an extension of this fact tothe case of Tracy-Singh products.

Corollary 23. Let A ∈ B(H) and let M > 1 be a constant. If A satisfies oneof the following properties, then the same property hold for A I and I A:paranormal, M -paranormal, ∗-paranormal, M∗-paranormal.




4 Tracy-Singh products and operators of typenilpotent, Hermitian, and isometry

In this section, we discuss relationship between Tracy-Singh products and cer-tain classes of operators, namely, nilpotent operators, (skew)-Hermitian oper-ators, (co)isometry operators, and unitary operators. Recall that an operatorT ∈ B(H) is said to be nilpotent if T k = 0 for some natural number k.

Proposition 24. Let A ∈ B(H) and B ∈ B(K). Then AB is nilpotent if andonly if A or B is nilpotent.

Proof. It follows directly from Lemmas 1 and 4.

Recall that an operator T ∈ B(H) is Hermitian if T ∗ = T , and T is skew-Hermitian if T ∗ = −T . It follows from Lemma 1 that the Tracy-Singh productof Hermitian operators is also Hermitian. The Tracy-Singh product of twoskew-Hermitian operators is Hermitian. The Tracy-Singh product between aHermitian operator and a skew-Hermitian operator is skew-Hermitian.

Proposition 25. Let A ∈ B(H) and B ∈ B(K) be nonzero operators.

1. Assume AB is Hermitian. Then A is Hermitian (resp. skew-Hermitian)if and only if B is Hermitian (resp. skew-Hermitian).

2. Assume A B is skew-Hermitian. Then A is Hermitian (resp. skew-Hermitian) if and only if B is skew-Hermitian (resp. Hermitian).

Proof. It follows directly from Lemmas 1 and 4.

Recall that an operator T ∈ B(H) is a coisometry if TT ∗ = I. A unitaryoperator is an operator which is both an isometry and a coisometry. Stochel [21]gave a necessary and sufficient condition for A ⊗ B to be an isometry (resp. acoisometry, unitary). Now, we will extend this result to the case of Tracy-Singhproducts.

Proposition 26. Let A = [Aij ]m,mi,j=1 ∈ B(H) and B = [Bkl]

n,nk,l=1 ∈ B(K) be

operator matrices such that Aij and Bkl are nonzero operators for all i, j =1, . . . ,m and k, l = 1, . . . , n. Then AB is an isometry (resp. a coisometry) ifand only if so are αA and α−1B for some α ∈ C \ 0.

Proof. If αA and α−1B are isometries, then by Lemma 1,

(AB)∗(AB) = A∗AB∗B

= (αA)∗(αA) (α−1B)∗(α−1B)

= I I.

Suppose that A B is an isometry. Then A∗A B∗B = I I. Thus, byProposition 6, there exists β ∈ C \ 0 such that βA∗A = I and β−1B∗B = I.Setting α =

√β, we obtain (αA)∗(αA) = I and (α−1B)∗(α−1B) = I. Hence

αA and α−1B are isometries. The proof for the case of coisometry is similar tothat of isometry.






matrices such that Aij and Bkl are nonzero operators for all i, j = 1, . . . ,m andk, l = 1, . . . , n. Then A B is unitary if and only if so are αA and α−1B forsome α ∈ C \ 0.

Proof. If αA and α−1B are unitary, then Lemma 1 implies

(AB)∗(AB) = A∗AB∗B = (αA)∗(αA) (α−1B)∗(α−1B) = I.

Similarly, we have (A B)(A B)∗ = I. Conversely, suppose that A Bis unitary. We know that A B is both an isometry and a coisometry. ByProposition 26, there exist α, β ∈ C\0 such that αA and α−1B are isometries,and βA and β−1B are coisometries. We have (αA)∗(αA) = I = (βA)(βA)∗ and

(α−1B)∗(α−1B) = I = (β−1B)(β−1B)∗.

Since AB is normal, so are A and B (Theorem 9). Then α2AA∗ = α2A∗A =β2AA∗ and α−2BB∗ = α−2B∗B = β−2BB∗. Since α, β > 0, it comes to theconclusion that α = β. Hence αA and α−1B are unitary.

5 Tracy-Singh products and class-A type oper-ators

The following classes of operators bring attention to operator theorists; see moreinformation in [8, 9, 11, 12, 20].

Definition 28. Let k ∈ N. An operator T ∈ B(H) is said to be

• class A if |T 2| > |T |2 ;

• class A(k) if(T ∗|T |2kT

) 1k+1 > |T |2 ;

• quasi-class A if T ∗|T 2|T > T ∗|T |2T ;

• quasi-class (A, k) if T ∗k|T 2|T k > T ∗k|T |2T k ;

• ∗-class A if |T 2| > |T ∗|2 ;

• quasi-∗-class A if T ∗|T 2|T > T ∗|T ∗|2T ;

• quasi-∗-class (A, k) if T ∗k|T 2|T k > T ∗k|T ∗|2T k.

The next theorem shows that such classes of operators are preserved underTracy-Singh products.

Theorem 29. Let A ∈ B(H), B ∈ B(K), and let k ∈ N. If both A and Bsatisfy one of the following properties, then the same property holds for AB:class A(k), class A, quasi-class (A, k), quasi-class A, ∗-class A, quasi-∗-classA, quasi-∗-class (A, k).




Proof. Assume that A and B are class A(k). By Lemmas 1 and 3, we get

[(AB)∗|AB|2k(AB)]1

k+1 =[(A∗ B∗)

(|A|2k |B|2k

)(AB)

] 1k+1

=(A∗|A|2kAB∗|B|2kB

) 1k+1

=(A∗|A|2kA

) 1k+1

(B∗|B|2kB

) 1k+1

> |A|2 |B|2

= |AB|2.

Hence AB is a class A(k) operator. Now, assume that A and B are quasi-classA(k). Applying Lemmas 1 and 3, we get

(AB)k∗|(AB)2|(AB)k = (Ak∗ Bk∗)(|A2| |B2|)(Ak Bk)

= Ak∗|A2|Ak Bk∗|B2|Bk

> Ak∗|A|2Ak Bk∗|B|2Bk

= (Ak∗ Bk∗)(|A|2 |B|2)(Ak Bk)

= (AB)k∗|AB|2(AB)k.

Hence, AB is a quasi-class A(k) operator. The proof for class A (resp. quasi-class A) is done by replacing k = 1 in the case of class A(k) (resp. quasi-class(A, k)). The proof for the case of quasi ∗-class (A, k) is similar to that of quasi-class (A, k). Similarly, the proof for ∗-class A (resp. quasi-∗-class A) is done byreplacing k = 0 (resp. k = 1) in the case of quasi-∗-class (A, k).

Acknowledgement. This research was supported by Thailand Research Fundgrant no. MRG6080102.

References

[1] A. Aluthge, D. Wang, w-hyponormal operators II. Integral Equations Op-erator Theory, 37, 324-331 (2000).

[2] T. Ando, Operators with a norm condition. Acta Sci. Math. (Szeged), 33,169-178 (1972).

[3] S. C. Arora, J. K. Thukral, On a class of operators. Glasnik Math., 41,381-386 (1986).

[4] A. Bucar, Posinormality versus hyponormality for Cesoro operators. Gen.Math., 11, 33-46 (2003).

[5] B. P. Duggal, Tensor products of operators-strong stability and p-hyponormality. Glasg. Math. J., 42, 371-381 (2000).

[6] M. Fujii, J. F. Jiang, E. Kamei, K. Tanahashi, A Characterization ofChaotic Order and a Problem. J. lnequal. Appl., 2, 149-156 (1998).




[7] T. Furuta, Invitation to Linear Operators: From Matrices to Bounded Lin-ear Operators on a Hilbert Space, Taylor & Francis, New York, 2001.

[8] T. Furuta, M. Ito, T. Yamazaki, A subclass of paranormal operators in-cluding class of log-hyponormal and several related classes. Sci. Math., 1,389-403 (1998).

[9] A. Gupta, N. Bhatia, On (n, k)-quasiparanormal weighted composition op-erators. Int. J. Pure Appl. Math., 91, 23-32 (2014).

[10] I. H. Jeon, B. P. Duggal, On operators with an absolute value condition.J. Korean Math. Soc., 41, 617-627 (2004).

[11] I. H. Jeon, I. H. Kim, On operators satisfying T ∗|T 2|T > T ∗|T |2T , LinearAlgebra Appl., 418, 854-862 (2006).

[12] G. H. Kim, J. H. Jeon, A study on generalized quasi-class A operators.Korean J. Math., 17, 155-159 (2009).

[13] I. H. Kim, Tensor products of log-hyponormal operators. Bull. KoreanMath. Soc., 42, 269-277 (2005).

[14] M. M. Kutkut, B. Kashkari, On the class of class M -paranormal (M∗-paranormal) operators. M. Sci. Bull. (Nat. Sci), 20, 129-144 (1993).

[15] A. Ploymukda, P. Chansangaim, W. Lewkeeratiyutkyl, Algebraic and orderproperties of Tracy-Singh product for operator matrices. J. Comput. Anal.Appl., 24, 656-664 (2018).

[16] A. Ploymukda, P. Chansangaim, W. Lewkeeratiyutkyl, Analytic propertiesof Tracy-Singh product for operator matrices. J. Comput. Anal. Appl., 24,665-674 (2018).

[17] H. C. Rhaly, B. E. Rhoades, Posinormal factorable matrices with a constantmain diagonal. Rev. Un. Mat. Argentina, 55, 19-24 (2014).

[18] D. Senthilkumar, T. Prasad,M class Q composition operators. Sci. Magna,6, 25-30 (2010).

[19] A. Sekar, C. V. Seshaiah, D. Senthil Kumar, P. Maheswari Naik, Isolatedpoints of spectrum for quasi-∗-class A operators. Appl. Math. Sci., 6, 6777-6786 (2012).

[20] J. L. Shen, F. Zuo, C. S. Yang, On operators satisfying T ∗|T 2|T >T ∗|T ∗|2T . Acta Math. Sinica (Eng. Ser.), 26, 2109-2116 (2010).

[21] J. Stochel, Seminormality of operators from their tensor product. Proc.Amer. Math. Soc., 124, 135-140 (1996).

[22] J. Zanni, C. S. Kubrsly, A note on compactness of tensor products. ActaMath. Univ. Comenian. (N.S.), 84, 59-62 (2015).



On unicity theorems of difference of entire or meromorphic

functions

YONG LIU

ABSTRACT. In this article, we investigate the uniqueness problems of differences ofmeromorphic functions and obtain some results which can be viewed as discrete analoguesof the results given by Yi. An example is given to show the results in this paper are bestpossible.

1 INTRODUCTION

In this paper, we assume that the reader is familiar with the fundamental results and thestandard notations of the Nevanlinna theory (see, e.g., [7, 18]). Let f(z) and g(z) betwo non-constant meromorphic functions in the complex plane. By S(r, f), we denote anyquantity satisfying S(r, f) = o(T (r, f)) as r → ∞, possibly outside a set of r with finitelinear measure. Then the meromorphic function α is called a small function of f(z), ifT (r, α) = S(r, f). If f(z)−α and g(z)−α have same zeros, counting multiplicity (ingoringmultiplicity), then we say f(z) and g(z) share the small function α CM (IM). For a smallfunction α related to f(z), we define

δ(α, f) = lim infr→∞

m(r, 1

f−α

)T (r, f)

.

In 1976, Yang [17] proposed the following problem:

Suppose that f(z) and g(z) are two entire functions such that f(z) and g(z) share 0 CMand f ′(z) and g′(z) share 1 CM . What can be said about the relationship between f(z)and g(z)?

2010 Mathematics Subject Classification. Primary 30D35, 39B12.∗The work was supported by the NNSF of China (No.10771121, 11301220, 11401387, 11661052), the NSF of

Zhejiang Province, China (No. LQ 14A010007), the NSF of Shandong Province, China (No. ZR2012AQ020)

and the Fund of Doctoral Program Research of Shaoxing College of Art and Science(20135018).

Key words:meromorphic functions, difference equations, uniqueness, finite order.

1


1414 YONG LIU 1414-1427

In [13], Yi answered the question posed by C. C. Y. These results may be stated as follows:

Theorem A. Let f(z) and g(z) be two nonconstant entire functions. Assume that f(z)and g(z) share 0 CM , f ′(z) and g′(z) share 1 CM and δ(0, f) > 1

2 . Then f ′(z)g′(z) ≡ 1unless f(z) ≡ g(z).

Currently, there has been an increasing interest in studying difference equations in thecomplex plane. For example, Halburd and Korhonen [3, 4] established a version of Nevan-linna theory based on difference operators. Ishizaki and Yanagihara [7] deveploped a versionof Wiman-Valiron theory for difference equations of entire functions of small growth. AlsoChiang and Feng [1] has a difference version of Wiman-Valiron.

The main purpose of this paper is to establish partial difference counterparts of Theo-rem A. Our results can be stated as follows:

Theorem 1.1. Let cj , aj , bj(j = 1, 2, · · · , k) be complex constants, and let f(z) andg(z) be two nonconstant entire functions of finite order. Assume that f(z) and g(z) share0 CM , L(f) =

∑ki=1 aif(z + ci) and L(g) =

∑ki=1 big(z + ci) share 1 CM and δ(0, f) > 2

3 .Then L(f)L(g) ≡ 1 or L(f) ≡ L(g).

Theorem 1.2. Let c ∈ C \ 0, and let f(z) and g(z) be two nonconstant meromor-phic functions of finite order satisfying f(z + c) and g(z + c) share 1 CM, f(z) and g(z)share ∞ CM . If

N(r,

1

f

)+N

(r,1

g

)+ 2N(r, f) < (λ+ o(1))T (r), (1.1)

where λ < 1 and T (r) = maxT (r, f), T (r, g), then f(z)g(z) ≡ 1 or f(z) ≡ g(z).

The following example shows that Theorem 1.2 is exact.

Example 1.1. Let f(z) = e2z + ez, g(z) = e−2z − e−z. We have that f(z + c) and g(z + c)share 1 CM, f(z) and g(z) share ∞ CM and

N(r,

1

f

)+N

(r,1

g

)+ 2N(r, f) = (λ+ o(1))T (r),

but f(z) ≡ g(z) and f(z)g(z) ≡ 1.

2 Proof of Theorem 1.1

In order to prove Theorem 1.1, we need the following lemmas.The following lemma is a difference analogue of the logarithmic derivative lemma.Lemma 2.1 [3] Let f(z) be a meromorphic function of finite order and let c be a non-zero

2


1415 YONG LIU 1414-1427

complex number. Then we have

m(r,f(z + c)

f(z)

)= S(r, f).

Lemma 2.2 Let f(z) be a nonconstant entire function of finite order, and let ci, ai(i =1, 2, · · · , k) be complex constants. Then

T(r,

k∑i=1

aif(z + ci))≤ T (r, f(z)) + S(r, f),

N(r,

1∑ki=1 aif(z + ci)

)≤ N

(r,

1

f(z)

)+ S(r, f).

Proof of Lemma 2.2. By Lemma 2.1, we have

T(r,

k∑i=1

aif(z + ci))= m

(r,

k∑i=1

aif(z + ci))= m

(r, f(z)

∑ki=1 aif(z + ci)

f(z)

)≤

k∑i=1

m(r,f(z + ci)

f(z)

)+m(r, f(z)) +O(1) = T (r, f(z)) + S(r, f). (2.1)

m(r,

1

f(z)

)= m

(r,


∑ki=1 aif(z + ci)

f(z)

)≤ m

(r,


)+S(r, f).

(2.2)From the first main theory and (2.2), we obtain

T (r, f(z))−N(r,

1

f(z)

)≤ T

(r,

k∑i=1

aif(z+ ci))−N

(r,


)+S(r, f). (2.3)

By (2.1) and (2.3), we deduce

N(r,


)≤ N

(r,

1

f(z)

)+ S(r, f). (2.4)

Lemma 2.3 Assume that the conditions of Theorem 1.1 are satisfied. Then

T (r, f(z)) = O(T (r,

k∑i=1

aif(z + ci)))

for r ∈ E,

T (r, g(z)) = O(T (r,

k∑i=1

aif(z + ci)))

for r ∈ E,

T(r,

k∑i=1

big(z + ci))= O

(T (r,

k∑i=1

aif(z + ci)))

for r ∈ E,

3


1416 YONG LIU 1414-1427

where E is a set of finite linear measure.Proof of Lemma 2.3. By the first main theory and (2.2), we have

(δ(0, f) + o(1))T (r, f(z)) ≤ m(r,

1

f(z)

)+ S(r, f)

≤ m(r,


)+ S(r, f) ≤ T

(r,

k∑i=1

aif(z + ci))+ S(r, f).

And so

T (r, f(z)) ≤( 1

δ(0, f)+ o(1)

)T(r,

k∑i=1

aif(z + ci))+ S(r, f). (2.5)

Hence, we have

T (r, f(z)) = O(T (r,

k∑i=1

aif(z + ci))

r ∈ E.

By the second main theorem, the first main theory, Lemma 2.2 and (2.5), we have

T(r,

k∑i=1

big(z + ci))

< N(r,

1∑ki=1 big(z + ci)

)+N

(r,

1∑ki=1 big(z + ci)− 1

)+ S

(r,

k∑i=1

big(z + ci))

≤ N(r,

1

g(z)

)+N

(r,

1∑ki=1 big(z + ci)− 1

)+ S(r, g)

= N(r,

1

f(z)

)+N

(r,

1∑ki=1 aif(z + ci)− 1

)+ S(r, g)

≤ (1− δ(0, f) + o(1))T (r, f(z)) + T(r,

k∑i=1

aif(z + ci))+ S(r, g)

≤( 1

δ(0, f)+ o(1)

)T(r,

k∑i=1

aif(z + ci))+ S(r, f) + S(r, g). (2.6)

Using the method similar to the proof of (2.3), we have

T (r, g(z))−N(r,

1

g(z)

)≤ T

(r,

k∑i=1

big(z + ci))+ S(r, g). (2.7)

4


1417 YONG LIU 1414-1427

From (2.5)-(2.7), we obtain

T (r, g(z)) ≤ N(r,

1

g(z)

)+ T

(r,

k∑i=1

big(z + ci))+ S(r, g)

≤ N(r,

1

f(z)

)+( 1

δ(0, f)+ o(1)

)T(r,

k∑i=1


≤ (1− δ(0, f) + o(1))T (r, f(z)) +( 1

δ(0, f)+ o(1)

)T(r,

k∑i=1


≤( 2

δ(0, f)− 1 + o(1)

)T(r,

k∑i=1

aif(z + ci))+ S(r, g) + S(r, f),

that is

T (r, g(z)) = O(T (r,

k∑i=1

aif(z + ci)))

r ∈ E. (2.8)

From Lemma 2.2 and (2.8), we get

T(r,

k∑i=1

big(z + ci))= O

(T (r,

k∑i=1

aif(z + ci)))

r ∈ E.

Lemma 2.3 thus is be proved.Lemma 2.4 [10] Let f1, f2 and f3 be three entire functions satisfying

3∑i=1

fi ≡ 1.

If f1 ≡ constant, and

3∑i=1

N(r,

1

fi

)≤ (λ+ o(1))T (r) (r ∈ E),

where T (r) = maxT (r, fi)|i = 1, 2, 3), and λ < 1, then f2 ≡ 1 or f3 ≡ 1.

Proof of Theorem 1.1.Since L(f) =

∑ki=1 aif(z + ci) and L(g) =

∑ki=1 big(z + ci) share 1 CM, we have

L(f)− 1

L(g)− 1= ep(z), (2.9)

where p(z) is polynomial.Let f1 = L(f), f2 = ep(z), f3 = −ep(z)L(g), by (2.9) and Lemma 2.2, we have

f1 + f2 + f3 ≡ 1,

5


1418 YONG LIU 1414-1427

N(r,

1

f1

)≤ N

(r,

1

f

)+ S(r, f), (2.10)

N(r, f2) = N(r,

1

ep(z)

)= 0, (2.11)

N(r, f3) = N(r,

1

−epL(g)

)= N

(r,

1

L(g)

)≤ N

(r,1

g

)+S(r, g) = N

(r,

1

f

)+S(r, g). (2.12)

By Lemma 2.3, (2.5), (2.10)-(2.12) and δ(0, f) > 23 , we have

3∑i=1

N(r,

1

fi

)≤ 2N

(r,

1

f

)+ S(r, f) + S(r, g)

≤ 2(1− δ(0, f) + o(1))T (r, f) + S(r, f) + S(r, g)

≤ 2( 1

δ(0, f)− 1 + o(1)

)T (r, L(f)) + S(r, L(f))

= (λ+ o(1))T (r, L(f)) r ∈ E,

where λ = 2(

1δ(0,f) − 1

)< 1. From Lemma 2.4, we have

f2 ≡ 1 or f3 ≡ 1.

If f2 ≡ 1, by (2.9), we obtain L(f)−1L(g)−1 = ep(z) ≡ 1. Hence, we have L(f) ≡ L(g).

If f3 ≡ 1, that is −ep(z)L(g) ≡ 1. So L(g) = −e−p(z). By L(f) + 1 + ep(z) = 1, we haveL(f) = −ep(z). So L(f)L(g) ≡ 1. Theorem 1.1 thus is proved.

3 Proof of Theorem 1.2

In order to prove Theorem 1.2, we need the following lemmas.Lemma 3.1 [14] Let f1 and f2 be two nonconstant meromorphic functions, and let c1, c2and c3 be three nonzero constants. If c1f1 + c2f2 = c3, then

T (r, f1) < N(r,

1

f1

)+N

(r,

1

f2

)+N(r, f1) + S(r, f1).

Lemma 3.2 [12] Let f1, f2, · · · , fn be linearly independent meromorphic functions satisfying∑ni=1 fi ≡ 1. Then for j = 1, 2, · · · , n, we have

T (r, fj) <n∑

i=1

N(r,

1

fi

)+N(r, fj) +N(r,D)−

n∑i=1

N(r, fi)−N(r,1

D)

+O(log r + log Tn(r)) for r ∈ E,

where D denotes the Wronskian

D =

∣∣∣∣∣∣∣∣f1, f2, · · · , fnf ′1, f ′2, · · · , f ′n

· · · , · · · , · · · , · · ·f(n−1)1 , f

(n−1)2 , · · · , f

(n−1)n

∣∣∣∣∣∣∣∣6


1419 YONG LIU 1414-1427

and Tn(r) denotes the maximum of T (r, fi), i = 1, 2, · · · , n.

Lemma 3.3 [14] Let f1, f2 and f3 be three nonconstant meromorphic functions satisfying∑3i=1 fi ≡ 1, and let g1 = −f3

f2, g2 =

1f2, g3 = −f1

f2. If f1, f2 and f3 are linearly independent,

then g1, g2 and g3 are linearly independent.

Lemma 3.4 [8] Let f(z) be a meromorphic function of finite order, c = 0 be fixed. Then

N(r, f(z + c)) ≤ N(r, f(z)) + S(r, f),

N(r,

1

f(z + c)

)≤ N

(r,

1

f(z)

)+ S(r, f).

Remark. 1 Using the same method of Lemma 3.4, we obtain

N(r,

1

f(z + c)

)≥ N

(r,

1

f(z)

)+ S(r, f).

From this and Lemma 3.4, we deduce

N(r,

1

f(z + c)

)= N

(r,

1

f(z)

)+ S(r, f).

Lemma 3.5 [5, 6] Let f(z) be a nonconstant finite order meromorphic function and letc = 0 be an arbitrary complex number. Then

T (r, f(z + |c|)) = T (r, f(z)) + S(r, f).

Remark. 2 It is shown in [2, p. 66], that for c ∈ C \ 0, we have

(1 + o(1))T (r − |c|, f(z)) ≤ T (r, f(z + c)) ≤ (1 + o(1))T (r + |c|, f(z))

hold as r → ∞, for a general meromorphic. By this and Lemma 3.5, we obtain

T (r, f(z + c)) = T (r, f(z)) + S(r, f)

Lemma 3.6 [7] Let P (z) be a polynomial of degree m, and let F (z) = R(z)Q(z) , where Q(z) is

a polynomial of degree n ≥ 1 and R(z) is a polynomial of degree less than n which has nocommon factor with Q(z). Then

T (r, P ) = N(r,1

P) +O(1) = m log r +O(1),

T (r, F ) = n log r +O(1),

N(r,1

F) ≤ (n− 1) log r +O(1),

T (r, F + P ) = N(r,1

F + P) +O(1) = (n+m) log r +O(1).

Lemma 3.7 Let f(z) be an nonconstant finite order meromorphic function, and let g(z) bea rational function such that f(z+c) and g(z+c) share 1 CM, f(z) and g(z) share ∞ CM .If

N(r) < (λ+ o(1))T (r), (3.1)

7


1420 YONG LIU 1414-1427

where N(r) = maxN(r, 1f ), N(r, 1g ) , and λ(< 1) is a positive constant, then f(z) ≡ g(z).Proof of Lemma 3.7. If f(z) is a transcendental function, then

T (r, g) = S(r, f).

By this, f(z + c) and g(z + c) share 1 CM, f(z) and g(z) share ∞ CM , we obtain

N(r,

1

f(z + c)− 1

)= N

(r,

1

g(z + c)− 1

)≤ T (r, g(z + c)) +O(1) = S(r, f),

N(r, f) = N(r, g) ≤ T (r, g) = S(r, f).

By the second main theorem and Remark 1, we deduce that

T (r, f(z)) < N(r,

1

f(z)

)+N(r, f(z)) +N

(r,

1

f(z)− 1

)+ S(r, f)

= N(r,

1

f(z)

)+N

(r,

1

f(z + c)− 1

)+ S(r, f)

= N(r,

1

f(z)

)+ S(r, f). (3.2)

(3.1) and (3.2) imply thatT (r) < (λ+ o(1))T (r),

a contradiction.If f(z) is a polynomial, by f(z) and g(z) share ∞ CM , we see that g(z) is a polynomial.By Lemma 3.6, we obtain that

T (r, f) = N(r,

1

f

)+O(1),

T (r, g) = N(r,1

g

)+O(1),

and hence T (r) = N(r) +O(1). From this and (3.1), we get

T (r) < (λ+ o(1))T (r),

a contradiction.If f(z) is a rational function which is not a polynomial, then f(z + c) is not a constant.Since f(z + c) and g(z + c) share 1 CM, f(z) and g(z) share ∞ CM, we deduce

f(z + c)− 1

g(z + c)− 1≡ α,

where α is a constant. And so

f(z + c)− αg(z + c) ≡ 1− α. (3.3)

Let g(z) = m(z)n(z) + P1(z), where P1(z) is a polynomial, n(z) is a polynomial of degree n ≥ 1

and m(z) is a polynomial of degree less than n which has no common factor with n(z). By(3.3), we have

f(z) =αm(z)

n(z)+ P2(z),

8


1421 YONG LIU 1414-1427

where P2(z) = 1−α+αP1(z). By Lemma 3.6 and (3.1), we have P1(z) ≡ 0 and P2(z) ≡ 0.Hence α = 1. So f(z) ≡ g(z). Lemma 3.7 is thus be proved.

Proof of Theorem 1.2.By Lemma 3.7, it may be assumed that f(z) and g(z) are two transcendental meromorphicfunctions. By f(z + c) and g(z + c) share 1 CM, f(z) and g(z) share ∞ CM , we obtain

f(z + c)− 1 = eh(z)(g(z + c)− 1),

where h(z) is a polynomial, and so

f(z + c) + eh(z) − eh(z)g(z + c) = 1. (3.4)

Let f1 = f(z + c), f2 = eh(z), f3 = −eh(z)g(z + c) and let T1(r) = maxT (r, fi)|i = 1, 2, 3.By (3.4), we have

3∑i=1

fi ≡ 1, (3.5)

3∑i=1

N(r,

1

fi

)= N

(r,

1

f(z + c)

)+N

(r,

1

g(z + c)

), (3.6)

T1(r) = O(T (r)). (3.7)

We divide the proof in two parts:Case 1. Suppose that f1, f2 and f3 are linearly independent. From Lemma 3.2 and (3.7),we deduce

T (r, f(z + c)) <3∑

i=1

N(r,

1

fi

)+N(r,D)−N(r, f2)−N(r, f3) + o(T (r)), (3.8)

for r ∈ E, where

D =

∣∣∣∣∣∣f1, f2, f3f ′1, f ′2, f ′3f ′′1 , f ′′2 , f ′′3

∣∣∣∣∣∣By the above and (3.5), we obtain

D =

∣∣∣∣ f ′2, f ′3f ′′2 , f ′′3

∣∣∣∣Now combining this and Lemma 3.4, we deduce

N(r,D)−N(r, f2)−N(r, f3) ≤ N(r, g′′(z + c))−N(r, g(z + c))

= 2N(r, g(z + c)) ≤ 2N(r, g(z)) + S(r, g) = 2N(r, f(z)) + S(r, g). (3.9)

By Remark 1, we have

N(r,

1

f(z + c)

)= N

(r,

1

f(z)

)+ S(r, f), N

(r,

1

g(z + c)

)= N

(r,

1

g(z)

)+ S(r, g). (3.10)

9


1422 YONG LIU 1414-1427

By (3.6), (3.10), we have

3∑i=1

N(r,

1

fi

)= N

(r,

1

f(z + c)

)+N

(r,

1

g(z + c)

)= N

(r,

1

f(z)

)+N

(r,

1

g(z)

)+ o(T (r)),

(3.11)for r ∈ E. By (3.8), (3.9), (3.11) and Remark 2, we have

T (r, f(z)) = T (r, f(z + c)) + S(r, f) < N(r,

1

f(z)

)+N

(r,

1

g(z)

)+ 2N(r, f(z)) +O(T (r)),

(3.12)for r ∈ E.Let g1 = −f3

f2= g(z + c), g2 =

1f2

= e−h(z), g3 =−f1f2

= −e−h(z)f(z + c). By (3.4), we have

3∑i=1

gi ≡ 1.

From Lemma 3.3, we see that g1, g2 and g3 are linearly independent. Using the similarmethod as above, we obtain that

T (r, g(z)) < N(r,

1

f(z)

)+N

(r,

1

g(z)

)+ 2N(r, f(z)) + o(T (r)), (3.13)

for r ∈ E. From (3.12) and (3.13), we know that

T (r) < N(r,

1

f

)+N

(r,1

g

)+ 2N(r, f) + o(T (r)),

for r ∈ E. From (1.1) and (3.13), we have

T (r) < (λ+ o(1))T (r), (3.14)

(3.14) is impossible.Case 2. Suppose that f1, f2, f3 are linearly dependent. Then there exist three constants(c1, c2, c3) = (0, 0, 0) satisfying

3∑i=1

cifi = 0. (3.15)

If c1 = 0, (3.15) yields c2 = 0, c3 = 0 and f3 = − c2c3f2. So g(z + c) = c2

c3, a contradiction.

Hence we have, c1 = 0 and

f1 = −c2c1f2 −

c3c1f3. (3.16)

(3.5) and (3.16) imply that (1− c2

c1

)f2 +

(1− c3

c1

)f3 = 1. (3.17)

Next we deal with the following three subcases.Subcase 2.1. Assume that c1 = c2. Then by (3.17), we have c1 = c3 and

f3 =c1

c1 − c3.

10


1423 YONG LIU 1414-1427

And sog(z + c) = − c1

c1 − c3e−h(z). (3.18)

Hence g(z) is an entire function, and N(r, 1

g(z+c)

)= 0. By (3.4) and (3.18), we obtain that

f(z + c)− c1c1 − c3

1

g(z + c)= − c3

c1 − c3. (3.19)

If c3 = 0, by (3.18), (3.19), Lemma 3.1 and the first main theory, we deduce that

T (r, f(z + c)) < N(r,

1

f(z + c)

)+ S(r, f), (3.20)

T (r, g(z + c)) < N(r,

1

f(z + c)

)+ S(r, f). (3.21)

From (3.20), (3.21), Remark 1 and Remark 2, we have

T (r, f(z)) = T (r, f(z + c)) + S(r, f) < N(r,

1

f(z + c)

)+ S(r, f) = N

(r,

1

f(z)

)+ S(r, f),

and

T (r, g(z)) = T (r, g(z + c)) + S(r, f) < N(r,

1

f(z + c)

)+ S(r, f) = N

(r,

1

f(z)

)+ S(r, f).

By the above, we deduce that

T (r) < N(r,

1

f

)+ o(T (r)) for r ∈ E,

which contradicts our assumption (1.1). Thus c3 = 0. Hence from (3.19), we have f(z +c)g(z + c) ≡ 1, so, f(z)g(z) ≡ 1.Subcase 2.2. Assume that c1 = c3. From (3.17), we have c1 = c2 and f2 =

c1c1−c2

.And so

eh(z) =c1

c1 − c2. (3.22)

By (3.4) and (3.22), we obtain that

f(z + c)− c1c1 − c2

g(z + c) = − c2c1 − c2

. (3.23)

If c2 = 0, from (3.23), Lemma 3.1, Remark 1 and Remark 2, we obtain that

T (r, f(z)) = T (r, f(z + c)) + S(r, f)

< N(r,

1

f(z + c)

)+N

(r,

1

g(z + c)

)+N(r, f(z)) + S(r, f)

= N(r,

1

f(z)

)+N

(r,

1

g(z)

)+N(r, f(z)) + S(r, f). (3.24)

11


1424 YONG LIU 1414-1427

Using the similar method as above, we can get

T (r, g) < N(r,

1

f

)+N

(r,1

g

)+N(r, f) + S(r, f). (3.25)

By (3.24) and (3.25), we obtain that

T (r) < N(r,

1

f

)+N

(r,1

g

)+N(r, f) + o(T (r)) for r ∈ E,

a contradiction. Thus c2 = 0, and by (3.23), we obtain that f(z + c) = g(z + c). Hencef(z) = g(z).Case 2.3. Assume that c1 = c2 and c1 = c3. From (3.17), we obtain

(c1 − c3)g(z + c) + c1e−h(z) = c1 − c2. (3.26)

By (3.26), Lemma 3.1, Remark 1 and Remark 2, we obtain that

T (r, g(z)) = T (r, g(z + c)) + S(r, g) < N(r,

1

g(z + c)

)+ S(r, g) = N

(r,

1

g(z)

)+ S(r, g).

(3.27)By (3.4) and (3.26), we have

(c3 − c1)f(z + c) + (c3 − c2)eh(z) = c3. (3.28)

If c3 = 0, by (3.28), Lemma 3.1, Remark 1 and Remark 2, we obtain that

T (r, f(z)) = T (r, f(z + c)) + S(r, f) < N(r,

1

f(z + c)

)+ S(r, f) = N

(r,

1

f(z)

)+ S(r, f).

(3.29)By (3.27) and (3.29), we see

T (r) < N(r,1

f) +N

(r,1

g

)+ o(T (r)) for r ∈ E,

which contradicts the assumption (1.1).If c3 = 0, by (3.28), we obtain

f(z + c) = −c2c1eh(z). (3.30)

Hence f(z) is an entire function, and N(r, 1f(z+c)) = 0. From (3.26) and (3.30), we have

c21

f(z + c)− c1g(z + c) = c2 − c1. (3.31)

By (3.31), Lemma 3.1, Remark 1 and Remark 2, we have

T (r, f(z)) = T (r, f(z + c)) + S(r, f) < N(r,

1

g(z + c)

)+ S(r, f) = N

(r,

1

g(z)

)+ S(r, f).

(3.32)By (3.27) and (3.32), we have

T (r) < N(r,1

g

)+ o(T (r)) for r ∈ E,

which contradicts the assumption (1.1).

12


1425 YONG LIU 1414-1427

Acknowledgements

The author would like to thank the referee for his/her helpful suggestions and comments.

References

[1] Y. M. Chiang, S. J. Feng, On the growth of logarithmic differences, difference quotientsand logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. 361(2009), 3767-3791.

[2] A. A. Goldberg, I. V. Ostrovskii, Distribution of Values of Meromorphic Functions.Nauka, Mosow, 1970.

[3] R. G. Halburd, R. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad.Sci. Fenn. Math. 94 (2006), 463-478.

[4] R. G. Halburd, R. Korhonen, Difference analogue of the lemma on the logarithmicderivative with applications to difference equatons, J. Math. Anal. Appl. 314 (2006),477-487.

[5] R. G. Halburd, R. Korhonen, Finite order solutions and the discrete Painleve equations,Proc. London Math. Soc. 94 (2007), 443-474.

[6] R. G. Halburd, R. Korhonen, Meromorphic solutions of difference equations, integra-bility and the discrete Painleve equations. J- Phys. A. 40 (2007), 1-38.

[7] W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

[8] K. Ishizaki, N. Yanagihara, Wiman-Valiron method for difference equations, NagoyaMath. J. 175 (2004), 75-102.

[9] X. D. Luo, W. C. Lin, Value sharing results for shifts of meromorphic functions, J.Math. Anal. Appl. 377 (2011), 441-449.

[10] J. T. Li, P. Li, Uniqueness of entire functions concerning differential polynomials.Commun. Korean Math. Soc. 30 (2015), no. 2, 93-101.

[11] L. Indrajit, Linear differential polynomials sharing the same 1-points with weight two.Ann. Polon. Math. 79 (2002), no. 2, 157-170.

[12] R. Nevanlinna, Le Thorems de Picard-Borel et la Theorie des Fonctions Meromorphes,GauthierVillars, Paris, 1929.

[13] H. X. Yi, A question of C. C. Yang on the uniqueness of entire functions, Kodai Math.J., 13 (1990), 39-46.

[14] H. X. Yi, Meromorphic functions that share two or three values, Kodai Math. J. 13(1990), 363-372.

[15] H. X. Yi, Unicity theorems for entire or meromorphic functions. Acta Math. Sinica(N.S.), 10 (1994), 121-131.

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1426 YONG LIU 1414-1427

[16] H. X. Yi, C. C. Yang, A uniqueness theorem for meromorphic functions whose nthderivatives share the same 1-points. J. Anal. Math. 62 (1994), 261-270.

[17] C. C. Yang, On two entire functions which together with their first derivative have thesame zeros, J. Math. Anal. Appl. 56 (1) (1976), 1-6.

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YONG LIUDepartment of Mathematics, Shaoxing College of Arts and Sciences, Shaoxing, Zhejiang312000, ChinaE-mail address: [email protected]

14


1427 YONG LIU 1414-1427

On the existence and behavior of the solutions for somedifference equations systems

M. M. El-Dessoky1,2 and Aatef Hobiny11King Abdulaziz University, Faculty of Science,Mathematics Department,

P. O. Box 80203, Jeddah 21589, Saudi Arabia.2Department of Mathematics, Faculty of Science,Mansoura University, Mansoura 35516, Egypt.

E-mail: [email protected]; [email protected]

ABSTRACT

The main goal of this paper, is to study the existence of solutions for a class of nonlinear systems of differenceequations of order four, in four-dimensional with the initial conditions are real numbers. Moreover, we studysome behavior such as the periodicity and boundedness of solutions for such systems. Finally, some numericalexamples are presented and graphed by Matlab.

Keywords: Difference equations, Recursive sequences, Periodic solutions, System of difference equations.

Mathematics Subject Classification: 39A10, 39A11, 39A99, 34C99.

–––––––––––––––––––

1. INTRODUCTION

In this paper, we are concerned with the existence of solutions for the rational systems of difference equations oforder four in four-dimensional case

xn+1 = xn−3±1±xn−3yn−2zn−1tn , yn+1 =

yn−3±1±yn−3zn−2tn−1xn ,

zn+1 = zn−3±1±zn−3tn−2xn−1yn , tn+1 =

tn−3±1±tn−3xn−2yn−1zn ,

with the initial conditions are real numbers. Although we study the dynamics of these solutions such as theperiodicity and boundedness and give some numerical examples for the systems.

The study of nonlinear difference equations systems of order greater than one is quite challenging and re-warding because some prototypes for the development of the basic theory of the global behavior of nonlineardifference equations systems of order greater than one come from the results for rational difference equations[1-45]. Therefore, the study of rational difference equations systems of order greater than one is worth furtherconsideration.

Recently, Din et al. [3] studied the local asymptotic stability of an equilibrium point, periodicity behaviorof positive solutions, and global character of an equilibrium point of a fourth-order system of rational differenceequations

xn+1 =αxn−3

β+γyn−3yn−2yn−1yn, yn+1 =

α1yn−3β1+γ1xn−3xn−2xn−1xn

.

El-Dessoky [4] has investigated the solutions of the rational equation systems

xn+1 =yn−1yn−2

xn(±1±yn−1yn−2) , yn+1 =xn−1xn−2

yn(±1±xn−1xn−2) .

Yalçınkaya [5] got the sufficient conditions for the global asymptotic stability of the system of nonlineardifference equations

xn+1 =xn+yn−1xnyn−1−1 , yn+1 =

yn+xn−1ynxn−1−1 .


1428 El-Dessoky ET AL 1428-1439

In [6] Yang et al. studied the system of high order rational difference equations

xn =a

yn−p, yn =

byn−pxn−qyn−q

.

In [7], Kulenovic et al. obtained the global asymptotic behavior of solutions of the system of differenceequations

xn+1 =xn+ayn+b

, yn+1 =yn+czn+d

, zn+1 =zn+exn+f

.

In [8] El-Dessoky obtained the form of the solutions and periodicity of some systems of rational differenceequations

xn+1 =zn−3

a1+b1znyn−1xn−2zn−3, yn+1 =

xn−3a2+b2xnzn−1yn−2xn−3

, zn+1 =yn−3

a3+b3ynxn−1zn−2yn−3.

2. SYSTEMS AND THE FORMULATION OF THEIR SOLUTIONSHere we interest to investigate the following system of some rational difference equations

xn+1 = xn−31+xn−3yn−2zn−1tn

, yn+1 =yn−3

1+yn−3zn−2tn−1xn,

zn+1 = zn−31+zn−3tn−2xn−1yn

, tn+1 =tn−3

1+tn−3xn−2yn−1zn. (1)

where n ∈ N0 and the initial conditions are arbitrary real numbers.Theorem 2.1. Suppose that xn, yn, zn, tn are solutions of system (1). Then for n = 0, 1, 2, ..., we see that

x4n−3 = x−3n−1Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+1)t0z−1y−2x−3)

, x4n−2 = x−2n−1Qi=0

(1+(4i+1)z0y−1x−2t−3)(1+(4i+2)z0y−1x−2t−3)

,

x4n−1 = x−1n−1Qi=0

(1+(4i+2)y0x−1t−2z−3)(1+(4i+3)y0x−1t−2z−3)

, x4n = x0n−1Qi=0

(1+(4i+3)x0t−1z−2y−3)(1+(4i+4)x0t−1z−2y−3)

,

y4n−3 = y−3n−1Qi=0

(1+(4i)x0t−1z−2y−3)(1+(4i+1)x0t−1z−2y−3)

, y4n−2 = y−2n−1Qi=0

(1+(4i+1)t0z−1y−2x−3)(1+(4i+2)t0z−1y−2x−3)

,

y4n−1 = y−1n−1Qi=0

(1+(4i+2)z0y−1x−2t−3)(1+(4i+3)z0y−1x−2t−3)

, y4n = y0n−1Qi=0

(1+(4i+3)y0x−1t−2z−3)(1+(4i+4)y0x−1t−2z−3)

,

z4n−3 = z−3n−1Qi=0

(1+(4i)y0x−1t−2z−3)(1+(4i+1)y0x−1t−2z−3)

, z4n−2 = z−2n−1Qi=0

(1+(4i+1)x0t−1z−2y−3)(1+(4i+2)x0t−1z−2y−3)

,

z4n−1 = z−1n−1Qi=0

(1+(4i+2)t0z−1y−2x−3)(1+(4i+3)t0z−1y−2x−3)

, z4n = z0n−1Qi=0

(1+(4i+3)z0y−1x−2t−3)(1+(4i+4)z0y−1x−2t−3)

,

t4n−3 = t−3n−1Qi=0

(1+(4i)z0y−1x−2t−3)(1+(4i+1)z0y−1x−2t−3)

, t4n−2 = t−2n−1Qi=0

(1+(4i+1)y0x−1t−2z−3)(1+(4i+2)y0x−1t−2z−3)

,

t4n−1 = t−1n−1Qi=0

(1+(4i+2)x0t−1z−2y−3)(1+(4i+3)x0t−1z−2y−3)

, t4n = t0n−1Qi=0

(1+(4i+3)t0z−1y−2x−3)(1+(4i+4)t0z−1y−2x−3)

,

where−1Qi=0

Bi = 1.

Proof: For n = 0 the result holds. Now suppose that n > 1 and that our assumption holds for n− 1, that is,

x4n−7 = x−3n−2Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+1)t0z−1y−2x−3)

, x4n−6 = x−2n−2Qi=0

(1+(4i+1)z0y−1x−2t−3)(1+(4i+2)z0y−1x−2t−3)

,

x4n−5 = x−1n−2Qi=0

(1+(4i+2)y0x−1t−2z−3)(1+(4i+3)y0x−1t−2z−3)

, x4n−4 = x0n−2Qi=0

(1+(4i+3)x0t−1z−2y−3)(1+(4i+4)x0t−1z−2y−3)

,

y4n−7 = y−3n−2Qi=0

(1+(4i)x0t−1z−2y−3)(1+(4i+1)x0t−1z−2y−3)

, y4n−6 = y−2n−2Qi=0

(1+(4i+1)t0z−1y−2x−3)(1+(4i+2)t0z−1y−2x−3)

,



y4n−5 = y−1n−2Qi=0

(1+(4i+2)z0y−1x−2t−3)(1+(4i+3)z0y−1x−2t−3)

, y4n−4 = y0n−2Qi=0

(1+(4i+3)y0x−1t−2z−3)(1+(4i+4)y0x−1t−2z−3)

,

z4n−7 = z−3n−2Qi=0

(1+(4i)y0x−1t−2z−3)(1+(4i+1)y0x−1t−2z−3)

, z4n−6 = z−2n−2Qi=0

(1+(4i+1)x0t−1z−2y−3)(1+(4i+2)x0t−1z−2y−3)

,

z4n−5 = z−1n−2Qi=0

(1+(4i+2)t0z−1y−2x−3)(1+(4i+3)t0z−1y−2x−3)

, z4n−4 = z0n−2Qi=0

(1+(4i+3)z0y−1x−2t−3)(1+(4i+4)z0y−1x−2t−3)

,

t4n−7 = t−3n−2Qi=0

(1+(4i)z0y−1x−2t−3)(1+(4i+1)z0y−1x−2t−3)

, t4n−6 = t−2n−2Qi=0

(1+(4i+1)y0x−1t−2z−3)(1+(4i+2)y0x−1t−2z−3)

,

t4n−5 = t−1n−2Qi=0

(1+(4i+2)x0t−1z−2y−3)(1+(4i+3)x0t−1z−2y−3)

, t4n−4 = t0n−2Qi=0

(1+(4i+3)t0z−1y−2x−3)(1+(4i+4)t0z−1y−2x−3)

.

We deduce from system (1) that

x4n−3 = x4n−71+t4n−4z4n−5y4n−6x4n−7

=x−3

n−2Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+1)t0z−1y−2x−3)

1+

⎡⎢⎢⎢⎢⎢⎣t0

n−2Qi=0

(1+(4i+3)t0z−1y−2x−3)(1+(4i+4)t0z−1y−2x−3)

z−1n−2Qi=0

(1+(4i+2)t0z−1y−2x−3)(1+(4i+3)t0z−1y−2x−3)

y−2n−2Qi=0

(1+(4i+1)t0z−1y−2x−3)(1+(4i+2)t0z−1y−2x−3)

x−3n−2Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+1)t0z−1y−2x−3)

⎤⎥⎥⎥⎥⎥⎦

=x−3

n−2Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+1)t0z−1y−2x−3)

1+

⎡⎣t0z−1y−2x−3 n−2Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+4)t0z−1y−2x−3)

⎤⎦ =x−3

n−2Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+1)t0z−1y−2x−3)

1+t0z−1y−2x−3

(1+(4i−4)t0z−1y−2x−3)

= x−3n−2Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+1)t0z−1y−2x−3)

(1+(4i−4)t0z−1y−2x−3)(1+(4i−3)t0z−1y−2x−3) = x−3

n−1Qi=0

(1+(4i)t0z−1y−2x−3)(1+(4i+1)t0z−1y−2x−3)

,

y4n−3 = y4n−71+x4n−4t4n−5z4n−6y4n−7

=y−3

n−2Qi=0

(1+(4i)x0t−1z−2y−3)(1+(4i+1)x0t−1z−2y−3)

1+

⎡⎢⎢⎢⎢⎢⎣x0

n−2Qi=0

(1+(4i+3)x0t−1z−2y−3)(1+(4i+4)x0t−1z−2y−3)

t−1n−2Qi=0

(1+(4i+2)x0t−1z−2y−3)(1+(4i+3)x0t−1z−2y−3)

z−2n−2Qi=0

(1+(4i+1)x0t−1z−2y−3)(1+(4i+2)x0t−1z−2y−3)

y−3n−2Qi=0

(1+(4i)x0t−1z−2y−3)(1+(4i+1)x0t−1z−2y−3)

⎤⎥⎥⎥⎥⎥⎦

=y−3

n−2Qi=0

(1+(4i)x0t−1z−2y−3)(1+(4i+1)x0t−1z−2y−3)

1+

⎡⎣x0t−1z−2y−3 n−2Qi=0

(1+(4i)x0t−1z−2y−3)(1+(4i+4)x0t−1z−2y−3)

⎤⎦

=y−3

n−2Qi=0

(1+(4i)x0t−1z−2y−3)(1+(4i+1)x0t−1z−2y−3)

1+x0t−1z−2y−3

(1+(4i−4)x0t−1z−2y−3)= y−3

n−1Qi=0

(1+(4i)x0t−1z−2y−3)(1+(4i+1)x0t−1z−2y−3)

.



z4n−3 = z4n−71+y4n−4x4n−5t4n−6z4n−7

=z−3

n−2Qi=0

(1+(4i)y0x−1t−2z−3)(1+(4i+1)y0x−1t−2z−3)

1+

⎡⎢⎢⎢⎢⎢⎣y0

n−2Qi=0

(1+(4i+3)y0x−1t−2z−3)(1+(4i+4)y0x−1t−2z−3)

x−1n−2Qi=0

(1+(4i+2)y0x−1t−2z−3)(1+(4i+3)y0x−1t−2z−3)

t−2n−2Qi=0

(1+(4i+1)y0x−1t−2z−3)(1+(4i+2)y0x−1t−2z−3)

z−3n−2Qi=0

(1+(4i)y0x−1t−2z−3)(1+(4i+1)y0x−1t−2z−3)

⎤⎥⎥⎥⎥⎥⎦

=z−3

n−2Qi=0

(1+(4i)y0x−1t−2z−3)(1+(4i+1)y0x−1t−2z−3)

1+y0x−1t−2z−3

n−2Qi=0

(1+(4i)y0x−1t−2z−3)(1+(4i+4)y0x−1t−2z−3)

= z−3n−2Qi=0

(1+(4i)y0x−1t−2z−3)(1+(4i+1)y0x−1t−2z−3)

.

Finally from Eq.(1), we see that

t4n−3 = t4n−71+z4n−4y4n−5x4n−6t4n−7

=t−3

n−2Qi=0

(1+(4i)z0y−1x−2t−3)(1+(4i+1)z0y−1x−2t−3)

1+

⎡⎢⎢⎢⎢⎢⎣z0

n−2Qi=0

(1+(4i+3)z0y−1x−2t−3)(1+(4i+4)z0y−1x−2t−3)

y−1n−2Qi=0

(1+(4i+2)z0y−1x−2t−3)(1+(4i+3)z0y−1x−2t−3)

x−2n−2Qi=0

(1+(4i+1)z0y−1x−2t−3)(1+(4i+2)z0y−1x−2t−3)

t−3n−2Qi=0

(1+(4i)z0y−1x−2t−3)(1+(4i+1)z0y−1x−2t−3)

⎤⎥⎥⎥⎥⎥⎦

=t−3

n−2Qi=0

(1+(4i)z0y−1x−2t−3)(1+(4i+1)z0y−1x−2t−3)

1+

⎡⎣z0 n−2Qi=0

(1+(4i)z0y−1x−2t−3)(1+(4i+4)z0y−1x−2t−3)

⎤⎦ = t−3n−1Qi=0

(1+(4i)z0y−1x−2t−3)(1+(4i+1)z0y−1x−2t−3)

.

Similarly we can prove the other relations. This completes the proof.

Lemma 1. Let xn, yn, zn, tn be a positive solution of system (1), then every solution of system (1) isbounded and converges to zero.

Proof: It follows from System (1) that


< xn−3, yn+1 =yn−3

1+yn−3zn−2tn−1xn< yn−3,


< zn−3, tn+1 =tn−3

1+tn−3xn−2yn−1zn< tn−3,

we see thatxn+1 < xn−3, yn+1 < yn−3, zn+1 < zn−3, tn+1 < tn−3,

Then the subsequences x4n−3∞n=0, x4n−2∞n=0, x4n−1∞n=0, x4n∞n=0 are decreasing and so are bounded fromabove by M = maxx−3, x−2, x−1, x0. Also, for the other subsequences of the main sequences yn, zn,and tn∞n=0.

Lemma 2. If xi, yi, zi, ti, i = −3, − 2, − 1, 0 arbitrary real numbers and let xn, yn, zn, tn are solutionsof system (1) then the following statements are true:-

(i) If x−3 = 0, then we have x4n−3 = 0 and y4n−2 = y−2, z4n−1 = z−1, t4n = t0.

(ii) If x−2 = 0, then we have x4n−2 = 0 and t4n−3 = t−3, z4n = z0, y4n−1 = y−1.

(iii) If x−1 = 0, then we have x4n−1 = 0 and z4n−3 = z−3, t4n−2 = t−2, y4n = y0.

(iv) If x0 = 0, then we have x4n = 0 and y4n−3 = y−3, z4n−2 = z−2, t4n−1 = t−1.

(v) If y−3 = 0, then we have y4n−3 = 0 and z4n−2 = z−2, t4n−1 = t−1, x4n = x0 .



(vi) If y−2 = 0, then we have y4n−2 = 0 and x4n−3 = x−3, z4n−1 = z−1, t4n = t0.

(vii) If y−1 = 0, then we have y4n−1 = 0 and t4n−3 = t−3, z4n = z0, x4n−2 = x−2.

(viii) If y0 = 0, then we have y4n = 0 and z4n−3 = z−3, t4n−2 = t−2, x4n−1 = x−1.

(ix) If z−3 = 0, then we have z4n−3 = 0 and t4n−2 = t−2, y4n = y0, x4n−1 = x−1.

(x) If z−2 = 0, then we have z4n−2 = 0 and y4n−3 = y−3, t4n−1 = t−1, x4n = x0.

(xi) If z−1 = 0, then we have z4n−1 = 0 and x4n−3 = x−3, y4n−2 = y−2, t4n = t0.

(xii) If z0 = 0, then we have z4n = 0 and t4n−3 = t−3, y4n−1 = y−1, x4n−2 = x−2.

(xiii) If t−3 = 0, then we have t4n−3 = 0 and z4n = z0, y4n−1 = y−1, x4n−2 = x−2.

(ivx) If t−2 = 0, then we have t4n−2 = 0 and z4n−3 = z−3, y4n = y0, x4n−1 = x−1.

(vx) If t−1 = 0, then we have t4n−1 = 0 and y4n−3 = y−3, z4n−2 = z−2, x4n = x0.

(vxi) If t0 = 0, then we have t4n = 0 and x4n−3 = x−3, y4n−2 = y−2, z4n−1 = z−1.

Proof: The proof follows directly from the expressions of the solutions of system (1).

Theorem 2.2. Let xn, yn, zn, tn are solutions of the system

xn+1 = xn−3−1+xn−3yn−2zn−1tn , yn+1 =

yn−3−1+yn−3zn−2tn−1xn ,


, tn+1 =tn−3

1−tn−3xn−2yn−1zn , (2)

with the initial values are arbitrary real numbers satisfies t0z−1y−2x−3 = y0x−1t−2z−3 6= ±1, z0y−1x−2t−3 =x0t−1z−2y−3 6= 1, z0y−1x−2t−3 = x0t−1z−2y−3 6= 1

2 . Then the solution are given by the following formulae forn = 0, 1, 2, ...,

x4n−3 = x−3(−1+t0z−1y−2x−3)n , x4n−2 =

(−1)nx−2(−1+z0y−1x−2t−3)n(−1+2z0y−1x−2t−3)n ,

x4n−1 = x−1(−1+y0x−1t−2z−3)n , x4n = x0 (−1 + x0t−1z−2y−3)

n,

y4n−3 = y−3(−1+x0t−1z−2y−3)n , y4n−2 = y−2 (−1 + t0z−1y−2x−3)

n ,

y4n−1 = (−1)ny−1(−1+2z0y−1x−2t−3)n(−1+z0y−1x−2t−3)n , y4n = y0 (−1 + y0x−1t−2z−3)

n ,

z4n−3 = z−3(1+y0x−1t−2z−3)

n , z4n−2 =z−2(−1+x0t−1z−2y−3)n(−1+2x0t−1z−2y−3)n ,

z4n−1 = z−1(1+t0z−1y−2x−3)

n , z4n = (−1)n z0 (−1 + z0y−1x−2t−3)n ,

t4n−3 = (−1)nt−3(−1+z0y−1x−2t−3)n , t4n−2 = t−2 (1 + y0x−1t−2z−3)

n,

t4n−1 = t−1(−1+2x0t−1z−2y−3)n(−1+x0t−1z−2y−3)n , t4n = t0 (1 + t0z−1y−2x−3)

n.

Proof: As the proof of Theorem 1.

Theorem 2.3. Suppose that xn, yn, zn, tn are solutions of the system

xn+1 = xn−3−1+xn−3yn−2zn−1tn , yn+1 =

yn−3−1+yn−3zn−2tn−1xn ,

zn+1 = zn−3−1+zn−3tn−2xn−1yn , tn+1 =

tn−3−1+tn−3xn−2yn−1zn , (3)

with t0z−1y−2x−3 = z0y−1x−2t−3 = y0x−1t−2z−3 = x0t−1z−2y−3 6= 1,then all solutions of the system areunbounded if t0z−1y−2x−3 = z0y−1x−2t−3 = y0x−1t−2z−3 = x0t−1z−2y−3 6= 2, and takes the form

x4n−3 = x−3(−1+t0z−1y−2x−3)n , x4n−2 = x−2 (−1 + z0y−1x−2t−3)

n ,

x4n−1 = x−1(−1+y0x−1t−2z−3)n , x4n = x0 (−1 + x0t−1z−2y−3)

n ,

y4n−3 = y−3(−1+x0t−1z−2y−3)n , y4n−2 = y−2 (−1 + t0z−1y−2x−3)

n,

y4n−1 = y−1(−1+z0y−1x−2t−3)n , y4n = y0 (−1 + y0x−1t−2z−3)

n ,



z4n−3 = z−3(−1+y0x−1t−2z−3)n , z4n−2 = z−2 (−1 + x0t−1z−2y−3)

n ,

z4n−1 = z−1(−1+t0z−1y−2x−3)n , z4n = z0 (−1 + z0y−1x−2t−3)

n,

and

t4n−3 = t−3(−1+z0y−1x−2t−3)n , t4n−2 = t−2 (−1 + y0x−1t−2z−3)

n,

t4n−1 = t−1(−1+x0t−1z−2y−3)n , t4n = t0 (−1 + t0z−1y−2x−3)

n .

Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n− 1. That is

x4n−7 = x−3(−1+t0z−1y−2x−3)n−1

, x4n−6 = x−2 (−1 + z0y−1x−2t−3)n−1

,

x4n−5 = x−1(−1+y0x−1t−2z−3)n−1

, x4n−4 = x0 (−1 + x0t−1z−2y−3)n−1

,

y4n−7 = y−3(−1+x0t−1z−2y−3)n−1

, y4n−6 = y−2 (−1 + t0z−1y−2x−3)n−1

,

y4n−5 = y−1(−1+z0y−1x−2t−3)n−1

, y4n−4 = y0 (−1 + y0x−1t−2z−3)n−1 ,

z4n−7 = z−3(−1+y0x−1t−2z−3)n−1

, z4n−6 = z−2 (−1 + x0t−1z−2y−3)n−1 ,

z4n−5 = z−1(−1+t0z−1y−2x−3)n−1

, z4n−4 = z0 (−1 + z0y−1x−2t−3)n−1 ,

t4n−7 = t−3(−1+z0y−1x−2t−3)n−1

, t4n−6 = t−2 (−1 + y0x−1t−2z−3)n−1

,

t4n−5 = t−1(−1+x0t−1z−2y−3)n−1

, t4n−4 = t0 (−1 + t0z−1y−2x−3)n−1

.

It follows from System (3) that

x4n−3 = x4n−7−1+t4n−4z4n−5y4n−6x4n−7 =

x−3(−1+t0z−1y−2x−3)

n−1

−1+ t0z−1y−2x−3(−1+t0z−1y−2x−3)n−1(−1+t0z−1y−2x−3)

n−1

(−1+t0z−1y−2x−3)n−1(−1+t0z−1y−2x−3)

n−1

=

x−3(−1+t0z−1y−2x−3)

n−1

[−1+t0z−1y−2x−3] = x−3(−1+t0z−1y−2x−3)n ,

y4n−2 = y4n−6−1+x4n−3t4n−4z4n−5y4n−6 =

y−2(−1+t0z−1y−2x−3)n−1

−1+ t0z−1y−2x−3(−1+t0z−1y−2x−3)n−1(−1+t0z−1y−2x−3)

n−1

(−1+t0z−1y−2x−3)n(−1+t0z−1y−2x−3)

n−1

= y−2(−1+t0z−1y−2x−3)n−1

−1+ t0z−1y−2x−3(−1+t0z−1y−2x−3)

= y−2 (−1 + t0z−1y−2x−3)n ,

z4n−1 = z4n−5−1+y4n−2x4n−3t4n−4z4n−5 =

z−1(−1+t0z−1y−2x−3)n−1

−1+ t0z−1y−2x−3(−1+t0z−1y−2x−3)n(−1+t0z−1y−2x−3)n−1(−1+t0z−1y−2x−3)n(−1+t0z−1y−2x−3)n−1

=

z−1(−1+t0z−1y−2x−3)n−1[−1+t0z−1y−2x−3] = z−1

(−1+t0z−1y−2x−3)n ,

t4n = t4n−4−1+z4n−1y4n−2x4n−3t4n−4 =

t0(1+t0z−1y−2x−3)n−1

−1+ t0z−1y−2x−3(−1+t0z−1y−2x−3)n(−1+t0z−1y−2x−3)n−1(−1+t0z−1y−2x−3)n(−1+t0z−1y−2x−3)n

= t0(−1+t0z−1y−2x−3)n−1

−1+ t0z−1y−2x−3(−1+t0z−1y−2x−3)

= t0 (−1 + t0z−1y−2x−3)n .

Also, we can prove the other relations similarly. The proof is complete.

Theorem 2.4. If the sequences xn, yn, zn, tn are solutions of difference equation system (3) such thatt0z−1y−2x−3 = z0y−1x−2t−3 = y0x−1t−2z−3 = x0t−1z−2y−3 = 2.Then all solutions of system (3) are periodic



with period four and takes the form

x4n−3 = x−3, x4n−2 = x−2, x4n−1 = x−1, x4n = x0.

y4n−3 = y−3, y4n−2 = y−2, y4n−1 = y−1, y4n = y0.

z4n−3 = z−3, z4n−2 = z−2, z4n−1 = z−1, z4n = z0.

t4n−3 = t−3, t4n−2 = t−2, t4n−1 = t−1, t4n = t0.

Or

xn = x−3, x−2, x−1, x0, x−3, x−2, ... .yn = y−3, y−2, y−1, y0, y−3, y−2, ... .zn = z−3, z−2, z−1, z0, z−3, z−2, ... .tn = t−3, t−2, t−1, t0, t−3, t−2, ... .

Proof: The proof follows from the previous Theorem and will be omitted.

The following theorems can be proved similarly.

3. OTHER SYSTEMS:

In this section, we get the solutions of the following systems of the difference equations


, yn+1 =yn−3

1+yn−3zn−2tn−1xn,


, tn+1 =tn−3

1−tn−3xn−2yn−1zn . (4)


, yn+1 =yn−3

1+yn−3zn−2tn−1xn

zn+1 = zn−31−zn−3tn−2xn−1yn , tn+1 =

tn−31−tn−3xn−2yn−1zn . (5)

xn+1 = xn−31−xn−3yn−2zn−1tn , yn+1 =

yn−31−yn−3zn−2tn−1xn ,


, tn+1 =tn−3

1+tn−3xn−2yn−1zn. (6)

xn+1 = xn−3−1+xn−3yn−2zn−1tn , yn+1 =

yn−31+yn−3zn−2tn−1xn

,

zn+1 = zn−31−zn−3tn−2xn−1yn , tn+1 =

tn−3−1+tn−3xn−2yn−1zn . (7)

where n ∈ N0 and the initial conditions are arbitrary real numbers.Theorem 3.1. If xn, yn, zn, tn are solutions of difference equation system (4). Then for n = 0, 1, 2, ...,

x4n−3 = x−3n−1Qi=0

(1+(2i)t0z−1y−2x−3)(1+(2i+1)t0z−1y−2x−3)

, x4n−2 = x−2n−1Qi=0

(1+(2i−1)z0y−1x−2t−3)(1+(2i)z0y−1x−2t−3)

,

x4n−1 = x−1n−1Qi=0

(1+(2i)y0x−1t−2z−3)(1+(2i+1)y0x−1t−2z−3)

, x4n = x0n−1Qi=0

(1+(2i+1)x0t−1z−2y−3)(1+(2i+2)x0t−1z−2y−3)

,

y4n−3 = y−3n−1Qi=0

(1+(2i)x0t−1z−2y−3)(1+(2i+1)x0t−1z−2y−3)

, y4n−2 = y−2n−1Qi=0

(1+(2+1)t0z−1y−2x−3)(1+(2i+2)t0z−1y−2x−3)

,

y4n−1 = y−1n−1Qi=0

(1+(2i)z0y−1x−2t−3)(1+(2i+1)z0y−1x−2t−3)

, y4n = y0n−1Qi=0

(1+(2i+1)y0x−1t−2z−3)(1+(2i+2)y0x−1t−2z−3)

,

z4n−3 = z−3n−1Qi=0

(1+(2i)y0x−1t−2z−3)(1+(2i+1)y0x−1t−2z−3)

, z4n−2 = z−2n−1Qi=0

(1+(2i+1)x0t−1z−2y−3)(1+(2i+2)x0t−1z−2y−3)

,



z4n−1 = z−1n−1Qi=0

(1+(2i+2)t0z−1y−2x−3)(1+(2i+3)t0z−1y−2x−3)

, z4n = z0n−1Qi=0

(1+(2i+1)z0y−1x−2t−3)(1+(2i+4)z0y−1x−2t−3)

,

t4n−3 = t−3n−1Qi=0

(1+(2i)z0y−1x−2t−3)(1+(2i−1)z0y−1x−2t−3) , t4n−2 = t−2

n−1Qi=0

(1+(2i+1)y0x−1t−2z−3)(1+(2i)y0x−1t−2z−3)

,

t4n−1 = t−1n−1Qi=0

(1+(2i+2)x0t−1z−2y−3)(1+(2i+1)x0t−1z−2y−3)

, t4n = t0n−1Qi=0

(1+(2i+3)t0z−1y−2x−3)(1+(2i+2)t0z−1y−2x−3)

,

where−1Qi=0

Bi = 1.

Theorem 3.2. The form of the solutions of system (5) are given by the following formulae:

x4n−3 = x−3(1+t0z−1y−2x−3)

n , x4n−2 = (−1)nx−2 (−1 + z0y−1x−2t−3)n ,

x4n−1 = x−1(−1+2y0x−1t−2z−3)n(−1+y0x−1t−2z−3)n , x4n = (−1)nx0 (−1 + x0t−1z−2y−3)

n,

y4n−3 = y−3(1+x0t−1z−2y−3)

n , y4n−2 =y−2(1+t0z−1y−2x−3)

n

(1+2t0z−1y−2x−3)n ,

y4n−1 = y−1(1+z0y−1x−2t−3)

n , y4n = (−1)ny0 (−1 + y0x−1t−2z−3)n,

z4n−3 = (−1)nz−3(−1+y0x−1t−2z−3)n , z4n−2 = z−2 (1 + x0t−1z−2y−3)

n,

z4n−1 = z−1(1+2t0z−1y−2x−3)n

(1+t0z−1y−2x−3)n , z4n = z0 (1 + z0y−1x−2t−3)

n,

t4n−3 = (−1)nt−3(−1+z0y−1x−2t−3)n , t4n−2 =

t−2(−1+y0x−1t−2z−3)n(−1+2y0x−1t−2z−3)n ,

t4n−1 = (−1)nt−1(−1+x0t−1z−2y−3)n , t4n = t0 (1 + t0z−1y−2x−3)

n ,

where t0z−1y−2x−3 6= −1, t0z−1y−2x−3 6= −12 , y0x−1t−2z−3 6= 1, y0x−1t−2z−3 6=12 , z0y−1x−2t−3 = x0t−1z−2y−3 6=

±1.

Theorem 3.3. Let xn, yn, zn, tn are solutions of difference equation system (6) with t0z−1y−2x−3 6=1, t0z−1y−2x−3 6= 1

2 , x0t−1z−2y−3 = z0y−1x−2t−3 6= ±1, y0x−1t−2z−3 6= −1, y0x−1t−2z−3 6= −12 ,then forn = 0, 1, 2, ...,

x4n−3 = (−1)nx−3(−1+t0z−1y−2x−3)n , x4n−2 = x−2 (1 + z0y−1x−2t−3)

n,

x4n−1 = x−1(1+2y0x−1t−2z−3)n

(1+y0x−1t−2z−3)n , x4n = x0 (1 + x0t−1z−2y−3)

n ,

y4n−3 = (−1)ny−3(−1+x0t−1z−2y−3)n , y4n−2 =

y−2(−1+t0z−1y−2x−3)n(−1+2t0z−1y−2x−3)n ,

y4n−1 = (−1)ny−1(−1+z0y−1x−2t−3)n , y4n = y0 (1 + y0x−1t−2z−3)

n ,

z4n−3 = z−3(1+y0x−1t−2z−3)

n , z4n−2 = (−1)nz−2 (−1 + x0t−1z−2y−3)n ,

z4n−1 = z−1(−1+2t0z−1y−2x−3)n(−1+t0z−1y−2x−3)n , z4n = (−1)nz0 (−1 + z0y−1x−2t−3)

n ,

t4n−3 = t−3(1+z0y−1x−2t−3)

n , t4n−2 =t−2(1+y0x−1t−2z−3)

n

(1+2y0x−1t−2z−3)n ,

t4n−1 = t−1(1+x0t−1z−2y−3)

n , t4n = (−1)nt0 (−1 + t0z−1y−2x−3)n .

Theorem 3.4. Suppose that the initial conditions of the system (7) are arbitrary real numbers satisfies t0z−1y−2x−3 6=1, t0z−1y−2x−3 6= 1

2 , x0t−1z−2y−3 = z0y−1x−2t−3 6= ±1, y0x−1t−2z−3 6= 1, y0x−1t−2z−3 6= 12 , and if

xn, yn, zn, tn are solutions of system (7). Then for n = 0, 1, 2, ...,

x4n−3 = x−3(−1+t0z−1y−2x−3)n , x4n−2 = x−2 (−1 + z0y−1x−2t−3)

n ,

x4n−1 = (−1)nx−1(−1+2y0x−1t−2z−3)n(−1+y0x−1t−2z−3)n , x4n = x0 (−1 + x0t−1z−2y−3)

n ,



y4n−3 = y−3(1+x0t−1z−2y−3)

n , y4n−2 =y−2(−1+t0z−1y−2x−3)n(−1+2t0z−1y−2x−3)n ,

y4n−1 = y−1(1+z0y−1x−2t−3)

n , y4n = (−1)ny−2 (−1 + t0z−1y−2x−3)n,

z4n−3 = (−1)nz−3(−1+y0x−1t−2z−3)n , z4n−2 = z−2 (1 + x0t−1z−2y−3)

n,

z4n−1 = z−1(−1+2t0z−1y−2x−3)n(−1+t0z−1y−2x−3)n , z4n = z0 (1 + z0y−1x−2t−3)

n ,

t4n−3 = t−3(−1+z0y−1x−2t−3)n , t4n−2 =

(−1)nt−2(−1+y0x−1t−2z−3)n(−1+2y0x−1t−2z−3)n ,

t4n−1 = t−1(−1+x0t−1z−2y−3)n , t4n = t0 (−1 + t0z−1y−2x−3)

n.

4. NUMERICAL EXAMPLES

Here we consider some numerical examples for the previous systems to illustrate the results.

Example 1. We consider the system (1) with the initial conditions x−3 = .16, x−2 = −.3, x−1 = 7, x0 =−1.3, y−3 = .2, y−2 = −.4, y−1 = .51, y0 = 1, z−3 = −.8, z−2 = .4, z−1 = 5, z0 = .74, t−3 = .18, t−2 =.64, t−1 = −.5 and t0 = 1.9. (See Fig. 1). Also, see Figure 2 to see the behavior of the solutions of System(1) with initials conditions x−3 = .16, x−2 = .3, x−1 = 0, x0 = 1.3, y−3 = .2, y−2 = .4, y−1 = .51,y0 = 1, z−3 = .8, z−2 = .4, z−1 = .5, z0 = .74, t−3 = .8, t−2 = .64, t−1 = .5 and t0 = 1.9.

0 10 20 30 40 50 60 70 80−2

−1

0

1

2

3

4

5

6

7

n

x(n),y(

n),Z(n),

T(n)

x(n)y(n)z(n)t(n)

Figure 1. Plot of the system (1).

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

n

x(n),y(

n),z(n),

t(n)

x(n)y(n)z(n)t(n)

Figure 2. Draw the behavior of the solution of the system (1).



Example 2. See Figure (3) for an example for the system (2) with the initial values x−3 = .6, x−2 = .3, x−1 =.19, x0 = −.3, y−3 = .2, y−2 = .4, y−1 = .56, y0 = .91, z−3 = .28, z−2 = .4, z−1 = .65, z0 = .37,t−3 = .8, t−2 = .64, t−1 = .5 and t0 = .7.

0 10 20 30 40 50 60−3

−2

−1

0

1

2

3

4

n

x(n),y(

n),z(n),

t(n)

x(n)y(n)z(n)t(n)

Figure 3. Sketch the behavior of the solution of the system (2).

Example 3. If we take the initial conditions as follows x−3 = .6, x−2 = .3, x−1 = −.19, x0 = −.3, y−3 =.2, y−2 = .04, y−1 = .56, y0 = .91, z−3 = .28, z−2 = −.4, z−1 = .49, z0 = .37, t−3 = −.8, t−2 = −.64, t−1 =.5 and t0 = .7, for the difference system (3), see Fig. 4.

0 5 10 15 20 25 30 35 40 45 50−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

n

x(n),y(

n),z(n),

t(n)

x(n)y(n)z(n)t(n)

Figure 4. Plot of system (3).

Example 4. Figure (5) shows the periodicity behavior of the solution of the difference system (3) with theinitial conditions x−3 = 6, x−2 = −.3, x−1 = 9, x0 = −8, y−3 = 1/9, y−2 = −9, y−1 = 5, y0 = .1, z−3 =20, z−2 = 6, z−1 = −.7, z0 = .2, t−3 = −20/3, t−2 = 1/9, t−1 = −3/8 and t0 = 10/189.

0 5 10 15 20 25 30−10

−5

0

5

10

15

20

n

x(n),y(

n),z(n),

t(n)

x(n)y(n)z(n)t(n)

Figure 5. Plot the behavior of the solution of the difference system (3).



AcknowledgementsThis Project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah,

Saudi Arabia under grant no. (G - 572 - 130 - 38). The authors, therefore, acknowledge with thanks DSR fortechnical and financial support.

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40. M. M. El-Dessoky, M. Mansour, E. M. Elsayed, Solutions of some rational systems of difference equations,Utilitas Mathematica, 92, (2013), 329-336.

41. E. O. Alzahrani, M. M. El-Dessoky, E. M. Elsayed and Y. Kuang, Solutions and Properties of Some Degen-erate Systems of Difference Equations, J. Comput. Anal. Appl., 18(2), (2015), 321-333.

42. Y. Yazlik, D. T. Tollu, N. Taskara, On the Behaviour of Solutions for Some Systems of Difference Equations,J. Comput. Anal. Appl., 18(1), (2015),166-178.

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44. M. M. El-Dessoky, E. M. Elsayed and E. O. Alzahrani, The form of solutions and periodic nature for somerational difference equations systems, J. Nonlinear Sci. Appl., 9(10), (2016), 5629—5647.

45. M. M. El-Dessoky, Solution of a rational systems of difference equations of order three, Mathematics, 4(3),(2016), 1-12.



Applications of soft sets to BCC-ideals in BCC-algebras

Sun Shin Ahn

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Abstract. The notions of a union soft ideal and a union soft BCC-ideal of a BCC-algebra are introduced and

some related properties of them are investigated. A quotient structure of BCC-algebra using a uni-soft BCC-ideal

is constructed and some related properties are studied.

1. Introduction

Y. Kormori [10] introduced a notion of a BCC-algebras, and W. A. Dudek [3] redefined the

notion of BCC-algebras by using a dual from of the ordinary definition of Y. Kormori. In [6], J.

Hao introduced the notion of ideals in a BCC-algebra and studied some related properties. W.

A. Dudek and X. Zhang [4] introdued a BCC-ideals in a BCC-algebra and described connections

between such BCC-ideals and congruences. S. S. Ahn and S. H. Kwon [1] defined a topological

BCC-algebra and investigated some properties of it.

Various problems in system identification involve characteristics which are essentially non-

probabilistic in nature [15]. In response to this situation Zadeh [16] introduced fuzzy set theory

as an alternative to probability theory. Uncertainty is an attribute of information. In order to

suggest a more general framework, the approach to uncertainty is outlined by Zadeh [17]. To

solve complicated problem in economics, engineering, and environment, we can’t successfully use

classical methods because of various uncertainties typical for those problems. There are three

theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we

can consider as mathematical tools for dealing with uncertainties. But all these theories have

their own difficulties. Uncertainties can’t be handled using traditional mathematical tools but

may be dealt with using a wide range of existing theories such as probability theory, theory of

(intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of

rough sets. However, all of these theories have their own difficulties which are pointed out in

[13]. Maji et al. [12] and Molodtsov [13] suggested that one reason for these difficulties may be

due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties,

Molodtsov [13] introduced the concept of soft set as a new mathematical tool for dealing with

uncertainties that is free from the difficulties that have troubled the usual theoretical approaches.

Molodtsov pointed out several directions for the applications of soft sets. Maji et al. [12] described

the application of soft set theory to a decision making problem. Maji et al. [11] also studied

02010 Mathematics Subject Classification: 06F35; 03G25; 06D72.0Keywords: γ-exclusive set, Union soft ideal, Union soft BCC-ideal.0E-mail: [email protected]


1440 Sun Shin Ahn 1440-1450

Sun Shin Ahn

several operations on the theory of soft sets. Jun [8] discussed the union soft sets with applications

in BCK/BCI-algebras. We refer the reader to the papers [2, 5, 7, 9, 14] for further information

regarding algebraic structures/properties of soft set theory.

In this paper, we introduce the notions of a union soft ideal and a union soft BCC-ideal

of a BCC-algebra and investigated some related properties of them. A quotient structure of

BCC-algebra using a uni-soft BCC-ideal is constructed and some related properties are studied.

2. Preliminaries

By a BCC-algebra [3] we mean an algebra (X, ∗, 0) of type (2,0) satisfying the following con-

ditions: for all x, y, z ∈ X,

(a1) ((x ∗ y) ∗ (z ∗ y)) ∗ (x ∗ z) = 0,

(a2) 0 ∗ x = 0,

(a3) x ∗ 0 = x,

(a4) x ∗ y = 0 and y ∗ x = 0 imply x = y.

For brevity, we also call X a BCC-algebra. In X, we can define a partial order “≤” by putting

x ≤ y if and only if x ∗ y = 0. Then ≤ is a partial order on X.

A BCC-algebra X has the following properties: for any x, y ∈ X,

(b1) x ∗ x = 0,

(b2) (x ∗ y) ∗ x = 0,

(b3) x ≤ y ⇒ x ∗ z ≤ y ∗ z and z ∗ y ≤ z ∗ x.

Any BCK-algebra is a BCC-algebra, but there are BCC-algebras which are not BCK-algebra

(see [3]). Note that a BCC-algebra is a BCK-algebra if and only if it satisfies:

(b4) (x ∗ y) ∗ z = (x ∗ z) ∗ y, for all x, y, z ∈ X.

Let (X, ∗, 0X) and (Y, ∗, 0Y ) be BCC-algebras. A mapping φ : X → Y is called a homomorphism

if φ(x ∗X y) = φ(x) ∗Y φ(y) for all x, y ∈ X. A non-empty subset S of a BCC-algebra X is called

a subalgebra of X if x ∗ y ∈ S whenever x, y ∈ S. A non-empty subset I of a BCI-algebra X is

called an ideal [6] of X if it satisfies:

(c1) 0 ∈ I,

(c2) x ∗ y, y ∈ I ⇒ x ∈ I for all x, y ∈ X.

I is called an BCC-ideal [4] of X if it satisfies (c1) and

(c3) (x ∗ y) ∗ z, y ∈ I ⇒ x ∗ z ∈ I, for all x, y, z ∈ X.

Theorem 2.1. [6] In a BCC-algebra, an ideal is a subalgebra.

Theorem 2.2. [4] In a BCC-algebra, a BCC-ideal is an ideal.

Corollary 2.3. [4] Any BCC-ideal of a BCC-algebra is a subalgebra.


1441 Sun Shin Ahn 1440-1450


Let X be a BCC-algebra and let I be a BCC-ideal of X. Define a relation ∼I on X by x ∼ y if

and only if x ∗ y, y ∗x ∈ I for any x, y ∈ X. Then it is a congruence relation on X [4]. Denote by

[x]I the equivalence class containing x, i.e., [x]I := y ∈ X|x ∼I y and let X/I := [x]I |x ∈ X.

Theorem 2.4. If I is a BCC-algebra X, then the quotient algebra X/I is a BCC-algebra.

A soft set theory is introduced by Molodtsov [13].

In what follows, let U be an initial universe set and E be a set of parameters. We say that the

pair (U,E) is a soft universe. Let P(U) denotes the power set of U and A,B,C, · · · ⊆ E.

Definition 2.5. [13] A soft set (f, A) over U is defined to be the set of ordered pairs (f, A) :=

(x, f(x)) : x ∈ E, f(x) ∈ P(U) , where f : E → P(U) such that f(x) = ∅ if x /∈ A.

For ϵ ∈ A, f(ϵ) may be considered as the set of ϵ-approximate elements of the soft set (f, A).

Clearly, a soft set is not a set. For a soft set (f, A) of X and a subset γ of U , the γ-exclusive set

of (f, A), defined to be the set eA(f ; γ) := x ∈ A|f(x) ⊆ γ.For any soft sets (f,X) and (g,X) of X, we call (f,X) a soft subset of (g,X) , denoted by

(f,X) ⊆ (g,X) , if f(x) ⊆ g(x) for all x ∈ X. The soft union of (f,X) and (g,X), denoted by

(f,X) ∪ (g,X) , is defined to be the soft set (f ∪ g,X) of X over U in which f ∪ g is defined by

(f ∪ g) (x) := f(x) ∪ g(x) for all x ∈ X. The soft intersection of (f,X) and (g,X) , denoted by

(f,X) ∩ (g,X) , is defined to be the soft set (f ∩ g,M) of X over U in which f ∩ g is defined by

(f ∩ g) (x) := f(x) ∩ g(x) for all x ∈M.

3. Uni-soft BCC-ideals

In what follows, let X be a BCC-algebra unless otherwise specified.

Definition 3.1. A soft set (f,X) over U is called a union soft subalgebra (briefly, uni-soft

subalgebra) of a BCC-algebra X over U if it satisfies:

(3.0) f(x ∗ y) ⊆ f(x) ∪ f(y) for all x, y ∈ X.

Proposition 3.2. Every uni-soft subalgebra (f,X) of a BCC-algebra X over U satisfies the

following inclusion:

(3.1) f(0) ⊆ f(x) for all x ∈ X.

Proof. Using (3.0) and (b1), we have f(0) = f(x ∗ x) ⊆ f(x) ∪ f(x) = f(x) for all x ∈ X.

Example 3.3. Let (U := Z, X) where X = 0, 1, 2, 3 is a BCC-algebra [6] with the following

table:∗ 0 1 2 3

0 0 0 0 0

1 1 0 0 1

2 2 1 0 1

3 3 3 3 0


1442 Sun Shin Ahn 1440-1450

Sun Shin Ahn

Let (f,X) be a soft set over U defined as follows:

f : X → P(U), x 7→

4Z if x = 0,

2Z if x ∈ 1, 2,Z if x = 3.

It is easy to check that (f,X) is a uni-soft subalgebra of X over U .

Theorem 3.4. A soft set (f,X) of a BCC-algebra X over U is a uni-soft subalgebra of X

over U if and only if the γ-exclusive set eX(f ; γ) is a subalgebra of X for all γ ∈ P(U) with

eX(f ; γ) = ∅.

Proof. Assume that (f,X) is a uni-soft subalgebra of X over U . Let x, y ∈ X and let γ ∈ P(U)

be such that x, y ∈ eX(f ; γ). Then f(x) ⊆ γ and f(y) ⊆ γ. It follows from (3.0) that f(x ∗ y) ⊆f(x) ∪ f(y) ⊆ γ Hence x ∗ y ∈ eX(f ; γ). Thus eX(f ; γ) is a subalgebra of X.

Conversely, suppose that eX(f ; γ) is a subalgebra X for all γ ∈ P(U) with eX(f ; γ) = ∅. Letx, y ∈ X, be such that f(x) = γx and f(y) = γy. Take γ = γx ∪ γy. Then x, y ∈ eX(f ; γ) and so

x ∗ y ∈ eX(f ; γ) by assumption. Hence f(x ∗ y) ⊆ γ = γx ∪ γy = f(x) ∪ f(y). Thus (f,X) is a

uni-soft subalgebra of X over U .

Theorem 3.5. Every subalgebra of a BCC-algebra X can be represented as a γ-exclusive set of

a uni-soft subalgebra of X over U .

Proof. Let A be a subalgebra of a BCC-algebra X. For a subset γ of U , define a soft set (f,X)

over U by

f : X → P(U), x 7→γ if x ∈ A,

U if x /∈ A.

Obviously, A = eX(f ; γ). We now prove that (f,X) is a uni-soft subalgebra of X over U . Let

x, y ∈ X. If x, y ∈ A, then x ∗ y ∈ A because A is a subalgebra of X. Hence f(x) = f(y) =

f(x ∗ y) = γ, and so f(x ∗ y) ⊆ f(x) ∪ f(y). If x ∈ A and y /∈ A, then f(x) = γ and f(y) = U

which imply that f(x ∗ y) ⊆ f(x) ∪ f(y) = γ ∪ U = U . Similarly, if x /∈ A and y ∈ A, then

f(x ∗ y) ⊆ f(x) ∪ f(y). Obviously, if x /∈ A and y /∈ A, then f(x ∗ y) ⊆ f(x) ∪ f(y). Therefore

(f,X) is a uni-soft subalgebra of X over U .

Any subalgebra of a BCC-algebra X may not be represented as a γ-exclusive set of a uni-soft

subalgebra (f,X) of X over U in general (see Example 3.6).

Example 3.6. Let E = X be the set of parameters, and let U = X be the initial universe set

where X = 0, 1, 2, 3 is a BCC-algebra as in Example 3.3. Consider a soft set (f,X) which is

given by


1443 Sun Shin Ahn 1440-1450


f : X → P(U), x 7→

0 if x = 0,

0, 3 if x ∈ 1, 2, 3.

It is easy to check that (f,X) is a uni-soft subalgebra of X over U . The γ-exclusive set of (f,X)

are described as follows:

eX(f ; γ) =

0 if γ = 0,X if γ ∈ 0, 3, 0, 2, 3, X,∅ otherwise.

The subalgebra 0, 2 cannot be a γ-exclusive set eX(f ; γ) since there is no γ ⊆ U such that

eX(f ; γ) = 0, 2.

Definition 3.7. A soft set (f,X) over U is called a union soft ideal (briefly, uni-soft ideal) of a

BCC-algebra X over U if it satisfies (3.1) and

(3.2) f(x) ⊆ f(x ∗ y) ∪ f(y) for all x, y ∈ X.

Proposition 3.8. Every uni-soft ideal of a BCC-algebra X over U is a uni-soft subalgebra of

X over U .

Proof. Put x := x ∗ y and y := x in (3.2). Then we have f(x ∗ y) ⊆ f((x ∗ y) ∗ x) ∪ f(x). Using(b2) and (3.1), we obtain f(x∗y) ⊆ f((x∗y)∗x)∪f(x) = f(0)∪f(x) ⊆ f(y)∪f(x) = f(x)∪f(y)for all x, y ∈ X. Hence (f,X) is a uni-soft subalgebra of X over U .

Theorem 3.9. A soft set (f,X) of a BCC-algebra X over U is a uni-soft ideal of X over U if

and only if the γ-exclusive set eX(f ; γ) is a ideal of X for all γ ∈ P(U) with eX(f ; γ) = ∅.

Proof. Similar to Theorem 3.4.

Proposition 3.10 Every uni-soft ideal (f,X) of a BCC-algebra over U satisfies the following

properties:

(i) (∀x ∈ X)(x ≤ y ⇒ f(x) ⊆ f(y)),

(ii) (∀x, y, z ∈ X)(x ∗ y ≤ z ⇒ f(x) ⊆ f(y) ∪ f(z)).

Proof. (i) Let x, y ∈ X be such that x ≤ y. Then x ∗ y = 0. It follows from (3.2) and (3.1) that

f(x) ⊆ f(x ∗ y) ∪ f(y) = f(0) ∪ f(y) = f(y).

(ii) Let x, y, z ∈ X be such that x ∗ y ≤ z. By (3.2) and (3.1), we have f(x ∗ y) ⊆ f((x ∗ y) ∗ z)∪f(z) = f(0) ∪ f(z) = f(z). Hence f(x) ⊆ f(x ∗ y) ∪ f(y) ⊆ f(z) ∪ f(y) = f(y) ∪ f(z).

The following corollary is easily proved by induction.

Corollary 3.11. Every uni-soft ideal of aBCC-algebraX over U satisfies the following condition:

(3.3) (· · · (x ∗ a1) ∗ · · · ) ∗ an = 0 ⇒ f(x) ⊆∪n

k=1 f(ak) for all x, a1, · · · , an ∈ X.


1444 Sun Shin Ahn 1440-1450

Sun Shin Ahn

Theorem 3.12. If (f,X) and (g,X) are uni-soft ideals of a BCC-algebra X over U , then the

union (f,X)∪(g,X) of (f,X) and (g,X) is a uni-soft ideal of X over U .

Proof. For any x ∈ X, we have (f ∪g)(0) = f(0) ∪ g(0) ⊆ f(x) ∪ g(x) = (f ∪g)(x). Let

x, y ∈ X. Then we have (f ∪g)(x) = f(x) ∪ g(x) ⊆ (f(x ∗ y) ∪ f(y)) ∪ (g(x ∗ y) ∪ g(y)) =

(f(x ∗ y) ∪ g(x ∗ y)) ∪ (f(y) ∪ g(y)) = (f ∪g)(x ∗ y) ∪ (f ∪g)(y). Hence (f,X)∪(g,X) is a uni-soft

ideal of X over U .

The soft intersection of uni-soft ideals of a BCC-algebra X may not be a uni-soft ideal of X

over U (see Example 3.13).

Example 3.13. Let E = X be the set of parameters, and let U := Z be the initial universe set

where X = 0, 1, 2, 3 is a BCC-algebra with the following table:

∗ 0 1 2 3

0 0 0 0 0

1 1 0 0 1

2 2 1 0 2

3 3 3 3 0

Let (f,X) and (g,X) be soft sets over U = Z defined as follows:

f : X → P(U), x 7→

9Z if x ∈ 0, 1, 2,3Z if x ∈ 2, 3,

and

g : X → P(U), x 7→

12Z if x = 0,

3Z if x = 3,

Z if x ∈ 1, 2.

Then (f,X) and (g,X) are uni-soft ideals of X over U . But (f,X)∩(g,X) is not a uni-soft

ideal of X over U , since (f ∩g)(2) = f(2) ∩ g(2) = 3Z ∩ Z = 3Z ⊈ (f ∩g)(2 ∗ 1) ∪ (f ∩g)(1) =

(f(1) ∩ g(1)) ∪ (f(1) ∩ g(1)) = f(1) ∩ g(1) = 9Z ∩ Z = 9Z.

Definition 3.14. A soft set (f,X) over U is called a union soft BCC-ideal (briefly, uni-soft

BCC-ideal) of a BCC-algebra X over U if it satisfies (3.1) and

(3.4) f(x ∗ z) ⊆ f((x ∗ y) ∗ z) ∪ f(y) for all x, y, z ∈ X.

Lemma 3.15. Every uni-soft BCC-ideal of a BCC-algebra X over U is a uni-soft ideal of X

over U .

Proof. Put z := 0 in (3.4). Using (a3), we have f(x∗0) = f(x) ⊆ f((x∗y)∗0)∪f(y) = f(x∗y)∪f(y)for all x, y ∈ X. Hence (f,X) is a uni-soft ideal of X over U .


1445 Sun Shin Ahn 1440-1450


Corollary 3.16. Every uni-soft BCC-ideal of a BCC-algebra X over U is a uni-soft subalgebra

of X over U .

The converse of Proposition 3.8 and Lemma 3.15 need not be a true, in general (see Example

3.17).

Example 3.17. Let (U := Z, X) whereX = 0, 1, 2, 3, 4 is a BCC-algebra [4] with the following

table:∗ 0 1 2 3 4

0 0 0 0 0 0

1 1 0 1 0 0

2 2 2 0 0 0

3 3 3 1 0 0

4 4 3 4 3 0


f : X → P(U), x 7→

3Z if x ∈ 0, 1, 2, 3,Z if x = 4.

It is easy to check that (f,X) is a uni-soft subalgebra of X over U , but not a uni-soft ideal of

X over U , since f(4) = Z ⊈ f(4 ∗ 3) ∪ f(3) = f(3) ∪ f(3) = 3Z. Consider a uni-soft set (g,X)

which is given by

g : X → P(U), x 7→

2Z if x ∈ 0, 1,Z if x ∈ 2, 3, 4.

It is easy to show that (g,X) is a uni-soft ideal of X over U . But it is not a uni-soft BCC-ideal

of X over U , since g(4 ∗ 3) = g(3) = Z ⊈ g((4 ∗ 1) ∗ 3) ∪ g(1) = g(0) ∪ g(1) = 2Z.

Example 3.18. Let (U := Z, X) where X = 0, 1, 2, 3, 4, 5 is a BCC-algebra [4] with the

following table:

∗ 0 1 2 3 4 5

0 0 0 0 0 0 0

1 1 0 0 0 0 1

2 2 2 0 0 1 1

3 3 2 1 0 1 1

4 4 4 4 4 0 1

5 5 5 5 5 5 0


f : X → P(U), x 7→

5Z if x ∈ 0, 1, 2, 3, 4,Z if x = 5.

It is easy to check that (f,X) is a uni-soft BCC-ideal of X over U .


1446 Sun Shin Ahn 1440-1450

Sun Shin Ahn

Theorem 3.19. A soft set (f,X) of a BCC-algebra X over U is a uni-soft BCC-ideal of X

over U if and only if the γ-exclusive set eX(f ; γ) is a BCC-ideal of X for all γ ∈ P(U) with

eX(f ; γ) = ∅.

Proof. Suppose that (f,X) is a uni-softBCC-ideal ofX over U . Let x, y, z ∈ X and γ ∈ P([0, 1])

be such that (x ∗ y) ∗ z ∈ eX(f ; γ) and y ∈ eX(f ; γ). Then f((x ∗ y) ∗ z) ⊆ γ and f(y) ⊆ γ. It

follows from (3.1) and (3.4) that f(0) ⊆ f(x ∗ z) ⊆ f((x ∗ y) ∗ z) ∪ f(y) ⊆ γ. Hence 0 ∈ eX(f ; γ)

and x ∗ z ∈ eX(f ; γ), and therefore eX(f ; γ) is a BCC-ideal of X.

Conversely, assume that eX(f ; γ) is a BCC-ideal of X for all γ ∈ P([0, 1]) with eX(f ; γ) = ∅.For any x ∈ X, let f(x) = γ. Then x ∈ eX(f ; γ). Since eX(f ; γ) is a BCC-ideal of X, we have

0 ∈ eX(f ; γ) and so f(0) ⊆ f(x) = γ. For any x, y, z ∈ X, let f((x ∗ y) ∗ z) = γ(x∗y)∗z and

f(y) = γy. Let γ := γ(x∗y)∗z ∪ γy. Then (x ∗ y) ∗ z ∈ eX(f ; γ) and y ∈ eX(f ; γ) which imply that

x ∗ z ∈ eX(f ; γ). Hence f(x ∗ z) ⊆ γ = γ(x∗y)∗z) ∪ γy = f((x ∗ y) ∗ z) ∪ f(y). Thus (f,X) is a

uni-soft BCC-ideal of X over U .

Proposition 3.20. Let (f,X) be a uni-soft BCC-ideal of a BCC-algebra X over U . Then

Xf := x ∈ X|f(x) = f(0) is a BCC-ideal of X.

Proof. Clearly, 0 ∈ Xf . Let (x ∗ y) ∗ z, y ∈ Xf . Then f((x ∗ y) ∗ z) = f(0) and f(y) = f(0). It

follows from (3.4) that f(x ∗ z) ⊆ f((x ∗ y) ∗ z) ∪ f(y) = f(0). By (3.1), we get f(x ∗ z) = f(0).

Hence x ∗ z ∈ Xf . Therefore Xf is a BCC-ideal of X.

4. Quotient BCC-ideals induced by uni-soft BCC-ideals

Let (f,X) be a uni-soft BCC-ideal of a BCC-algebra X over U . For any x, y ∈ X, we define

a binary operation “ ∼f ” on X as follows:

x ∼f y ⇔ f(x ∗ y) = f(y ∗ x) = 0.

Lemma 4.1. The operation “ ∼f ” is an equivalence relation on a BCC-algebra X.

Proof. Obviously, ∼f is both reflexive and symmetric. Let x, y, z ∈ X be such that x ∼f y and

y ∼f z. Then f(x∗y) = f(0) = f(y∗x) and f(y∗z) = f(0) = f(z∗y). Since (x∗z)∗(y∗z) ≤ x∗yand (z∗x)∗(y∗x) ≤ z∗y, it follows from Proposition 3.10(ii) that f(x∗z) ⊆ f(y∗z)∪f(x∗y) = f(0)

and f(z ∗ x) ⊆ f(y ∗ x) ∪ f(z ∗ y) = f(0). By (3.1), we have f(x ∗ z) = f(0) = f(z ∗ x) and so

x ∼f z. Therefore “ ∼f ” is an equivalence relation on X.

Lemma 4.2. For any x, y in a BCC-algebra X, if x ∼f y, then x ∗ z ∼f y ∗ z and z ∗ x ∼f z ∗ yfor all z ∈ X.

Proof. Let x, y, z ∈ X be such that x ∼f y. Then f(x∗y) = f(0) = f(y∗x). Since (x∗z)∗(y∗z) ≤x∗y and (y∗z)∗(x∗z) ≤ y∗x, it follows from Proposition 3.10(i) that f((x∗z)∗(y∗z)) ⊆ f(x∗y) =f(0) and f((y ∗z)∗ (x∗z)) ⊆ f(y ∗x) = f(0). Thus f((x∗z)∗ (y ∗z)) = f(0) = f((y ∗z)∗ (x∗z)),and so x∗z ∼f y∗z. Since ((z∗x)∗(y∗x))∗(z∗y) = 0, we have f((z∗x)∗(z∗y)) ⊆ f(((z∗x)∗(y∗


1447 Sun Shin Ahn 1440-1450


x))∗ (z ∗ y))∪ f(y ∗x) = f(0)∪ f(y ∗x) = f(y ∗x) = f(0). Since ((z ∗ y)∗ (x∗ y))∗ (z ∗x) = 0, we

have f((z ∗y)∗ (z ∗x)) ⊆ f(((z ∗y)∗ (x∗y))∗ (z ∗x))∪f(x∗y) = f(0)∪f(x∗y) = f(x∗y) = f(0).

By (3.1), we have f((z ∗x)∗ (z ∗y)) = f(0) and f((z ∗y)∗ (z ∗x)) = f(0). Therefore x∗z ∼f y ∗zand z ∗ x ∼f z ∗ y.

Using Lemma 4.2 and the transitivity of ∼f , we have the following Lemma.

Lemma 4.3. For any x, y, u, v in a BCC-algebra X, if x ∼f y and u ∼f v, then x ∗ u ∼f y ∗ v.

By Lemmas 4.1, 4.2 and 4.3, the operation “ ∼f ” is a congruence relation on a BCC-algebra

X. Denote by fx the equivalence class containing x ∈ X, and by X/f the set of all equivalence

classes of X, i.e., fx := y ∈ X|y ∼f x and X/f := fx|x ∈ X. Define a binary operation •on X/f as follows: for all fx, fy ∈ X/f, fx • fy := fx∗y. Then this operation is well-defined by

Lemma 4.3.

Theorem 4.4. If (f,X) is a uni-soft BCC-ideal of a BCC-algebra X over U , then the quotient

X/f := (X/f, •, f0) is a BCC-algebra.

Proof. Straightforward. Proposition 4.5. Let µ : (X, ∗, 0X) → (Y, ∗, 0Y ) be an epimorphism of BCC-algebras. If (g, Y )

is a uni-soft BCC-ideal of Y over U , then (g µ,X) is a uni-soft BCC-ideal of X over U .

Proof. For any x ∈ X, we have (g µ)(0) = g(µ(0X)) = g(0Y ) ⊆ g(µ(x)) = (g µ)(x). For any

x, y ∈ X, we have (gu)(x∗z) = g(µ(x∗z)) = g(µ(x)∗µ(z)) ⊆ g((µ(x)∗Y a)∗µ(z))∪g(a) for anya ∈ Y . Let y be any preimage of a under µ. Then (g µ)(x ∗ z) ⊆ g((µ(x) ∗Y a) ∗ µ(z)) ∪ g(a) =g((µ(x)∗Y µ(y))∗µ(z))∪g(µ(y)) = g(µ((x∗X y)∗X z))∪g(µ(y)) = (gµ)((x∗X y)∗X z)∪(gµ)(y).Hence g µ is a uni-soft BCC-ideal of X over U . Theorem 4.6. Let µ : (X, ∗, 0X) → (Y, ∗, 0Y ) be an epimorphism of BCC-algebras. If (g, Y ) is a

uni-soft BCC-ideal of Y over U , then the quotient algebra X/(g µ) := (X/(g µ), •X , (g µ)0X )is isomorphic to the quotient algebra Y/g := (Y/g, •Y , g0Y ).

Proof. By Theorem 4.4 and Proposition 4.5, X/(g µ) := (X/(g µ), •X , (g µ)0X ) and and

Y/g := (Y/g, •Y , g0Y ) are BCC-algebras. Define a map

η : X/(g µ) → Y/g, (g µ)x 7→ gµ(x)

for all x ∈ X. Then the function η is well-defined. In fact, assume that (g µ)x = (g µ)y for all

x, y ∈ X. Then we have g(µ(x)∗Y µ(y)) = g(µ(x∗Xy)) = (gµ)(x∗Xy) = (gµ)(0X) = g(µ(0X)) =

g(0Y ) and g(µ(y) ∗Y µ(x)) = g(µ(y ∗X x)) = (g µ)(y ∗X x) = (g µ)(0X) = g(µ(0X)) = g(0Y ).

Hence gµ(x) = gµ(y).

For any (g µ)x, (g µ)y ∈ X/(g µ), we have η((g µ)x •X (g µ)y) = η((g µ)x∗y) =

gµ(x∗Xy) = gµ(x)∗Y µ(y) = gµ(x) •Y gµ(y) = η((g µ)x)•Y η((g µ)y)). Therefore η is a homomorphism.

Let ga ∈ Y/g. Then there exists x0 ∈ X such that µ(x0) = a since µ is surjective. Hence

η((g µ)x0) = gµ(x0) = ga and so η is surjective.


1448 Sun Shin Ahn 1440-1450

Sun Shin Ahn

Let x, y ∈ X be such that gµ(x) = gµ(y). Then we have (g µ)(x ∗X y) = g(µ(x ∗X y)) =

g(µ(x) ∗Y µ(y)) = g(0Y ) = g(µ(0X)) = (g µ)(0X) It follows that (g µ)x = (g µ)y. Thus η is

injective. This completes the proof.

The homomorphism π : X → X/g, x → gx, is called the natural homomorphism of X onto

X/g. In Theorem 4.6, if we define natural homomorphisms πX : X → X/g µ and πY : Y → Y/g

then it is easy to show that η πX = πY µ, i.e., the following diagram commutes.

Xµ−−−→ Y

πX

y πY

yX/(g µ) η−−−→ Y/g.

Proposition 4.7. Let (f,X) be a uni-soft BCC-ideal of a BCC-algebra X over U . The mapping

γ : X → X/f , given by γ(x) := fx, is a surjective homomorphism, and Kerγ = x ∈ X|γ(x) =f0 = Xf .

Proof. Let fx ∈ X/f . Then there exists an element x ∈ X such that γ(x) = fx. Hence γ is

surjective. For any x, y ∈ X, we have γ(x ∗ y) = fx∗y = fx • fy = γ(x) • γ(y). Thus γ is a

homomorphism. Moreover, Ker γ = x ∈ X|γ(x) = f0 = x ∈ X|x ∼f 0 = x ∈ X|f(x) =f(0) = Xf .

Proposition 4.8. Let (f,X) be a uni-soft BCC-ideal of a BCC-algebra X over U . If J is a

BCC-ideal of X, then J/f is a BCC-ideal of X/f .

Proof. Let (f,X) be a uni-soft BCC-ideal of X over U and let J be a BCC-ideal of X. Since

0 ∈ J , we have f0 ∈ J/f . For any x, y, z ∈ J , (x ∗ y) ∗ z ∈ J and y ∈ J , we get x ∗ z ∈ J . Let

(fx • fy) • fz, fy ∈ J/f . Then (fx • fy) • fz = f(x∗y)∗z ∈ J/f and fy ∈ J/f imply fx • fz ∈ J/f .

Thus J/f is a BCC-ideal of X/f .

Theorem 4.9. Let (f,X) be a uni-soft BCC-ideal of a BCC-algebra X over U . If J∗ is a

BCC-ideal of a BCC-algebra X/f , then there exists a BCC-ideal J = x ∈ X|fx ∈ J∗ in X

such that J/f = J∗.

Proof. Since J∗ is a BCC-ideal of X/f , (fx • fy) • fz = f(x∗y)∗z, fy ∈ J∗ imply fx • fz = fx∗z ∈ J∗

for any fx, fy, fz ∈ X/f . Thus (x∗y)∗z, y ∈ J imply x∗z ∈ J for any x, y, z ∈ X. Therefore J is

a BCC-ideal of X. By Proposition 4.8, we have J/f = fj|j ∈ J = fj|∃fx ∈ J∗ such that j ∼f

x = fj|∃fx ∈ J∗ such that fx = fj = fj|fj ∈ J∗ = J∗.

Theorem 4.10. Let (f,X) be a uni-soft BCC-ideal of a BCC-algebra X over U . If J is a

BCC-ideal of X, thenX/f

J/f∼= X/J .


1449 Sun Shin Ahn 1440-1450


Proof. Note thatX/f

J/f= [fx]J/f |f ∈ X/f. If we define φ :

X/f

J/f→ X/J by φ([fx]J/f ) =

[x]J = y ∈ X|x ∼J y, then it is well defined. In fact, suppose that [fx]J/f = [fy]J/f . Then

fx ∼J/f fy and so fx∗y = fx • fy ∈ J/f and fy∗x = fy • fx ∈ J/f . Hence x ∗ y ∈ J and y ∗ x ∈ J .

Therefore x ∼J y, i.e., [x]J = [y]J . Given [fx]J/f , [fy]J/f ∈ X/f

J/f, we have φ([fx]J/f • [fy]J/f ) =

φ([fx • fy]J/f ) = [x ∗ y]J = [x]J ∗ [y]J = φ([fx]J/f ) ∗ φ([fy]J/f ). Hence φ is a homomorphism.

Obviously, φ is onto. Finally, we show that φ is one-to-one. If φ([fx]J/f ) = φ([fy]J/f ), then

[x]J = [y]J , i.e., x ∼J y. If fa ∈ [fx]J/f , then fa ∼J/f fx and hence fa∗x, fx∗a ∈ J/f . It follows

that a ∗ x, x ∗ a ∈ J , i.e., a ∼J x. Since ∼J is an equivalence relation, a ∼J y and so Ja = Jy.

Hence a ∗ y, y ∗ a ∈ J and so fa∗y, fy∗a ∈ J/f . Therefore fa ∼J/f fy. Hence fa ∈ [fy]J/f .

Thus [fx]J/f ⊆ [fy]J/f . Similarly, we obtain [fy]J/f ⊆ [fx]J/f . Therefore [fx]J/f = [fy]J/f . This

completes the proof.

References

[1] S. S. Ahn and S. H. Kwon, Toplogical properties in BCC-algerbras, Commun. Korean Math. Soc. 23(2)

(2008), 169-178.

[2] S. S. Ahn, J. M. Ko and K. S. So, Union soft p-ideals and union soft sub-implicative ideals in BCI-algebbras,

J. Comput. Anal. Appl. 23(1) (2017), 152-165.

[3] W. A. Dudek, On constructions of BCC-algebras, Selected Papers on BCK- and BCI-algebras 1 (1992),

93-96.

[4] W. A. Dudek and X. Zhang, On ideals and congruences in BCC-algeras, Czecho Math. J. 48 (1998), 21-29.

[5] J. S. Han and S. S. Ahn, Applicationa of soft sets to q-ideals and a-ideals in BCI-algebras, J. Computational

Analysis and Applications 17(3) (2014), 10-21.

[6] J. Hao, Ideal Theory of BCC-algebras, Sci. Math. Japo. 3 (1998), 373-381.

[7] Y. S. Hwang and S. S. Ahn, Soft q-ideals of soft BCI-algebras, J. Comput. Anal. Appl. 16(3) (2014), 571-582.

[8] Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras, Bull. Korean Math. Soc. 50 (2013),

1937-1956.

[9] Y. B. Jun, S. Z. Song and S. S. Ahn, Union soft sets applied to commutative BCI-algebras, J. Comput. Anal.

Appl. 16(3) (2014), 468-477.

[10] Y. Kormori, The class of BCC-algebras is not a varity, Math. Japo. 29 (1984), 391-394.

[11] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003), 555-562.

[12] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput.

Math. Appl. 44 (2002), 1077-1083.

[13] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999), 19-31.

[14] K. S. Yang and S. S. Ahn, Union soft q-ideals in BCI-algebras, Applied Mathematical Sciences 8 (2014),

2859-2869.

[15] L. A. Zadeh, From circuit theory to system theory, Proc. Inst. Radio Eng. 50 (1962), 856-865.

[16] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), 338-353.

[17] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) - an outline, Inform. Sci. 172 (2005), 1-40.


1450 Sun Shin Ahn 1440-1450

FIXED POINT THEOREMS FOR VARIOUS CONTRACTION CONDITIONS

IN DIGITAL METRIC SPACES

CHOONKIL PARK1, OZGUR EGE2∗, SANJAY KUMAR3, DEEPAK JAIN4, JUNG RYE LEE5∗

Abstract. In this paper, we prove the existence of fixed points for Kannan contraction, Chat-terjea contraction and Reich contraction in setting of digital metric spaces. These digital con-tractions are the applications of metric fixed point theory contractions.

1. Introduction

The basic tool of metric fixed point theory is the Banach contraction principle, which statesthat “Let T be a mapping from a complete metric space (X, d) into itself satisfying

d(Tx, Ty) ≤ αd(x, y)

for all x, y ∈ X, where 0 ≤ α < 1. Then T has a unique fixed point.”This principle gives existence and uniqueness of fixed points and methods for obtaining ap-

proximate fixed points. This principle was generalized by several authors by using differenttypes of minimal commutative along with continuity one of the mappings. In finding commonfixed point generally we include the following steps:

(i) A commutative type condition,(ii) Completeness of the space or completeness of the range space of one or more mappings,(iii) A relation between the ranges of mappings,(iv) Continuity of one or more mappings,(v) A contractive type condition.

This principle was further generalized by using different types of properties such as E.A.property, Common Limit Range property along with containment of range spaces instead ofcontinuity of mappings.

The topological fixed point theory involves the study of spaces with the fixed point property.Moreover, topology is the study of geometric problems that does not depend only on the exactshape of the objects, but rather it acts on a space. In topology, generally we consider infinitelymany points in arbitrary small neighborhood of a point. To consider finite number of points ina neighborhood, the concept of digital topology was introduced by Rosenfeld [13].

In fact, digital topology is the study of geometric and topological properties of digital imageusing geometric and algebraic topology. The digital images have been used in computer sciencessuch as image processing, computer graphics. For detail one can refer to [1, 8, 11]. Digitaltopology also provides a mathematical basis for image processing operations. Further, digitaltopology is a developing area in 2D and 3D digital images. For a difference in general topologyand digital topology, see Figure 1.

2010 Mathematics Subject Classification. Primary 47H10; Secondary 54E35, 68U10.Key words and phrases. digital image, fixed point, digital contraction, digital continuity.∗Corresponding authors.


1451 CHOONKIL PARK ET AL 1451-1458

C. PARK, O. EGE, S. KUMAR, D. JAIN, J. LEE

Figure 1. Neighboorhood in general and digital topology

The elements of 2D image array are called pixels and the elements of 3D image array are calledvoxels. Each pixel or voxel is associated with lattice points (A point with integer coordinate)in the plane or in 3D-space. A lattice point associated with a pixel or voxel has values 0 and 1.The pixel or voxel that has value 0 is called a black point and the pixel or voxel that has value1 is called a white point.

2. Topological structure of digital metric spaces

Let Zn, n ∈ N, be the set of points in the Euclidean n-dimensional space with integer coordi-nates.

Definition 2.1. [4]. Let l, n be positive integers with 1 ≤ l ≤ n. Consider two distinct points

p = (p1, p2, ..., pn), q = (q1, q2, ..., qn) ∈ Zn.

The points p and q are kl-adjacent if there are at most l indices i such that |pi − qi| = 1, andfor all other indices j, |pj − qj | 6= 1, pj = qj .

(i) Two points p and q in Z are 2-adjacent if |p− q| = 1 (see Figure 2).

Figure 2. 2-adjacency

(ii) Two points p and q in Z2 are 8-adjacent if the points are distinct and differ by at most1 in each coordinate.

(iii) Two points p and q in Z2 are 4-adjacent if the points are 8-adjacent and differ in exactlyone coordinate (see Figure 3).

(iv) Two points p and q in Z3 are 26-adjacent if the points are distinct and differ by at most1 in each coordinate.

(v) Two points p and q in Z3 are 18-adjacent if the points are 26-adjacent and differ by atmost 2 coordinates.



DIGITAL METRIC SPACES

Figure 3. 4-adjacency and 8-adjacency

(vi) Two points p and q in Z3 are 26-adjacent if the points are 18-adjacent and differ inexactly one coordinate (see Figure 4).

Figure 4. Adjacencies in Z3

One can easily note that the coordination number of Na in the crystal structure of NaCl is 6which is equal to adjacency relation in digital images of Figure 5.

Figure 5. Crystal structure of NaCI

Definition 2.2. A digital image is a pair (X,κ), where ∅ 6= X ⊂ Zn for some positive integer nand κ is an adjacency relation on X. Technically, a digital image (X,κ) is an undirected graphwhose vertex set is the set of members of X and whose edge set is the set of unordered pairsx0, x1 ⊂ X such that x0 6= x1 and x0 and x1 are κ-adjacent.




The notion of digital continuity in digital topology was developed by Rosenfeld [14] for study-ing 2D and 3D digital images. Boxer [2] gave the digital version of several notions of topologyand Ege and Karaca [6] studied various digital continuous functions.

Let N and R denote the sets of natural numbers and real numbers, respectively. Boxer [3]defined a κ-neighbor of p ∈ Zn which is a point of Zn that is κ-adjacent to p where κ ∈2, 4, 6, 8, 18, 26 and n ∈ 1, 2, 3. The set

Nκ(p) = q | q is κ− adjacent to pis called the κ-neighborhood of p. Boxer [2] defined a digital interval as

[a, b]Z = z ∈ Z | a ≤ z ≤ b,where a, b ∈ Z and a < b. A digital image X ⊂ Zn is κ-connected [9] if and only if for every pairof different points x, y ∈ X, there is a set x0, x1, . . . , xr of points of a digital image X suchthat x = x0, y = xr and xi and xi+1 are κ-neighbors where i = 0, 1, . . . , r − 1.

Definition 2.3. Let (X,κ0) ⊂ Zn0 and (Y, κ1) ⊂ Zn1 be digital images and f : X → Y be afunction.

(i) If for every κ0-connected subset U of X, f(U) is a κ1-connected subset of Y , then f issaid to be (κ0, κ1)-continuous [3].

(ii) f is (κ0, κ1)-continuous if for every κ0-adjacent points x0, x1 of X, either f(x0) = f(x1)or f(x0) and f(x1) are κ1-adjacent in Y [3].

(iii) If f is (κ0, κ1)-continuous, bijective and f−1 is (κ1, κ0)-continuous, then f is called(κ0, κ1)-isomorphism and denoted by X ∼=(κ0,κ1) Y .

Now we start with digital metric space (X, d, κ) with κ-adjacency where d is usual Euclideanmetric for Zn as follows.

Definition 2.4. [6] Let (X,κ) be a digital image set. Let d be a function from (X,κ)×(X,κ)→Zn satisfying all the properties of metric space. The triplet (X, d, κ) is called a digital metricspace.

Proposition 2.5. [8] A sequence xn of points of a digital metric space (X, d, κ) is a Cauchysequence if and only if there is α ∈ N such that d(xn, xm) 1 for all n,m α.

Theorem 2.6. [8] For a digital metric space (X, d, κ), if a sequence xn ⊂ X ⊂ Zn is a Cauchysequence then there is α ∈ N such that we have xn = xm for all n,m α.

Proposition 2.7. [8] A sequence xn of points of a digital metric space (X, d, κ) converges toa limit l ∈ X if for all ε 0, there is α ∈ N such that d(xn, l) ε for all n α.

Proposition 2.8. [8] A sequence xn of points of a digital metric space (X, d, κ) converges toa limit l ∈ X if there is α ∈ N such that xn = l for all n α.

Theorem 2.9. [8] A digital metric space (X, d, κ) is complete.

Definition 2.10. [6] Let (X, d, κ) be any digital metric space. A self map f on a digital metricspace is said to be a digital contraction if there exists a λ ∈ [0, 1) such that for all x, y ∈ X,

d(f(x), f(y)) ≤ λd(x, y).

Proposition 2.11. [6] Every digital contraction map f : (X, d, κ) → (X, d, κ) is digitally con-tinuous.

Proposition 2.12. [8] In a digital metric space (X, d, κ), consider two points xi, xj in a sequencexn ⊂ X such that they are κ-adjacent. Then they have the Euclidean distance d(xi, xj) which

is greater than or equal to 1 and at most√t depending on the position of the two points.




3. Main results

In 2015, Ege and Karaca [6] proved Banach contraction principle in the setting of digitalmetric spaces. With the motivation of Banach contraction principle in digital metric spaces, weprove Kannan, Chatterjea and Reich contraction fixed point theorems in the setting of digitalmetric spaces.

The following theorem is the digital version of Kannan contraction fixed point theorem [10].

Theorem 3.1. Let (X,κ) be a digital image where X ⊂ Zn and κ is an adjacency relationbetween the objects of X. Let (X, d, κ) be a digital metric space and S be a self map on Xsatisfying the following:

d(Sx, Sy) ≤ αd(x, Sx) + d(y, Sy) (3.1)

for all x, y ∈ X and 0 < α < 12 . Then S has a unique fixed point in X.

Proof. Let x0 ∈ X and consider the iterate of sequence xn+1 = Sxn. Now

d(x1, x2) = d(Sx0, Sx1) ≤ αd(x0, Sx0) + d(x1, Sx1),i.e.,

d(x1, x2) ≤α

1− αd(x0, x1).

Similarly, we have

d(x2, x3) ≤α

1− αd(x1, x2)

≤ (α

1− α)2d(x0, x1)

and so ond(xn, xn+1) ≤ (

α

1− α)nd(x0, x1),

d(xn+1, xn+2) ≤ (α

1− α)n+1d(x0, x1).

Let β = α1−α . Then we can rewrite the above statement as follows:

d(xn, xn+1) ≤ βnd(x0, x1),

d(xn+1, xn+2) ≤ βn+1d(x0, x1).

If we use the triangle inequality repeatedly, then we obtain the following:

d(xn, xn+k) ≤ d(xn, xn+1) + d(xn+1, xn+2) + . . .+ d(xn+k−1, xn+k)

≤ (βn + βn+1 + . . .+ βn+k−1)d(x0, x1)

≤ βn

1− βd(x0, x1).

Since 0 ≤ β < 1, βn

1−βd(x0, x1)→ 0 as n→∞. This implies that the sequence xn is a Cauchy

sequence in (X, d, κ). By Theorem 2.9, there exists a limit point v and due to (κ, κ)-continuityof S, we have

S(v) = limn→∞

S(xn) = limn→∞

xn+1 = v.

Therefore, S has a fixed point.Now we show that S has a unique fixed point. If a and b are fixed points of S, then

d(a, b) = d(Sa, Sb) ≤ αd(a, Sa) + d(b, Sb)= d(a, a) + d(b, b) = 0.

As a result, d(a, b) = 0 and so a = b.




Now we prove the digital version of Chatterjea fixed point theorem [5] as follows:

Theorem 3.2. Let (X,κ) be a digital image where X ⊂ Zn and κ is an adjacency relation inX. Let (X, d, κ) be a digital metric space and S be a self map on X satisfying the following:

d(Sx, Sy) ≤ αd(x, Sy) + d(y, Sx)

for all x, y ∈ X and 0 < α < 12 . Then S has a unique fixed point in X.

Proof. Let x0 ∈ X and consider the iterate sequence xn = Sxn−1. Now

d(x1, x2) = d(Sx0, Sx1) ≤ αd(x0, Sx1) + d(x1, Sx0)≤ αd(x0, x2) + d(x1, x1)≤ αd(x0, x1) + d(x1, x2),

i.e.,(1− α)d(x1, x2) ≤ αd(x0, x1),

d(x1, x2) ≤α

1− αd(x0, x1).

In a similar way, we get the following:

d(x2, x3) ≤α

1− αd(x1, x2) ≤ (

α

1− α)2d(x0, x1),

d(xn, xn+1) ≤ (α

1− α)nd(x0, x1),

d(xn+1, xn+2) ≤ (α

1− α)n+1d(x0, x1).

If we take β = α1−α , then we obtain

d(xn, xn+1) ≤ βnd(x0, x1),

d(xn+1, xn+2) ≤ βn+1d(x0, x1).

From the triangle inequality, we conclude


≤ (βn + βn+1 + . . .+ βn+k−1)d(x0, x1)

≤ βn

1− βd(x0, x1).


1−βd(x0, x1)→ 0 as n→∞. This implies that the sequence xn is a Cauchy

sequence in (X, d, κ) and (X, d, κ) is a digital complete metric space. So there is a limit point uand by the (κ, κ)-continuity of S, we have

S(u) = limn→∞

S(xn) = limn→∞

xn+1 = u.

Therefore, S has a fixed point.To show the uniqueness, let a and b be fixed points of S. Then from the hypothesis, we get

d(a, b) = d(Sa, Sb) ≤ αd(a, Sa) + d(b, Sb)= d(a, a) + d(b, b) = 0.

As a result, d(a, b) = 0 and so a = b.

Reich fixed point theorem [12] can be given as follows in digital images.




Theorem 3.3. If S is a mapping on a digital metric space (X, d, κ) into itself satisfying thefollowing

d(Sx, Sy) ≤ ad(x, Sx) + bd(y, Sy) + cd(x, y)

for all x, y ∈ X and all nonnegative real numbers a, b, c with a+ b+ c < 1. Then S has a uniquefixed point in X.

Proof. Let x0 ∈ X. Defining the sequence xn+1 = Sxn, we get the following:

d(x1, x2) = d(Sx0, Sx1) ≤ ad(x0, Sx0) + bd(x1, Sx1) + cd(x0, x1)

≤ ad(x0, x1) + bd(x1, x2) + cd(x0, x1)

≤ a+ c

1− bd(x0, x1).

Similarly, we haved(xn, xn + 1) ≤ βnd(x0, x1),

where β = a+c1−b and β < 1. The triangle inequality gives the following:


≤ (βn + βn+1 + . . .+ βn+k−1)d(x0, x1)

≤ βn

1− βd(x0, x1).


1−βd(x0, x1) as n → ∞. Then we can say that xn is a Cauchy sequence in

(X, d, κ). There exists a limit point w such that

S(w) = limn→∞

S(xn) = limn→∞

xn+1 = w

by the completeness of (X, d, κ). Hence S has a fixed point. It can be easily shown that thisfixed point is unique.

We give an example about Theorem 3.1.

Example 3.4. Consider the minimal simple closed 18-surface MSS′18 = ci : i ∈ [0, 5]Z (see

Figure 6).

Figure 6. MSS′18 [7]

Let S : MSS′18 → MSS

′18 be any digital map satisfying the inequality (3.1). Consider a point

such as c0 in MSS′18 and take S(c0) = c′ ∈MSS

′18. For the point ci ∈ N18(c0, 1), i ∈ 1, 3, 4, 5,

we have

d(S(ci), S(c0)) ≤ αd(ci, S(ci)) + d(c0, S(c0)) ≤ α√

2 +√

2 = 2√

2α




since the maximum distance between different 18-adjacent points in MSS′18 is√

2 by Proposition

2.12. Since 0 < α < 12 , we get d(S(ci), S(c0))

√2. As a result, d(S(ci), S(c0)) = 0 implies

that S(ci) = S(c0) = c′ from the property of MSS′18. This procedure can be applied all points

in MSS′18 since c0 is an arbitrary point. Therefore, S is a constant map. By Theorem 3.1, we

can say that S has a fixed point.

References

1. G. Bertrand, Simple points, topological numbers and geodesic neighborhoods in cubic grids, Pattern RecognitionLetters, 15 (1994), 1003–1011.

2. L. Boxer, Digitally continuous functions, Pattern Recognition Letters, 15 (1994), 833–839.3. L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vis., 10 (1999), 51–62.4. L. Boxer, Digital products, wedges and covering spaces, J. Math. Imaging Vis., 25 (2006), 159–171.5. S.K. Chatterjea, Fixed point theorems, C.R. Acad. Bulgare Sci., 25 (1972), 727–730.6. O. Ege and I. Karaca, Banach fixed point theorem for digital images, J. Nonlinear Sci. Appl., 8 (2015),

237–245.7. S.E. Han, Connected sum of digital closed surfaces, Inform. Sci., 176 (2006), 332–348.8. S.E. Han, Banach fixed point theorem from the viewpoint of digital topology, J. Nonlinear Sci. Appl., 9 (2015),

895–905.9. G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing, 55 (1993),

381–396.10. R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71–76.11. T. Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Sci., Amsterdam,

1996.12. S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971), 121–124.13. A. Rosenfeld, Digital topology, Amer. Math. Monthly, 86 (1979), 76–87.14. A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters, 4 (1986), 177–184.

1Research Institute for Natural Sciences, Hanyang University, Seoul 04763, KoreaE-mail address: [email protected]

2Department of Mathematics, Celal Bayar University, Muradiye Campus, Yunusemre, 45140,Manisa, Turkey


3Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology,Murthal, Sonepat, Haryana, India


4Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology,Murthal, Sonepat, Haryana, India


5Department of Mathematics, Daejin University, Kyunggi 11159, KoreaE-mail address: [email protected]



Fixed points of Cırıc type ordered F -contractions on partial metric spaces

Muhammad Nazam1, Muhammad Arshad1, Choonkil Park2∗ and Sungsik Yun3∗

1Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad, Pakistane-mail: [email protected]

2Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Koreae-mail: [email protected]

3Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Koreae-mail: [email protected]

Abstract. In this paper, by considering both F -contraction and fixed point results on ordered partial metric

spaces, we introduce a pair of Cırıc type ordered F -contractions on an ordered partial metric space. Then we

give a common fixed point theorem for such contractions. We give an example showing that our main theorem

is applicable, but both results of Durmaz et al. [11] and Wardowski [18] are not. We also discuss that this fixed

point result can be applied to show the existence of solution of an integral equation.

1. Introduction

Matthews [12] introduced the concept of partial metric spaces and proved an analogue of

Banach fixed point theorem in partial metric spaces. In fact, a partial metric space is a gener-

alization of metric space in which the self distances p(r1, r1) of elements of a space may not be

zero and follows the inequality p(r1, r1) ≤ p(r1, r2). After this remarkable contribution, many

authors took interest in partial metric spaces and its topological properties and presented several

well known fixed point results in the framework of partial metric spaces (see [1, 2, 5, 6, 7, 8, 14]

and references therein).

Banach presented a landmark fixed point result (Banach Contraction Principle). This result

proved a gateway for the fixed point researchers and opened a new door in metric fixed point

theory. A number of efforts have been made to enrich and generalize Banach Contraction Prin-

ciple (see [9, 10] and references therein). Following Banach, in 2012, Wardowski [18] presented

a new contraction (known as F -contraction). Since 2012, a number of fixed point results have

been established by using F -contraction or ordered F -contraction (see [3, 11, 13, 15, 17]).

Wardowski [18] presented the concept of F -contraction. Then some generalizations of F -

contractions including multivalued case are obtained in [4, 3]. In this article, we prove a common

fixed point theorem for a pair ordered F -contractions in complete partial metric spaces. An

example is constructed to illustrate this result and to show that our result generalizes the result

established by Durmaz et al. [11]. We apply the mentioned theorem to show the existence of

solution of implicit type integral equations.

02010 Mathematics Subject Classification: 47H09; 47H10; 54H25.0Keywords: common fixed point; Cırıc type ordered F -contraction, partial metric space.

∗Corresponding authors.


1459 Muhammad Nazam ET AL 1459-1470

2. Preliminaries

Throughout this paper, we denote (0,∞) by R+, [0,∞) by R+0 , (−∞,+∞) by R and the set of

natural numbers by N. Following concepts and results will be required for the proofs of main

results.

Definition 1. [18] A mapping T : M → M is said to be an F -contraction if it satisfies the

following condition

(d(T (r1), T (r2)) > 0⇒ τ + F (d(T (r1), T (r2)) ≤ F (d(r1, r2))) (2.1)

for all r1, r2 ∈ M and some τ > 0. Here F : R+ → R is a function satisfying the following

properties.

(F1) : F is strictly increasing.

(F2) : For each sequence rn of positive numbers limn→∞ rn = 0 if and only if limn→∞ F (rn) =

−∞.(F3) : There exists θ ∈ (0, 1) such that limα→0+(α)θF (α) = 0.

Wardowski [18] established the following result using F -contraction.

Theorem 1. [18] Let (M,d) be a complete metric space and T : M →M be an F -contraction.

Then T has a unique fixed point υ ∈ M and for every r0 ∈ M the sequence Tn(r0) for all

n ∈ N is convergent to υ.

Recently, Durmaz et al. [11] presented an ordered version of Theorem 1.

Theorem 2. Let (M,, d) be an ordered complete metric space and T : M →M be an ordered

F -contraction. Let T be a nondecreasing mapping and there exists r0 ∈M such that r0 T (r0).

If T is continuous or M is regular, then T has a fixed point.

We denote by ∆F the set of all functions satisfying the conditions (F1)− (F3).

Example 1. [18] Let F : R+ → R be given by the formula F (α) = lnα. It is clear that F satisfies

(F1) − (F3) for any κ ∈ (0, 1). Each mapping T : M → M satisfying (2.1) is an F -contraction

such that

d(T (r1), T (r2)) ≤ e−τd(r1, r2), for all r1, r2 ∈M,T (r1) 6= T (r2).

Obviously, for all r1, r2 ∈M such that T (r1) = T (r2), the inequality

d(T (r1), T (r2)) ≤ e−τd(r1, r2)

holds, that is, T is a Banach contraction.

Remark 1. From (F1) and (2.1) it is easy to conclude that every F -contraction is necessarily

continuous.

Definition 2. [12] Let M be a nonempty set and assume that the function p : M ×M → R+0

satisfies the following properties:

(p1) r1 = r2 ⇔ p (r1, r1) = p (r1, r2) = p (r2, r2) ,

(p2) p (r1, r1) ≤ p (r1, r2) ,

(p3) p (r1, r2) = p (r2, r1) ,



(p4) p (r1, r3) ≤ p (r1, r2) + p (r2, r3)− p (r2, r2)

for all r1, r2, r3 ∈ M . Then p is called a partial metric on M and the pair (M,p) is known as

partial metric space.

In [12], Matthews proved that every partial metric p on M induces a metric dp : M×M → R+0

defined by

dp (r1, r2) = 2p (r1, r2)− p (r1, r1)− p (r2, r2)

for all r1, r2 ∈M .

Notice that a metric on a set M is a partial metric p such that p(r, r) = 0 for all r ∈M and

p(r1, r2) = 0 implies r1 = r2 (using (p1) and (p2)).

Matthews [12] established that each partial metric p on M generates a T0 topology τ(p) on

M . The base of topology τ(p) is the family of open p-balls Bp (r, ε) : r ∈M, ε > 0, where

Bp (r, ε) = r1 ∈M : p (r, r1) < p (r, r) + ε for all r ∈ M and ε > 0. A sequence rnn∈N in

(M,p) converges to a point r ∈M if and only if p(r, r) = limn→∞ p(r, rn).

Definition 3. [12] Let (M,p) be a partial metric space.

(1) A sequence rnn∈N in (M,p) is called a Cauchy sequence if limn,m→∞ p(rn, rm) exists

and is finite.

(2) A partial metric space (M,p) is said to be complete if every Cauchy sequence rnn∈N in

M converges, with respect to τ(p), to a point r ∈ X such that p(r, r) = limn,m→∞ p(rn, rm).

The following lemma will be helpful in the sequel.

Lemma 1. [12]

(1) A sequence rn is a Cauchy sequence in a partial metric space (M,p) if and only if it is

a Cauchy sequence in metric space (M,dp)

(2) A partial metric space (M,p) is complete if and only if the metric space (M,dp) is

complete.

(3) A sequence rnn∈N in M converges to a point r ∈M , with respect to τ(dp) if and only

if limn→∞ p(r, rn) = p(r, r) = limn,m→∞ p(rn, rm).

(4) If limn→∞ rn = υ such that p(υ, υ) = 0 then limn→∞ p(rn, r) = p(υ, r) for every r ∈M .

In the following example, we shall show that there are mappings which are not F -contractions

in metric spaces, nevertheless, such mappings follow the conditions of F -contraction in partial

metric spaces.

Example 2. Let M = [0, 1] and define partial metric by p(r1, r2) = max r1, r2 for all r1, r2 ∈M .

The metric d induced by partial metric p is given by d(r1, r2) = |r1 − r2| for all r1, r2 ∈M . Define

F : R+ → R by F (r) = ln(r) and T by

T (r) =

r

5if r ∈ [0, 1);

0 if r = 1.



Then T is not an F -contraction in a metric space (M,d). Indeed, for r1 = 1 and r2 = 56 ,

d(T (r1), T (r2)) > 0 and we have

τ + F (d(T (r1), T (r2))) ≤ F (d(r1, r2)) ,

τ + F

(d(T (1), T (

5

6))

)≤ F

(d(1,

5

6)

),

τ + F

(d(0,

1

6)

)≤ F

(1

6

),

1

6<

1

6,

which is a contradiction for all possible values of τ . Now if we work in partial metric space

(M,p), we get a positive answer, that is,

τ + F (p(T (r1), T (r2))) ≤ F (p(r1, r2)) implies

τ + F

(1

6

)≤ F (1) ,

which is true.

Similarly, for all other points in M our claim proves true.

Definition 4. Let (M ) be a partially ordered set. Two mappings S, T : M →M are said to

be weakly increasing mappings if S(m) TS(m) and T (m) ST (m) hold for all m ∈M .

Example 3. Let M = R+ be endowed with usual order and usual topology. Let S, T : M → M

be given by

S(m) =

m

12 if m ∈ [0, 1]

m2 if m ∈ (1,∞)and T (m) =

m if m ∈ [0, 1]2m if m ∈ (1,∞).

Then the pair (S, T ) is weakly increasing mappings, where T is a discontinuous mapping.

3. Main results

We begin with the following definitions.

Definition 5. Let (M,) be an ordered set and p be a metric on M. Then the triplet (M,, p)is known as an ordered partial metric space. If (M,p) is complete, then (M,, p) is called an

ordered complete partial metric space. Moreover, M is regular if the ordered partial metric space

(M,, p) provides the following condition:If rn ⊂M is a nondecreasing (nonincreasing) sequence with rn → r,then rn r (r rn) for all n.

Definition 6. Let (M,, p) be an ordered partial metric space and S, T : M → M be two

mappings. Let

γ = (h, k) ∈M ×M : h k, p(S(h), T (k)) > 0 .We say the mappings S and T are a pair of Cırıc type ordered F -contractions if there exist

F ∈ ∆F and τ > 0 such that for all (h, k) ∈ γ,

τ + F (p(S(h), T (k))) ≤ F (M(h, k)) , (3.1)



where

M(h, k) = max

p(h, k), p(h, S(h)), p(k, T (k)),

p(k, S(h)) + p(h, T (k))

2

.

The following lemma will be useful in the sequel.

Lemma 2. Let (M,, p) be an ordered complete partial metric space and S, T be a pair of

Cırıc type ordered F -contractions. Then for each i = 0, 1, 2, 3, ... p(r2i, r2i+1) = 0 implies

p(r2i+1, r2i+2) = 0.

Proof. Let r0 ∈ M be an initial point and take r1 = S(r0) and r2 = T (r1). Then by induction

we can construct an iterative sequence rn of points in M such a way that r2i+1 = S(r2i) and

r2i+2 = T (r2i+1), where i = 0, 1, 2, . . .. We argue by contradiction that p(r2i+1, r2i+2) > 0. We

note that

M(r2i, r2i+1) = max

p(r2i, r2i+1), p(r2i, S(r2i)), p(r2i+1, T (r2i+1)),

p(r2i+1, S(r2i)) + p(r2i, T (r2i+1))

2

= max

p(r2i, r2i+1), p(r2i, r2i+1), p(r2i+1, r2i+2),

p(r2i+1, r2i+1) + p(r2i, r2i+2)

2

= max 0, p(r2i+1, r2i+2) = p(r2i+1, r2i+2).

Consider τ + F (p(r2i+1, r2i+2)) = τ + F (p(S(r2i), T (r2i+1))). From (3.1), we have

τ + F (p(r2i+1, r2i+2)) = τ + F (p(S(r2i), T (r2i+1)))

≤ F (M(r2i, r2i+1))

≤ F (p(r2i+1, r2i+2))

for all i ∈ N ∪ 0, which gives a contradiction. Hence p(r2i+1, r2i+2)) = 0.

The following theorem is one of the main results.

Theorem 3. Let (M,, p) be an ordered complete partial metric space and S, T : M →M be a

pair of Cırıc type ordered F-contractions. If S, T are two weakly increasing mappings and there

exists r0 ∈ M such that r0 S(r0), then there exists a point υ such that p(υ, υ) = 0. Assume

that either one of S, T is continuous or M is regular. Then S, T have a common fixed point.

Proof. We begin with the following observation:

M(h, k) = 0 if and only if h = k is a common fixed point of (S, T ).

Indeed, if h = k is a common fixed point of (S, T ), then T (k) = T (h) = h = k = S(k) = S(h)

and

M(h, k) = max

p(h, k), p(h, S(h)), p(k, T (k)),

p(k, S(h)) + p(h, T (k))

2

= p(h, h).

If p(h, h) > 0, then from the contractive condition (3.1), we get

τ + F (p(h, h)) = τ + F (p(S(h), T (k))) ≤ F (p(h, h)) ,

which is a contradiction. Thus p(h, h) = 0 entails M(h, h) = 0.



Conversely, if M(h, k) = 0, then it is easy to check that h = k is a common fixed point of S

and T .

If M(r1, r2) > 0 for all r1, r2 ∈ M , then by the given assumptions there exists r0 ∈ M such

that r0 S(r0). Take r1 = S(r0) and r2 = T (r1). Then by induction we can construct an

iterative sequence rn of points in M such a way that r2i+1 = S(r2i) and r2i+2 = T (r2i+1), where

i = 0, 1, 2, . . .. Since r0 S(r0) and S, T are weakly increasing mappings, we obtain

r1 = S(r0) TS(r0) = T (r1) = r2 = T (r1) ST (r1) = S(r2) = r3.

Iteratively, we obtain

r0 r1 r2 · · · rn−1 rn rn+1 · · · .

Now if p(S(r2i), T (r2i+1)) = 0, then using Lemma 2, we can conclude that r2i is a common

fixed point of S, T . If p(S(r2i), T (r2i+1)) > 0, then (r2i, r2i+1) ∈ γ, since r2i r2i+1. From the

contractive condition (3.1), we get

τ + F (p(r2i+1, r2i+2)) = τ + F (p(S(r2i), T (r2i+1)))

≤ F (M(r2i, r2i+1)) (3.2)

for all i ∈ N ∪ 0, where

M(r2i, r2i+1) = max

p(r2i, r2i+1), p(r2i, S(r2i)), p(r2i+1, T (r2i+1)),

p(r2i+1, S(r2i)) + p(r2i, T (r2i+1))

2

= max

p(r2i, r2i+1), p(r2i, r2i+1), p(r2i+1, r2i+2),

p(r2i+1, r2i+1) + p(r2i, r2i+2)

2

= max p(r2i, r2i+1), p(r2i+1, r2i+2) .

If M(r2i, r2i+1) = p(r2i+1, r2i+2), then due to (F1) and (3.2), we get a contradiction. Thus,

for M(r2i, r2i+1) = p(r2i, r2i+1), we have

F (p(r2i+1, r2i+2)) ≤ F (p(r2i, r2i+1))− τ (3.3)

for all i ∈ N ∪ 0. Also since r2i+1 r2i+2, p(S(r2i+2), T (r2i+1)) > 0. Otherwise, by Lemma 2,

r2i+1 is a common fixed point of S, T . Thus (r2i+1, r2i+2) ∈ γ and note that

M(r2i+2, r2i+1) = max

p(r2i+2, r2i+1), p(r2i+2, S(r2i+2)), p(r2i+1, T (r2i+1)),p(r2i+1, S(r2i+2)) + p(r2i+2, T (r2i+1))

2

= max

p(r2i+2, r2i+1), p(r2i+2, r2i+3), p(r2i+1, r2i+2),p(r2i+1, r2i+3) + p(r2i+2, r2i+2)

2

= max p(r2i+2, r2i+1), p(r2i+2, r2i+3) .

Again the case M(r2i+2, r2i+1) ≤ p(r2i+2, r2i+3) is not possible. So, for the other case, the

contractive condition (3.1) implies

F (p(r2i+2, r2i+3)) ≤ F (p(r2i+1, r2i+2))− τ (3.4)

for all i ∈ N ∪ 0. By (3.3) and (3.4), we have

F (p(rn+1, rn+2)) ≤ F (p(rn, rn+1))− τ (3.5)



for all n ∈ N ∪ 0. By (3.5), we obtain

F (p(rn, rn+1)) ≤ F (p(rn−2, rn−1))− 2τ.

Repeating these steps, we get

F (p(rn, rn+1)) ≤ F (p(r0, r1))− nτ. (3.6)

By (3.6), we obtain limn→∞ F (p(rn, rn+1)) = −∞. Since F ∈ ∆F ,

limn→∞

p(rn, rn+1) = 0. (3.7)

From the property (F3) of F -contraction, there exists κ ∈ (0, 1) such that

limn→∞

((p(rn, rn+1))κ F (p(rn, rn+1))) = 0. (3.8)

By (3.6), for all n ∈ N, we obtain

(p(rn, rn+1))κ (F (p(rn, rn+1))− F (p(r0, x1))) ≤ − (p(rn, rn+1))

κ nτ ≤ 0. (3.9)

Considering (3.7), (3.8) and letting n→∞ in (3.9), we have

limn→∞

(n (p(rn, rn+1))κ) = 0. (3.10)

Since (3.10) holds, there exists n1 ∈ N such that n (p(rn, rn+1))κ ≤ 1 for all n ≥ n1 or

p(rn, rn+1) ≤1

n1κ

for all n ≥ n1. (3.11)

Using (3.11), we get, for m > n ≥ n1,

p(rn, rm) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ...+ p(rm−1, rm)

−m−1∑j=n+1

p(rj , rj)

≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ...+ p(rm−1, rm)

=m−1∑i=n

p(ri, ri+1)

≤∞∑i=n

p(ri, ri+1)

≤∞∑i=n

1

i1k

.

The convergence of the series∑∞

i=n

1

i1κ

entails limn,m→∞ p(rn, rm) = 0. Hence rn is a Cauchy

sequence in (M,p). Due to Lemma 1, rn is a Cauchy sequence in (M,dp). Since (M,p) is a

complete partial metric space, (M,dp) is a complete metric space and as a result there exists

υ ∈M such that limn→∞ dp(rn, υ) = 0. Moreover, by Lemma 1

limn→∞

p(υ, rn) = p(υ, υ) = limn,m→∞

p(rn, rm). (3.12)



Since limn,m→∞ p(rn, rm) = 0, from (3.12), we deduce that

p(υ, υ) = 0 = limn→∞

p(υ, rn). (3.13)

Now from (3.13) it follows that r2n+1 → υ and r2n+2 → υ as n → ∞ with respect to τ(p).

Suppose that T is continuous. Then

υ = limn→∞

rn = limn→∞

r2n+1 = limn→∞

r2n+2 = limn→∞

T (r2n+1) = T ( limn→∞

r2n+1) = T (υ).

Now we show that υ = S(υ). Suppose on contrary that p(υ, S(υ)) > 0. Regarding υ υ

together with the contractive condition (3.1), we obtain

τ + F (p(υ, S(υ))) = τ + F (p(S(υ), T (υ)))

≤ F (M(υ, υ)),

F (p(υ, S(υ))) < F (p(υ, υ)),

which is a contradiction. Thus p(υ, S(υ)) = 0 and due to (p1), (p2) we conclude that υ = S(υ).

Consequently, we have S(υ) = T (υ) = υ, that is, (S, T ) have a common fixed point υ.

In the other case, using the assumption that M is regular, we have that rn υ for all n ∈ N.

To show that υ is a common fixed point of S, T , we split the proof into two cases.

(1) rn = υ for some n. Then there exists i0 ∈ N such that r2i0 = υ. Consider S(υ) =

S(r2i0) = r2i0+1 υ and also υ = r2i0 r2i0+1 = S(υ). Thus υ = S(υ) and from (3.1),

we have υ = T (υ).

(2) rn 6= υ for all n. Suppose that p(υ, S(υ)) > 0. Since limn→∞ r2i = υ, there exists N ∈ Nsuch that

p(r2i+1, S(υ)) > 0 and p(r2i, υ) <p(υ, S(υ))

2for all i ≥ N .

Moreover,

M(r2i, υ) = max

p(r2i, υ), p(r2i, S(r2i)), p(υ, T (υ)),

p(υ, S(r2i)) + p(r2i, T (υ))

2

,

M(r2i, υ) ≤ p(υ, S(υ))

2for all i ≥ N .

So (r2i, υ) ∈ γ and S and T satisfy the generalized rational type ordered F -contraction.

Thus

τ + F (p(r2i+1, S(υ))) = τ + F (p(S(r2i), T (υ)))

≤ F (M(r2i, υ)),

F (p(υ, S(υ))) < F (p(υ, S(υ))

2) as i→∞,

which is a contradiction. Therefore, p(υ, S(υ)) = 0 and due to (p1), (p2) we conclude that

υ = S(υ) and from (3.1) we have υ = T (υ). Thus (S, T ) have a common fixed point υ.

We denote the set of common fixed points of S, T by Fix(S, T ).

Remark 2. If we assume that Fix(S, T ) in Theorem 3 is a chain along with existing conditions,

then it is a singleton set (common fixed point is unique). Indeed, if ω is another common fixed



point of S, T , then ω υ. Also p(S(υ), T (ω)) > 0 (otherwise υ = ω) and so (υ, ω) ∈ γ. From

the contractive condition (3.1), we have

τ + F (p(υ, ω)) = τ + F (p(S(υ), T (ω)))

≤ F (M(υ, ω)) , (3.14)

where

M(υ, ω) = max

p(υ, ω), p(υ, S(υ)), p(ω, T (ω)),

p(ω, S(υ)) + p(υ, T (ω))

2

= p(υ, ω).

From (3.14), we have

F (p(υ, ω)) < F (p(υ, ω)) ,

which leads to a contradiction. Hence υ = ω and υ is a unique common fixed point of a pair

(S, T ).

Remark 3. If Fix(S, T ) is not a chain and there exists z in M such that every element in the

orbit OT (z) = z, T (z), T 2(z), . . . is comparable to υ, ω, then υ = ω (υ is unique) provided that

S and T are Cırıc type ordered F -contractions.

Proof. Assume that υ, ω are in Fix(S, T ) and there exists an element z ∈ M such that every

element of OT (z) = z, T (z), T 2(z), . . . is comparable to υ, ω and hence (Tn−1(z), Sn−1(υ)) and

(Tn−1(z), Sn−1(ω)) are elements of γ for each n ≥ 1. Due to (3.1), we have

τ + F (p(υ, Tn(z))) = τ + F (p(Sn(υ), Tn(z))

≤ F (M(Sn−1(υ), Tn−1(z)

)), (3.15)

where

M(Sn−1(υ), Tn−1(z)

)= max

p(Sn−1(υ), Tn−1(z)

), p(Sn−1(υ), Sn(υ)

), p(Tn−1(z), Tn(z)

),

p(Tn−1(z), Sn(υ)

)+ p

(Sn−1(υ), Tn(z)

)2

= p

(Sn−1(υ), Tn−1(z)

)= p

(υ, Tn−1(z)

).

Thus, from (3.15), we deduce that p(υ, Tn(z)) is a nonnegative decreasing sequence which

in turn converges to 0.

Similarly, we can show that p(ω, Tn(z)) is a nonnegative decreasing sequence, which con-

verges to 0. Consequently, υ = ω.

The following example illustrates Theorem 3 and shows that the condition (3.1) is more

general than contractivity condition given by Durmaz et al. ([11]).

Example 4. Let M = [0, 1] and define p(r1, r2) = max r1, r2. Let ≺1 be defined by r1 ≺1 r2 if

and only if r2 ≤ r1 for all r1, r2 ∈M . Then r1 ≺1 r2 is a partial order on M and (M,≺1, p) is a

complete ordered partial metric space. Moreover, define d (r1, r2) = |r1 − r2|. Then (M,≺1, d)

is a complete ordered metric space. Define the mappings S, T : M →M as follows:

T (r) =

r

5if r ∈ [0, 1);

0 if r = 1

and S(r) =3r

7for all r ∈M.



Clearly, S, T are weakly increasing self mappings with respect to ≺1. Define the function F :

R+ → R by F (r) = ln(r) for all r ∈ R+ > 0. Let r1, r2 ∈ M such that p(S(r1), T (r2)) > 0 and

suppose that r2 ≺1 r1. Then

M(r1, r2) = max

r2,

r1r21 + r1

,r1r2

1 + max3r17 ,

r25

.Since r1

1+r1< 1 and r1

1+max

3r17,r25

< 1, we have that M(r1, r2) = r2.

In a similar way, if r1 ≺1 r2, then we obtain that M(r1, r2) = r1, i.e., M(r1, r2) = p(r1, r2).

Let τ ≤ ln(73). Since (r1, r2) ∈ γ

τ + (p(S(r1), T (r2))) = τ + ln

(max

3r17,r25

)≤ ln(

7

3) + ln

(max

3p(r1, r2)

7,p(r1, r2)

5

)= ln(

7

3) + ln

(3p(r1, r2)

7

)= ln (p(r1, r2))

= F (M(r1, r2)) .

Thus the contractive condition (3.1) is satisfied for all r1, r2 ∈ M with L = 0. Hence all the

hypotheses of Theorem 3 are satisfied. Note that (S, T ) have a unique common fixed point

r = 0. As we have seen in Example 2, T is not an F -contraction in (M,≺1, d). Thus we cannot

apply Theorem 1 and hence Theorem 2. The following corollary generalizes Theorem 2.

The following corollary generalizes the results in [13].

Corollary 1. Let (M,, p) be a complete ordered partial metric space and T : M → M be a

mapping such that r0 T (r0). Assume that

(1) either T is a continuous mapping or M is regular,

(2) T is a Cırıc type ordered F -contraction.

Then T has a unique fixed point υ in M such that p(υ, υ) = 0.

Proof. Setting S = T in Theorem 3, we obtain the required result.

4. Application of Theorem 3

This section contains an existence result which shows the usefulness of Theorem 3 in estab-

lishing existence of solution of implicit type integral equation:

A(t, u(r, t)) =

∫ a

0

∫ a

0H(t, θ, φ, u(θ, φ)) dθdφ, (4.1)

where u ∈ U = L [C([0, a])× [0, a]]=Lebesgue measurable space, t, θ, φ ∈ Ia = [0, a]. For u ∈ U ,

define norm as: ‖u‖ = maxt∈[0,a]

|u(t)|. Let U be endowed with the partial metric p : U ×U → R+0

defined by

p(u, v) = d(u, v) + c = maxt∈[0,a]

|u(r, t)− v(r, t)|+ c for all u, v ∈ U .



Also, U can be equipped with order ≺2 defined by u ≺2 v if and only if v(r, t) ≤ u(r, t).

Obviously, (U , ‖ · ‖) is a Banach space and (U ,≺2, p) is a complete ordered partial metric space.

Theorem 4. Assume that

(a) for all u, v ∈ U and κ = |u(r1, t)− v(r1, t)|+ c

|A(t, u(r1, t))−A(t, v(r1, t))|+ c ≤ (κ)e−τ for each t ∈ Ia,

(b) H(t, θ, φ, u(θ, φ)) ≤ 1a2u(r1, t) for all t ∈ Ia,

(c) for all t, θ, φ ∈ Ia,

A(t,

∫ a

0

∫ a

0H(t, θ, φ, u(θ, φ)) dθdφ) ≤

∫ a

0

∫ a

0H(t, θ, φ, u(θ, φ)) dθdφ,

(d) H(t, θ, φ,A(θ, u(θ, φ))) ≥ 1

a2A(t, u(r1, t)).

Then implicit integral equation (4.1) has a solution in U .

Proof. Firstly, define S(u(r1, t)) = A(t, u(r1, t)) and T (u(r1, t)) =∫ a0

∫ a0 H(t, θ, φ, u(θ, φ)) dθdφ.

We show that S,T are weakly increasing mappings. Consider

S(T (u(r1, t))) = A(t, T (u(r1, t)))

= A(t,

∫ a

0

∫ a

0H(t, θ, φ, u(θ, φ)) dθdφ

)≤

∫ a

0

∫ a

0H(t, θ, φ, u(θ, φ)) dθdφ = T (u(r1, t)) using (c)

and

T (S(u(r1, t))) =

∫ a

0

∫ a

0H(t, θ, φ, S(u(θ, φ))) dθdφ

=

∫ a

0

∫ a

0H(t, θ, φ,A(θ, u(θ, φ))) dθdφ

≤ 1

a2

∫ a

0

∫ a

0A(t, u(r1, t)) dθdφ = A(t, u(r1, t)) due to (b).

Thus S(T (u(r1, t))) ≤ T (u(r1, t)) and T (S(u(r1, t))) ≤ S(u(r1, t)) for all t ∈ Ia imply that S,T

are weakly increasing mappings with respect to ≺2.

Secondly, consider

p (S(u), T (v)) = maxt∈Ia|S(u(r1, t))− T (v(r2, t))|+ c

= maxt∈Ia

∣∣∣∣A(t, u(r1, t))−∫ a

0

∫ a

0H(t, θ, φ, v(θ, φ)) dθdφ

∣∣∣∣+ c

≤ maxt∈Ia

∣∣∣∣A(t, u(r1, t))−1

a2

∫ a

0

∫ a

0A(t, v(r1, t)) dθdφ

∣∣∣∣+ c using (d)

= maxt∈Ia|A(t, u(r1, t))−A(t, v(r1, t))|+ c

≤ maxt∈Ia

(κ)e−τ using (a)

≤ e−τp(u, v).



So

τ + ln(p (S(u), T (v))) ≤ ln(p(u, v)) ≤ ln(M(u, v)).

Thus by taking F (r) = ln(r), we have

τ + F (p (S(u), T (v))) ≤ F (M(u, v)).

Hence by Theorem 3 the integral equation (4.1) has a solution in L [C ([0, a])× [0, a]].

References

[1] T. Abdeljawad, E. Karapinar, K. Tas, Existence and uniqueness of a common fixed point on partial metricspaces, Appl. Math. Lett. 24 (2011), 1900-1904.

[2] O. Acar, I. Altun, Fixed point theorems for weak contractions in the sense of Berinde on partial metric spaces,Topology Appl. 159 (2012), 2642-2648.

[3] O. Acar, I. Altun, Multivalued F -contractive mappings with a graph and some fixed point results, Publ. Math.Debrecen 88 (2016), 305-317.

[4] O. Acar, G. Durmaz, G. Minak, Generalized multivalued F -contractions on complete metric spaces, Bull.Iranian Math. Soc. 40 (2014), 1469-1478.

[5] I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl. 157 (2010),2778-2785.

[6] I. Altun, S. Romaguera. Characterizations of partial metric completeness in terms of weakly contractivemappings having fixed point, Appl. Anal. Discrete Math. 6 (2012), 247-256.

[7] G.A. Anastassiou, I.K. Argyros, Approximating fixed points with applications in fractional calculus, J. Com-put. Anal. Appl. 21 (2016), 1225–1242.

[8] A. Batool, T. Kamran, S. Jang, C. Park, Generalized ϕ-weak contractive fuzzy mappings and related fixedpoint results on complete metric space, J. Comput. Anal. Appl. 21 (2016), 729–737.

[9] S. Chandok, Some fixed point theorems for (α, β)-admissible Geraghty type contractive mappings and relatedresults, Math. Sci. 9 (2015), 127-135.

[10] S. Cho, S. Bae, E. Karapinar, Fixed point theorems for α-Geraghty contraction type maps in metric spaces,Fixed Point Theory Appl. 2013, 2013:329.

[11] G. Durmaz, G. Minak, I. Altun, Fixed points of ordered F -contractions, Hacettepe J. Math. Stat. 45 (2016),15-21.

[12] S.G. Matthews, Partial metric topology, in Proceedings of the 11th Summer Conference on General Topologyand Applications, Vol. 728, pp.183-197, The New York Academy of Sciences, 1995.

[13] G. Minak, A. Helvaci, I. Altun, Ciric type generalized F -contractions on complete metric spaces and fixedpoint results, Filomat 28 (2014), 1143-1151.

[14] M. Nazam, M. Arshad, C. Park, Fixed point theorems for improved α-Geraghty contractions in partial metricspaces, J. Nonlinear Sci. Appl. 9 (2016), 4436-4449.

[15] H. Piri, P. Kumam, Some fixed point theorems concerning F -contraction in complete metric spaces, FixedPoint Theory Appl. 2014, 2014:210.

[16] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some application to matrixequations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.

[17] S. Shukla, S. Radenovic, Some common fixed point theorems for F -contraction type mappings on 0-completepartial metric spaces, J. Math. 2013, Art. ID 878730.

[18] D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces. FixedPoint Theory Appl. 2012, 2012:94.



The Theoretical Analysis of l1-TV Compressive Sensing

Model for MRI Image Reconstruction

Xiaoman Liu1

School of Mathematics, Southeast UniversityNanjing, 211189, P.R.China

July 27, 2017

Abstract

One of the main tasks in MRI image reconstruction is to catch the picture characteristicssuch as interfaces and textures from incomplete frequency data, and the iteration methodsare the most useful methods to the minimization problem. A new viewpoint of choosing theiteration stopping rules in image reconstruction problem is proposed. The reconstruction modelbased on compressive sensing theory consists of a data matching term and two penalty terms,wavelet sparse and total variation regularization term. Then the Bregman iteration with laggeddiffusivity fixed point iteration is used to solve the corresponding nonlinear Euler-Lagrangeequation of image reconstruction model with incomplete frequency data. A real MRI image isused to test the proposed method in numerical experiments with different stopping rules. Thetheoretical analysis illustrate that although the norm of objective functional decreases withrespect to the number of iteration, it cannot ensure the reconstructed image is the desiredoptimization image.

MSC(2010): 65M32, 65T50, 65T60, 65K10Keywords: image reconstruction; MRI image; total variation; wavelet transform; regularization

1 Introduction

Image processing can be roughly divided into three kinds of problems, namely, image deblurring,image enhancement and image restoration, and the main purpose is to obtain the clear image withinterfaces and textures from its noisy measurement. For a bounded connected domain Ω ⊂ R2 (arectangle in general [1]), let u(x),x = (x1,x2) be the grey function of an image defined in Ω. Ingeneral, we can get the degradation data bσ(x) with blurring noisy process, such as moving blurry,Gaussian blurry, white Gaussian noise, impulse noise (salt and pepper noise) as well as Poissonnoise [2, 3]. The optimization scheme is one of the classical way to reconstruct u from bσ, i.e.,minimizes the Tikhonov cost functional

J(u) =1

2‖K u− bσ‖2L2(Ω) + αL u (1)

1Corresponding author: X.M. Liu, email: [email protected]

1


1471 Xiaoman Liu 1471-1478

The Theoretical Analysis of l1-TV Compressive Sensing Model 2

with some penalty term L u and the regularization parameter α > 0, where the operator Lrepresents the a-priori regularity image u. Obviously, all of terms in (1) are continuous version,i.e., u(x) is defined in Ω everywhere, to unify the basic idea of this scheme with our numericalimplementations, we describe all these terms by the finite dimensional approximation of u(x) ∈RN×N with components ui,j(i, j = 1, · · · , N).

In many engineering configurations, instead of the spatial noisy data bσi,j for each pixel Ωi,j , thepractical measurement data may be the incomplete frequency data, or the finite number of discretefrequencies at the band-limited interval. For example, in the application of magnetic resonanceimaging (MRI) image reconstruction, data collected by an MR scanner are, roughly speaking, inthe frequency domain (called k -space) rather than the spatial domain. One of the main stage forMRI is the k -space data acquisition. In this stage, energy from a radio frequency pulse is directedto a small section of the targeted anatomy at a time. As a result, the protons within that area areforced to spin in a certain frequency and get aligned to the direction of the magnet. Upon stoppingthe radio frequency, the physical system gets back to its normal state and releases energy that isthen recorded for analysis. This process is repeated until enough data is collected for reconstructinga high quality image in the second stage. This process is based on the compressive sensing (CS), andthis kind of MRI image reconstruction problem is called CS-MRI image reconstruction. For moredetails about these contents see [4, 5, 6] and references therein. In this case, the data-matching

term in (1) should be replaced by ‖P F [u] − P bδ‖22, where F is the two-dimensional discreteFourier transform converting the spatial matrix u ∈ RN×N into frequency matrix F [u] ∈ CN×N ,

while P is a linear operator specifying the incomplete frequency data from CN×N , bδ ∈ CN×N isthe noisy frequency data, ‖ · ‖2 denotes the Euclidean norm.

In the case of CS-MRI, the recovery of u from bδ is equivalent to solving the l0 problem:

minu‖Ψ u‖0 : ‖P F [u]− P bδ‖22 ≤ δ2, (2)

where ‖ · ‖0 is the number of nonzero components of the objective, and orthogonal wavelet operator

Ψ : RN×N → RN2×1 is based on the orthogonal wavelet basis ψi,j(i, j = 1, · · · , N) [7]. However,it is well-known that (2) is a NP-hard problem, and as usually, we replace it by the l1-minimizingproblem:

minu‖Ψ u‖1, ‖P F [u]− P bδ‖22 ≤ δ2, (3)

which yields sparse solutions under some conditions [8], ‖ · ‖1 denotes the l1 norm.

2 A theoretical analysis for l1-TV optimization model

As usual, the image u has the obvious edges such as the interfaces in MRI images. So it is naturalto also cooperate this a-priori information into the reconstruction model by considering the totalvariation (TV) penalty. So it is natural to consider the following unconstraint cost functional:

J(u) :=1

2‖P F [u]− P bδ‖22 + α1‖Ψ u‖1 + α2|u|TV , (4)

where α1, α2 are positive regularization parameters that determine the penalty terms. Therefore,the image reconstruction problem is the following l1-TV optimization model

arg minu∈RN×N

J(u) =1

2‖P F [u]− P bδ‖22 + α1‖Ψ u‖1 + α2|u|TV . (5)


1472 Xiaoman Liu 1471-1478


Suppose u ∈ RN2×1 is a vector formed by stacking the columns of a two-dimensional MRI imagearray u := (ui,j), i, j = 1, · · · , N . Since (5) is not a differential, we added a small positive parameterβ [9, 10] and presented the optimization model as the following minimizing convex perturbed form

arg minu∈RN2×1

J(u) =1

2‖PFu−Pb

δ‖22 + α1‖Ψu‖1,β + α2|u|TV,β , (6)

in which two regularization terms are defined as

‖Ψu‖1,β =N2∑i=1

√((Ψ u)i)

2+ β, |u|TV,β =

N∑i,j=1

√|∇i,ju|2 + β,

where ∇i,ju =(∇xi,ju,∇

yi,ju)

is defined under periodic boundary condition

∇xi,ju =

ui+1,j − ui,j , if i < m,u1,j − um,j , if i = m.

∇yi,ju =

ui,j+1 − ui,j , if j < n,ui,1 − ui,n, if j = n.

for i, j = 1, · · · , N, |∇i,ju| =√

(∇xi,ju)2 + (∇yi,ju)2. P is an N2×N2 matrix consisting of sampling

matrix P (an N ×N matrix generating from the identity matrix I by setting its some rows as null

vectors), and F ∈ CN2×N2

is the two-dimensional discrete Fourier matrix defined in Fourier matrixF ∈ CN×N with the components Fm,n = e−i2πmn/N .

The objective function in the problem (6) is strictly convex and differentiable with respect tovariable u and its global minimizer is unique [11] when α1 = 0. The solution of (6) with smallenough β can better approximate to the solution of the minimizing (4). Because the solution of theminimization problem for (4) can be regarded as the limit of the solution of (6) when β → 0.

There are a number of numerical methods for solving the image reconstruction model (6), likefixed-point continuation method [12], split Bregman method [13], gradient project method [14],fast alternating minimization method [15], the variable splitting method [16], the operator-splittingalgorithm [17] and fast iterative shrinkage-thresholding algorithm [18]. Meanwhile the conjugategradient method (CGM) [19] is also very efficient approach to solve (6) in CS-MRI, and the formerwork [10] is better than CGM. In this paper, a fast scheme with different iteration stopping rulesis proposed to solve the objective problem (6), which is based on Bregman method [20] and laggeddiffusivity fixed point iteration [21]. The numerical experiments are shown to compare proposedmethod with the one in former work [10]. The fast iterative scheme for proposed model (6) is asfollows Algorithm 1.

Now we give the theoretical analysis based on regularization for the CS-MRI image reconstruc-tion problem. The objective optimization problem (6) can be rewritten as

arg minu∈RN×N

J(u) =1

2‖P F [u]− P bδ‖22 + α1‖Ψ u‖1,β + α2|u|TV,β . (7)

Assume Lα u = α1‖Ψ u‖1,β + α2|u|TV,β . In order to get the approximate solution u? to theabove problem, we change the optimization problem (7) to the equation below

((PF)∗PF + Lα) u? = (PF)∗P bδ, (8)

oru? = ((PF)∗PF + Lα)−1(PF)∗P bδ, (9)


1473 Xiaoman Liu 1471-1478


Algorithm 1 The fast iterative scheme based on Bregman iteration for minimizing J(u)

Input: frequency input bδi,j |i, j = 1, · · · , N, sampling matrix P ∈ RN2×N2

, and parametersα1, α2, β.Do iteration from k = 0 with b(0) = Θ, u(0) = Θ.While (the stopping rule is not satisfied) Compute:

b(k+1) = bδ +(b(k) −PFu

),

u(k+1) = arg minu∈RN2×1

α1‖Ψu‖1,β + α2|u|TV,β + 1

2‖PFu−Pb(k+1)

‖22

,

k ← k + 1. End dou? := u(k).End

where ∗ means the conjugate transpose. Let operator Rα = ((PF)∗PF + Lα)−1(PF)∗P. WhenRα is the regularization strategy [22], we have

‖u? − u‖2 ≤∥∥∥Rα bδ −Rα b∥∥∥

2+∥∥∥Rα b− u∥∥∥

2

≤ ‖Rα‖2 ·∥∥∥bδ − b∥∥∥

2+ ‖Rα (KF u)− u‖2

≤ δ ‖Rα‖2 + ‖RαKF u− u‖2 , (10)

where KF is an operator with Fourier transform in the classical way to reconstruct u from frequencydata b based on (1), i.e., KF u = b.

The iteration scheme can be based on Bregman iteration method [20], so the regularizationparameter is seen as discrete regularization parameter, like α1, α2, β and iteration number k. Withthe inequations above, the regularization error ‖u? − u‖2 can be seen as two parts: the ill-posed ofmodel, and the error when Rα tends to K−1

F . In [23], the error ‖u?− u‖2 is of the optimal value atsome iteration step. When iteration number k → ∞ which beyonds that iteration step, objectivefunction J (u?)→ 0 but the error ‖u? − u‖2 9 0.

Now, we provide the similar conclusion in finite dimension. As we all know, the penalty termsin (7) are the important functions to the objective problem. However, the ill-posed problem (6)

requires only the incomplete frequency data Pbδ. Therefore, we should optimize the objective

function ‖PFu − Pbδ‖22. In the other words, there also exists a u† (i.e. the exact solution) such

that ∥∥∥PFu† −Pbδ∥∥∥

2= inf

u∈RN2×1

∥∥∥PFu−Pbδ∥∥∥

2= 0. (11)

Noticing that the minimizing sequence uk,α : k = 1, 2, · · · only has the convergence

limk→∞

J(uk,α) = J(u?), (12)

there is no convergence for the norm ‖uk,α − u?‖ in general. So we need to identify the behaviorof uk,α as k →∞ and αi → 0(i = 1, 2).


1474 Xiaoman Liu 1471-1478


Theorem 2.1. There exists a subsequenceukj ,α : j = 1, 2, · · ·

⊂

uk,α : k = 1, 2, · · · such that

limj→∞

∥∥ukj ,α − uα∥∥

2= 0. (13)

Proof. Since (12), we easily know that

limk→∞

J(uk,α) = J(u?) = infu∈RN2×1

J(u).

According to the finite dimension domain, there exists a subsequence of uk,α denoted by ukj ,α :j = 1, 2, · · · such that ukj ,α → uα as j → ∞. Hence the limit of the norm

∥∥ukj ,α − uα∥∥

2equals

to 0. The proof is complete.

From Theorem 2.1, we have some sequence αm → 0 as m→∞, that means

limm→∞

limj→∞

ukj ,αm = limm→∞

uαm := u, (14)

limm→∞

limj→∞

J(ukj ,αm

)= 0. (15)

Noticing (11) above, (15) is equal to ‖PFu†−Pbδ‖2. It reveals the exact meaning of the approximate

solution to our problem by ukj ,α : j = 1, 2, · · · , while ukj ,α : j = 1, 2, · · · can converge to someu. But it is worth noting that u cannot be ensured theoretically to be the exact solution u†.Therefore the iteration number could not be too big. Even though the approximate solution u? isthe minimization for optimization model, it could not be the best solution for image reconstructionproblem, i.e., u? 9 u†.

3 Numerical Experiments

In this section, the proposed fast algorithm with different iteration stopping rules is shown to solvethe objective problem (6), which is compared with the method in [10]. All tests are performed inMATLAB 7.10 on a laptop with an Intel Core i5 CPU M460 processor and 2 GB of memory.

The signal to noise ratio (SNR) and relative error (ReErr) are used to measure the quality ofthe reconstructed images. The definitions of SNR and relative error are given as follows

SNR = 20 lg

(‖u‖2∥∥u− u(k)

∥∥2

), (16)

ReErr =

∥∥u(k) − u∥∥2

2

‖u‖22, (17)

where u(k) and u are the reconstructed and original images, respectively. The CPU time is used toevaluate the speed of MRI image reconstruction.

As usual, the iteration stopping rule is one of the following three conditions:

J(u(k)) ≤ δ,∥∥u(k) − u(k−1)

∥∥2∥∥u(k)

∥∥2

≤ δ, k = K0, (18)


1475 Xiaoman Liu 1471-1478


which mean the norm of objective function in (6), the relative difference between successive iterationfor the reconstructed image, and maximum number of iterations K0. In order to illustrate theefficient of the theoretical analysis in Section 2, we use the maximum iterations as the stoppingrule.

Firstly, the performance of Algorithm 1 in solving model (6) for a real MRI brain image is shownin Figure 1, which is compared with the efficient method in [10]. Let additive Gaussian noise levelin frequency domain is δ = 0.01, i.e., adding 1% additive noise on the frequency of original image.The parameters α1 = 0.01, α2 = β = 0.0001 which are the same as the comparison algorithm. Tothe sampling matrix P, we choose the radial sampling method with 22×8 views on frequency data.The tests results are shown in Figure 1(c)(d) which are based on stopping rule K0 = 60 in Algorithm1 and K0 = 100 in the comparison algorithm, respectively. The SNR in (c) and (d) is 38.2321dB,37.7443dB respectively, and the relative error is 4.0866 × 10−5, 1.8770 × 10−4 respectively. Fromthese data, the reconstruction is efficient with these parameters in the fast iteration scheme basedon the iterations K0 as stopping rule. However, ever though the iterations number in (d) is biggerthan the one in (c), the reconstructed image (d) is not clearer than (c).

Next our numerical experiment is to illustrate the relationship between reconstruction error(Err) and objective function J

(u(k)

). The reconstruction error between reconstructed image and

original image is used to evaluate the exactitude of ill-posed problem, which defined as follows

Err = ‖u(k) − u‖2. (19)

We take different iteration step by step, i.e., 10, 20, 30, · · · . The sampling mask in the frequencyspace separately take 22 × 8, 22 × 10, 22 × 12 views in radial sampling method. We still assumethe above mentioned parameters in the tests. The fitting curve of error ‖u(k) − u‖2 and J

(u(k)

)in

different iteration numbers are shown in Figure 2.From Figure 2, we find that although the stopping criterion is satisfied, the reconstruction error

‖u(k)−u‖2 is of the optimal value at some iteration step. It means that the more iteration is betterfor reconstruction is not true. Meanwhile, the convergence and error analysis are the same withtheoretical analysis in the above section. Therefore, the choice of iteration stopping rule, especiallythe iteration number K0, is one of the most important factor of MRI image reconstruction.

Figure 1: (a) Original image; (b) Sampling mask: 22 × 8 views; (c) Reconstructed image withK0 = 60; (d) Reconstructed image with K0 = 100.


1476 Xiaoman Liu 1471-1478


0 50 100 150 2000

5

10

15

20

25

30

35

40

k

||uk −

u||

22*822*1022*12

0 50 100 150 2000

1000

2000

3000

4000

5000

6000

7000

k

J(uk )

22*822*1022*12

Figure 2: Fitting curve of error and J(u(k)) in different iteration numbers.

4 Conclusion

The l1-TV optimization model based on compressive sensing was established to reconstruct MRIimages. Bregman method and lagged diffusivity fixed point iteration are used to solve the modifiedreconstruction model, and a fast iteration scheme with error estimate analysis is proposed. Basedon Tikhonov regularization theory, a theoretical analysis on iteration stopping rules is proposed. Areal MRI brain image is employed to test in the numerical experiments and the results demonstratethat proposed method and theoretical analysis is very efficient in CS-MRI image reconstruction.

Acknowledgements

This work is supported by NSFC (No. 11671082), and Postgraduate Research & Practice InnovationProgram of Jiangsu Province (No. KYCX17 0038).

References

[1] A. Gills. and V. Luminita, A variational method in image recovery, SIAM J. Numer. Anal.34(1997), pp. 1948-1979.

[2] Y.G. Zhu, X.M. Liu, A fast method for l1-l2 modeling for MR image compressive sensing, J.Inverse Ill-posed Prob. 23(2015), pp. 211-218.

[3] X.D. Wang, X. Feng, W. Wang, W. Zhang, Iterative reweighted total generalized variationbased poisson noise removal model, Appl. Math. Comput. 223 (2013), pp. 264-277.

[4] E.J. Candes, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstructionfrom highly incomplete frequency information, IEEE Trans. Inf. Theory. 52(2006), pp. 489-509.

[5] D.L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory. 52(2006), pp. 1289-1306.

[6] E.J. Candes, J. Romberg, Sparsity and incoherence in compressive sampling, Inverse Problems.23(2007), pp. 969-985.

[7] S. Mallat, A Wavelet Tour of Signal Processing (3rd), Academic Press, San Diego 2008.


1477 Xiaoman Liu 1471-1478


[8] J. Huang, S. Zhang and D. Metaxas, Efficient MR image reconstruction for compressed MRimaging, Medical Image Anal. 15(2011), pp. 670-679.

[9] C.R. Vogel, Computational Methods for Inverse Problems, SIAM Frontiers In Applied Math-ematics, Philadephia, 2002.

[10] X. Liu, Y. Zhu, A fast method for TV-L1-MRI image reconstruction in compressive sensing,J. Comput. Inform. Syst. 2(2014), pp. 1-9.

[11] R. Acar, C.R. Vogel, Analysis of total variation penalty methods for ill-posed problems, InverseProbl. 10(1994), pp. 1217-1229.

[12] E.T. Hale, W. Yin, Y. Zhang, A fixed-point continuation for l1-regularization with applicationto compressed sensing, Rice University CAAM Technical Report, TR07-07(2007)1-45.

[13] T. Goldstein, O. Osher, The split Bregman method for L1 regularized problems, SIAM J.Imag. Sci., 2(2)(2009)323-343.

[14] M. Figueiredo, R. Nowak, S. Wright, Gradient projection for sparse reconstruction: applicationto compressed sensing and other inverse problems, IEEE J-STSP, 1(2007), pp. 586-597.

[15] Y. Zhu, I. Chern, Fast alternating minimization method for compressive sensing MRI underwavelet sparsity and TV sparsity, Proceedings of 2011 Sixth International Conference on Imageand Graphics, (2011), pp. 356-361.

[16] J. Yang, Y. Zhang, W. Yin, A fast alternating direction method for TV-L1-L2 signal recon-struction from partial Fourier data, IEEE J-STSP, 4(2010), pp. 288-297.

[17] S. Ma, W. Yin, Y. Zhang, A. Chakraborty, An efficient algorithm for compressed MR imagingusing total variation and wavelets, IEEE Conf. Comput. Vis. Pattern Recognition, CVPR,(2008), pp. 1-8.

[18] A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse prob-lems, SIAM J. Imag. Sci. 2(2009), pp. 183-202.

[19] M. Lustig, D. Donoho, D. Pauly, Sparse MRI: the application of compressed sensing for rapidMR imaging, Magn. Reson. Med. 58(2007), pp. 1182-1195.

[20] W. Yin, S. Osher, D. Goldfarb, J. Darbon, Bregman iterative algorithms for l1-minimizationwith applications to compressed sensing, SIAM J. Imag. Sci. 1(2008), pp. 143-168.

[21] C.R. Vogel, M.E. Oman, Iterative Methods for Total Variation Denoising, SIAM J. Sci. Com-put. 17(1996), pp. 227-238.

[22] J. Liu, The Regularization Methods and Application for Ill-posed Problems, Science Publishing,Beijing, 2005. (in Chinese)

[23] B. Wang and J. Liu, Recovery of thermal conductivity in two-dimensional media with nonlinearsource by optimizations, Appl. Math Letters, 60(2016), pp. 73-80.


1478 Xiaoman Liu 1471-1478

On Fibonacci Z-sequences and their logarithm functions

Hee Sik Kim1, J. Neggers2 and Keum Sook So3,∗

1Department of Mathematics, Research Institute for Natural Sciences, Hanyang University,

Seoul, 04763, Korea2Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U. S. A

3,∗Department of Statistics and Financial Informatics, Hallym University, Chuncheon 24252,

Korea

Abstract. In this paper we discuss the concept of a Z-sequence and use it to define the Fibonacci Z-sequence.

Based on Z-sequences one may define analogs of logarithm functions. The special case of the logarithm function

associated with the Fibonacci Z-sequence is of interest, since the recursive property of this sequence permits a

more detailed study of these functions. They are similar to ordinary logarithm functions which may be based on

Z-sequences ann∈Z , where a > 1.

1. Introduction and Preliminaries

Fibonacci-numbers have been studied in many different forms for centuries and the literature on the subject is

consequently incredibly vast. One of the amazing qualities of these numbers is the variety of mathematical models

where they play some sort of role and where their properties are of importance in elucidating the ability of the

model under discussion to explain whatever implications are inherent in it. The fact that the ratio of successive

Fibonacci numbers approaches the Golden ratio (section) rather quickly as they go to infinity probably has a

good deal to do with the observation made in the previous sentence. Surveys and connections of the type just

mentioned are provided in [1] and [2] for a very minimal set of examples of such texts, while in [3] an application

(observation) concerns itself with a theory of a particular class of means which has apparently not been studied in

the fashion done there by two of the authors the present paper. Surprisingly novel perspectives are still available.

Kim and Neggers [6] showed that there is a mapping D : M → DM on means such that if M is a Fibonacci

mean so is DM , that if M is the harmonic mean, then DM is the arithmetic mean, and if M is a Fibonacci

mean, then limn→∞DnM is the golden section mean. Surprisingly novel perspectives are still available and will

presumably continue to be so for the future as long as mathematical investigations continue to be made.

Han et al. [4] considered several properties of Fibonacci sequences in arbitrary groupoids. They discussed

Fibonacci sequences in both several groupoids and groups. The present authors [7] introduced the notion of

generalized Fibonacci sequences over a groupoid and discussed these in particular for the case where the groupoid

contains idempotents and pre-idempotents. Using the notion of Smarandache-type P -algebras they obtained

several relations on groupoids which are derived from generalized Fibonacci sequences.

In [5] Han et al. discussed Fibonacci functions on the real numbers R, i.e., functions f : R→ R such that for all

x ∈ R, f(x+2) = f(x+1)+f(x), and developed the notion of Fibonacci functions using the concept of f -even and

f -odd functions. Moreover, they showed that if f is a Fibonacci function then limx→∞f(x+1)f(x) = 1+

√5

2 . The present

authors [8] discussed Fibonacci functions using the (ultimately) periodicity and we also discuss the exponential

0∗ Correspondence: Tel.: +82 33 248 2011, Fax: +82 33 256 2011 (K. S. So).


1479 Hee Sik Kim ET AL 1479-1486

Hee Sik Kim, J. Neggers and Keum Sook So∗

Fibonacci functions. Especially, given a non-negative real-valued function, we obtain several exponential Fibonacci

functions.

The present authors [9] introduced the notions of Fibonacci (co-)derivative of real-valued functions, and found

general solutions of the equations 4(f(x)) = g(x) and (4 + I)(f(x)) = g(x). Moreover, they [10] defined and

studied a function F : [0,∞) → R and extensions F : R → C, F : C → C which are continuous and such

that if n ∈ Z, the set of all integers, then F (n) = Fn, the nth Fibonacci number based on F0 = F1 = 1. If

x is not an integer and x < 0, then F (x) may be a complex number, e.g., F (−1.5) = 12 + i. If z = a + bi,

then F (z) = F (a) + iF (b − 1) defines complex Fibonacci numbers. In connection with this function (and in

general) they defined a Fibonacci derivative of f : R → R as (4f)(x) = f(x + 2) − f(x + 1) − f(x) so that if

(4f)(x) ≡ 0 for all x ∈ R, then f is a (real) Fibonacci function. A complex Fibonacci derivative 4 is given as

4f(a+ bi) = 4f(a) + i4 f(b− 1) and its properties are discussed in same detail.

2. Fibonacci logarithm

Let F = F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5, · · · , be the Fibonacci sequence where Fk denotes the kth

Fibonacci number. Let F∗ = F ∗0 = 1, F ∗1 = 2, F ∗2 = 3, F ∗3 = 5, · · · denote the short Fibonacci sequence and let

E(F∗) = F ∗0 = 1, F ∗1 = 2, F ∗−1 = 12 , F

∗2 = 3, F ∗−2 = 1

3 , · · · denote the extended short Fibonacci sequence or the

Fibonacci Z-sequence. In general, by a Z-sequence S = a0, a1, a−1, a2, a−2, · · · we mean a sequence of positive

real numbers satisfying ai < ai+1 for all i ∈ Z where limk→∞ ak =∞ and limk→−∞ ak = 0.

In general, for a Z-sequence S, we say that a positive real number x has S-characteristic k if k is the unique

integer such that ak ≤ x < ak+1. Thus, if S = E(F∗), then F ∗k ≤ x < F ∗k+1 means that x has E(F∗)-characteristic

k. Given the context, we shall refer to this number as the Fibonacci characteristic of x.

For example, if x = 1.2, then F ∗0 = 1 < 1.2 < 2 = F ∗1 , and hence its Fibonacci characteristic is 0. Again,

if x = 110 , then F ∗4 = F5 = 8 < 1

x = 10 < 13 = F ∗5 whence F ∗−5 < x < F ∗−4, i.e, x = 110 has Fibonacci

characteristic −5, while 10 has Fibonacci characteristic 4. In discussing characteristics we keep in mind that for

k ≥ 1, F ∗k = Fk+1, e.g., F ∗1 = F2 = 2.

We have a rule for computing Fibonacci characteristics of numbers 0 < x < 1. Indeed, compute the Fibonacci

characteristic of 1x . If it is n > 0, then the Fibonacci characteristic of x is −(n+ 1). Computing or estimating the

Fibonacci characteristics of numbers x ≥ 1 will be a topic of interest to be discussed below.

Suppose now that F ∗k ≤ x < F ∗k+1. Then the Fibonacci mantissa of x is defined as the number α such that

(F ∗k+1)α

(F ∗k )α−1= x

We note that if x = F ∗k , then (F ∗k+1/F∗k )α = 1, and hence α = 0. Also, if x = F ∗k+1, then (F ∗k+1/F

∗k )α−1 = 1, and

hence α = 1. Hence, 0 ≤ α < 1 for numbers x such that F ∗k ≤ x < F ∗k+1.

Finally, we define the Fibonacci logarithm logF (x) = k + α, where k is the Fibonacci characteristic of x and

where α is the Fibonacci mantissa of x. F is called the pseudo base of the logarithm. We simply denote logF (x)

by logF (x). It is our purpose in this paper to discuss the Fibonacci logarithm function of the positive real variable

x and to make several observations as a consequence.




3. Fibonacci logarithm logF (x)

We begin by noting that logF (x) is continuous everywhere. If F ∗k ≤ x < F ∗k+1, then logF (x) = k + α =

logF (F ∗k ) + α. We will show that logF (x) is differentiable at x. As we have seen above, if α for F ∗k+1 is computed

relative to F ∗k , then it equals 1 and hence logF (F ∗k )+1 = k+1 = logF (F ∗k+1) as well. Hence limx−→F∗k+1

logF (x) =

limx+→F∗k+1

logF (x) = k + 1, establishing continuity at that point.

Theorem 3.1. If logF (x) is the Fibonacci logarithm function, then its derivative is

(logF (x))′ =1

x

1

ln(F ∗k+1/F∗k )

(3.1)

when F ∗k < x < F ∗k+1 (where ln means the natural logarithm function.)

Proof. We compute logF (x) and logF (x + h), where x and x + h are both in the open interval (F ∗k , F∗k+1).

Accordingly both have the same Fibonacci characteristic k. Assume α and β are Fibonacci mantissas of x and

x+ h respectively. Then logF (x+ h)− logF (x) = β − α, the difference of the Fibonacci mantissas. Consider the

following.

x+ h

x=

(F ∗k+1)β/(F ∗k )β−1

(F ∗k+1)α/(F ∗k )α−1

= (F ∗k+1/F∗k )β−α

(3.2)

It follows that

ln(1 +h

x) = (β − α)ln(F ∗k+1/F

∗k ) (3.3)

Hence, we obtain

logF (x+ h)− logF (x) = β − α

=ln(1 + h

x )

ln(F ∗k+1/F∗k )

(3.4)

It follows that

limh→0

logF (x+ h)− logF (x)

h= limh→0

1h ln(1 + h

x )

ln(F ∗K+1/F∗k )

=1

x

1

ln(F ∗k+1/F∗k ),

(3.5)

proving the theorem.

If b := F ∗k+1/F∗k , then the usual logarithm function logb(x) with the base b has the derivative (logb(x))′ = 1

x1

ln(b) ,

and hence in Theorem 3.1, (logF (x))′ = (logb(x))′ on the open interval (F ∗k , F∗k+1), i.e., the functions logF (x)

and logb(x) differ by a constant. Let Ck := logF (F ∗k ) − logb(F∗k ). We need to find an upper bound for Ck.

Given any particular value of k, one can of course immediately determine Ck precisely. For example, if k = 5,

then F ∗5 = 13, F ∗6 = 21 and b = 21/13, so that C5 = logF (13) − log 2113

(13) = 5 − log 2113

(13) = −0.34840144.

However, in order to obtain an improved sense of the behavior of Ck as a function of k, it may be better to

determine a fairly simple bound for Ck which is a function of k itself. If we let logF (x) := logb(x) + Ck and

tk := logb(F∗k ), then btk = F ∗k and tk = ln(F ∗k )/ln(b) = ln(F ∗k )/ln(F ∗k+1/F

∗k ). Since 1 < F ∗k+1/F

∗k ≤ 2, it follows

that ln(F∗

k+1

F∗k

) ≤ ln2 ≤ ln(e) = 1, so that tk > ln(F ∗k ), and hence

Ck = k − tk < k − ln(F ∗k ) (3.6)




If −k < 0, then F ∗−k = 1/Fk∗ and F ∗−k+1 = 1/F ∗k−1, so that if we let b∗ := F ∗−k+1/F

∗−k = F ∗k /F

∗k−1, so that we

may determine (logF (x))′ = 1x

1ln(b∗) for F ∗−k ≤ x < F ∗−k+1 or 1/F ∗k ≤ x < 1/F ∗−k+1. If we let logF (x) := logb(x) +

C−k for some C−k where F ∗−k ≤ x < F ∗−k+1 and if we let t−k := logb∗(F ∗−k), then t−k =ln(F∗

−k)

ln(b∗) < −ln(F ∗k ).

Hence we obtain:

C−k = logF (F ∗−k)− logb∗(F ∗−k)

= −k − t−k

< −k + ln(F ∗k )

(3.7)

From this we have k − ln(F ∗k ) < −C−k. We summarize:

Proposition 3.2. If Ck := logF (F ∗k )− logb(F∗k ) then it has a bound Ck < k − ln(F ∗k ) < −C−k.

Note that logF (x) is not differentiable at the F ∗k ’s. There is both a left-derivative and right-derivative at that

point. Indeed, for the left derivative we get

limx−→F∗

k

1

x

1

ln(b∗)=

1

F ∗k

1

ln(b∗), (3.8)

while for the right derivative it is:

limx+→F∗

k

1

x

1

ln(b)=

1

F ∗k

1

ln(b), (3.9)

Hence, we define the saltus (jump) at F ∗k to be

∆(F ∗k ) =1

F ∗k[

1

ln(b)− 1

ln(b∗)]

=1

F ∗k[

ln( b∗

b )

ln(b)ln(b∗)]

where b∗/b = (F ∗k /F∗k−1)/(F ∗k+1/F

∗k ) = (F ∗k )2/F ∗k−1F

∗k+1. We recall that (F ∗k )2 = F ∗k−1F

∗k+1 + (−1)k+1, and thus

we have

b∗

b=

< 1 if k is even,

> 1 otherwise(3.10)

It follows that

ln(b∗

b) =

< 0 if k is even,

> 0 otherwise(3.11)

Since F ∗k ln(b)ln(b∗) > 0, ∆(F ∗k )’s sign is determined by the sign of ln( b∗

b ). Hence we obtain:

Proposition 3.3. If ∆(F ∗k ) is the saltus at F ∗k , then

∆(F ∗k ) =

< 0 if k is even,

> 0 otherwise(3.12)

When k = 0, b∗

b = (F ∗0 )2/F ∗−1F∗1 = 1 and so ln(b∗/b) = 0. Thus ∆(F ∗0 ) = 0, i.e., logF (x) is differentiable at

F ∗0 = 1.




For negative integers, replacing F ∗−k by 1F∗

k, the saltus formula ∆(F ∗−k) is transformed to

∆(F ∗−k) =1

F ∗−k

ln(b∗/b)

ln(b)ln(b∗)

= F ∗kln[F ∗k−1F

∗k+1/(F

∗k )2]

ln(F∗

k+1

F∗k

)ln(F∗

k

F∗k−1

)

(3.13)

For example, for k = 3, we obtainF∗

4 F∗2

(F∗3 )2 = 8·3

25 = 2425 < 1. Of course, (F−3)

2

F−4F−2= (1/5)2/( 1

8 )( 13 ) = ( 1

25 )/( 124 ) when

computed directly.

The Fibonacci number F ∗k is said to be elliptical if (F ∗k )2/F ∗k+1F∗k−1 > 1; parabolic if (F ∗k )2/F ∗k+1F

∗k−1 = 1;

hyperbolic if (F ∗k )2/F ∗k+1F∗k−1 < 1.

For k ≥ 1, F ∗k is elliptical if k is odd; hyperbolic if k is even. The only parabolic Fibonacci number is F ∗0 = 1.

If −k < 0, then F ∗−k is elliptical if k is even and F ∗−k is hyperbolic if k is odd.

Proposition 3.4. logF (x) + logF ( 1x ) = Ck + C−k.

Proof. If x 6= F ∗k for all k ∈ Z, then F ∗k ≤ x < F ∗k+1 yields logF (x) = k + α = logb(x) + Ck. Now,

F ∗−(k+1) <1x < F ∗−k and F ∗−k/F

∗−(k+1) = b, so that logF ( 1

x ) = logb(1x ) + C−k = − logb(x) + C−k, proving the

proposition.

4. Determining the Fibonacci Mantissa of x

Suppose that F ∗k ≤ x < F ∗k+1, i.e., x has Fibonacci characteristic k. If f(x) := logF (x), then we may determine

f(x) from its Taylor series around F ∗k provided we consider the derivatives at F ∗k to be the right-hand derivatives.

Thus, we will write f(x) = x+ α, with f(F ∗k ) = k and hence also

α =∞∑n=1

f (n)(F ∗k )(x− F ∗k )n

n!(4.1)

On the interval (F ∗k , F∗k+1), the derivative (logF (x))′ = f ′(x) = 1

x1

ln(b) , b = F ∗k+1/F∗k . Hence the nth derivative

fn(x) is (−1)n−1(n−1)!xn

1ln(b) . Thus, f (n)(F ∗k )

(x−F∗k )n

n! = (−1)n−1

nln(b) (x−F∗

k

F∗k

)n, whence the fact that x−F ∗k < F ∗k+1−F ∗k =

F ∗k−1 guarantees that |(x − F ∗k )/F ∗k | < |F ∗k−1/F ∗k | < 1, so that the series for α converges by the ratio-test. We

summarize:

Theorem 4.1. If f(x) = logF (x) with F ∗k ≤ x < F ∗k+1, then f(x) = k + α, where

α =∞∑n=1

f (n) =(−1)n−1

n ln(b)

[x− F ∗kF ∗k

]n

In particular, if x = F ∗k+1, then α = 1, and we obtain an expression

1 =1

ln(b)

[ ∞∑n=1

(−1)n−1[F ∗k+1 − F ∗k

F ∗k

]n], (4.2)




so that we obtain an expression for ln(b) as follows:

ln(b) = ln

[F ∗k+1

F ∗k

]=∞∑n=1

(−1)n−1

n

[F ∗k+1 − F ∗k

F ∗k

]n=∞∑n=1

(−1)n−1

n

[F ∗k−1F ∗k

]n (4.3)

From the expression for α as an infinite series, if we consider only the first term α1, where

α1 = f (1)(F ∗k )(x− F ∗k ) =1

F ∗k(x− F ∗k )

1

ln

[F∗

k+1

F∗k

] ,we may rewrite ln(F ∗k+1/F

∗k ) = ln(1 + F ∗k−1/F

∗k ). If we approximate this expression by F ∗k−1/F

∗k , i.e., ln(1 +

F ∗k−1/F∗k ) ∼ F ∗k−1/F ∗k , then we obtain a further approximation:

α1 =1

F ∗k

x− F ∗kF∗

k−1

F∗k

=x− F ∗kF ∗k−1

Now F ∗k−1 = F ∗k+1−F ∗k is the length of the interval (F ∗k , F∗k+1) and thus α∗1 represents the mantissa corresponding

to the straight line connecting (F ∗k , k) with (F ∗k+1, k + 1). See the following figure:

We may thus use the α∗1 estimate to construct a piecewise-linear average Fibonacci logarithm log∗F (c) = k + α∗1

for F ∗k ≤ x < F ∗k+1. We summarize:

Proposition 4.2. If we define log∗F (x) = k + α∗1 for F ∗k ≤ x < F ∗k+1, then

(i) log∗F (F ∗k ) = logF (F ∗k ) for all integers k,

(ii) log∗F (F ∗k ) is continuous and differentiable on non-Fibonacci numbers,

(iii) if x has Fibonacci characteristic k, then (log∗F (x))′ = 1F∗

k−1.




Thus, for log∗F (x) we have the saltus ∆∗(F ∗k ) as follow:

∆∗(F ∗k ) =1

F ∗k− 1

F ∗k−1

=F ∗k−1 − F ∗kF ∗kF

∗k−1

= −F ∗k−2F ∗kF

∗k−1

(4.4)

Hence, e.g., ∆∗(F ∗5 ) = − F∗3

F∗5 F

∗4

= − 5104 .

Returning to Theorem 4.1, a second level approximation for α is α1 + α2, where α2 = f (2)(F ∗k )(x − F ∗k )2 12! =

−12ln(b)

[x−F∗

k

F∗k

]2. Now ln(b) = ln(1 + F ∗k /F

∗k ), where in the first order approximation we approximate ln(b) =

ln(1 +A) by A itself. In order to be fair to the second order approximation we should be fair to ln(b) as well and

use ln(b) = A − A2/2 from its MacLaurin expansion. This affects the α1 estimate as well. Indeed, we obtain a

new estimate for α1 as:

α1 =1

F ∗k

x− F ∗kF∗

k−1

F∗k− 1

2 (F∗

k−1

F∗k

)2=

2F ∗k (x− F ∗k )

F ∗k−1(2F ∗k − F ∗k−1)

and for α2 we find that

α2 =−(x− F ∗k )2

F ∗k−1(2− F ∗k−1)=

(x− F ∗k )2

F ∗k (F ∗k−1 − 2)

In general, if we wish to make an n th order approximation for α, we write α = α1 + α2 + · · · + αn, where

αn = f (n)(F ∗k )(x− F ∗k )n/n! = (−1)n−1

nln(b)

[x−F∗

k

F∗k

]n.

At the same time we use an nth order approximation for ln(b) = ln(1+F ∗k−1/F∗k ) = ln(1+A) by setting it equal

to ln(b) = A−A2/2+ · · ·+(−1)n−1An/n, and recomputing α1, · · · , αn−1 in terms of the nth order approximation

of ln(b) as well.

5. Concluding remark and future works

In this paper we discussed Fibonacci Z-sequences and obtained some interesting results on Fibonacci logarithm

functions. Based on Fibonacci logarithm functions, we shall discuss on Fibonacci exponential functions and obtain

several properties on it.

Acknowledgement

∗This work was supported by Hallym University Research Fund HRF-201707-007.




References

[1] K. Atanassove et. al, New Visual Perspectives on Fibonacci numbers, World Sci. Pub. Co., New Jersey,

2002.

[2] R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Sci. Pub. Co., New Jersey, 1997.

[3] J. S. Han, H. S. Kim, J. Neggers, The Fibonacci norm of a positive integer n- observations and conjectures

-, Int. J. Number Th. 6 (2010), 371-385.

[4] J. S. Han, H. S. Kim and J. Neggers, Fibonacci sequences in groupoids, Advances in Difference Equations

2012 2012:19 (doi:10.1186/1687-1847-2012-19).

[5] J. S. Han, H. S. Kim and J. Neggers, On Fibonacci functions with Fibonacci numbers, Advances in

Difference Equations 2012 2012:126 (doi:10.1186/1687-1847-2012-126).

[6] H. S. Kim and J. Neggers, On Fibonacci means and golden section mean, Computers and Mathematics

with Applications 56 (2008), 228-232.

[7] H. S. Kim, J. Neggers and K. S. So, Generalized Fibonacci sequences in groupoids, Advances in Difference

Equations 2013 2013:26 (doi:10.1186/1687-1847-2013-26).

[8] H. S. Kim, J. Neggers and K. S. So, On Fibonacci functions with periodicity, Advances in Difference

Equations 2014 2014:293. (doi:10.1186/1687-1847-2014-293).

[9] H. S. Kim, J. Neggers and K. S. So, On continuous Fibonacci functions, J. Comput. & Appl. Math. 24

(2018), 1482-1490.

[10] H. S. Kim, J. Neggers and K. S. So, On Fibonacci derivative equations, J. Comput. & Appl. Math. 24

(2018), 628-635.



1

HERMITE-HADAMARD TYPE FRACTIONAL INTEGRAL

INEQUALITIES FOR MT(r;g,m,ϕ)-PREINVEX FUNCTIONS

MUHAMMAD ADIL KHAN, YU-MING CHU∗, ARTION KASHURI, ROZANA LIKO

Abstract. In the present paper, the notion of MT(r;g,m,ϕ)-preinvex func-

tion is introduced and some new integral inequalities for the left-hand side of

Gauss-Jacobi type quadrature formula involving MT(r;g,m,ϕ)-preinvex func-tions are given. Moreover, some generalizations of Hermite-Hadamard type

inequalities for MT(r;g,m,ϕ)-preinvex functions that are twice differentiable

via Riemann-Liouville fractional integrals are established. Some applicationsto special means are also given. These general inequalities give us some new

estimates for Hermite-Hadamard type fractional integral inequalities.

1. Introduction and Preliminaries

The following notations are used throughout this paper. We use I to denote aninterval on the real line R = (−∞,+∞) and I to denote the interior of I. Forany subset K ⊆ Rn,K is used to denote the interior of K. Rn is used to denote an-dimensional vector space. The set of integrable functions on the interval [a, b] isdenoted by L1[a, b].

The following inequality, named Hermite-Hadamard inequality, is one of the mostfamous inequalities in the literature for convex functions.

Theorem 1.1. Let f : I ⊆ R −→ R be a convex function on I and a, b ∈ I witha < b. Then the following inequality holds:

f

(a+ b

2

)≤ 1

b− a

∫ b

a

f(x)dx ≤ f(a) + f(b)

2. (1.1)

In (see [12],[22]), Tunc and Yildirim defined the following so-called MT-convexfunction:

Definition 1.2. A function f : I ⊆ R −→ R is said to belong to the class ofMT(I), if it is nonnegative and for all x, y ∈ I and t ∈ (0, 1) satisfies the followinginequality:

f(tx+ (1− t)y) ≤√t

2√

1− tf(x) +

√1− t2√tf(y). (1.2)

12010 Mathematics Subject Classification: Primary: 26A51; Secondary: 26A33, 26D07, 26D10,

26D15.Key words and phrases. Hermite-Hadamard type inequality, MT-convex function, Holder’s

inequality, power mean inequality, Minkowski inequality, Riemann-Liouville fractional integral,

m-invex, P -function.The research was supported by the Natural Science Foundation of China under Grants

61673169, 61374086, 11371125, 11401191, 11701176, the Tianyuan Special Funds of the National

Natural Science Foundation of China under Grant 11626101 and the Natural Science Foundationof the Department of Education of Zhejiang Province under GrantY201635325.

∗Corresponding author: Yu-Ming Chu, Email: [email protected].

1


1487 KHAN ET AL 1487-1503

2 MUHAMMAD ADIL KHAN, YU-MING CHU∗, ARTION KASHURI, ROZANA LIKO

In recent years, various generalizations, extensions and variants of such inequal-ities have been obtained. For other recent results concerning Hermite-Hadamardtype inequalities through various classes of convex functions, (please see [3],[4],[9]-[17], [24], [25]).

Fractional calculus (see [21]), was introduced at the end of the nineteenth cen-tury by Liouville and Riemann, the subject of which has become a rapidly growingarea and has found applications in diverse fields ranging from physical sciences andengineering to biological sciences and economics.

Definition 1.3. Let f ∈ L1[a, b]. The Riemann-Liouville integrals Jαa+f and Jαb−fof order α > 0 with a ≥ 0 are defined by

Jαa+f(x) =1

Γ(α)

∫ x

a

(x− t)α−1f(t)dt, x > a

and

Jαb−f(x) =1

Γ(α)

∫ b

x

(t− x)α−1f(t)dt, b > x,

where Γ(α) =

∫ +∞

0

e−uuα−1du is the classical gamma function (see [26]-[31], [32]

[33]-[35]). Here J0a+f(x) = J0

b−f(x) = f(x).In the case of α = 1, the fractional integral reduces to the classical integral.

Due to the wide application of fractional integrals, some authors extended to s-tudy fractional Hermite-Hadamard type inequalities for functions of different classes(please see [6]-[21]).

Definition 1.4. (see [2]) A nonnegative function f : I ⊆ R −→ [0,+∞) is said tobe P -function or P -convex, if

f(tx+ (1− t)y) ≤ f(x) + f(y), ∀x, y ∈ I, t ∈ [0, 1].

Definition 1.5. (see [5]) A set K ⊆ Rn is said to be invex with respect to themapping η : K ×K −→ Rn, if x+ tη(y, x) ∈ K for every x, y ∈ K and t ∈ [0, 1].

Notice that every convex set is invex with respect to the mapping η(y, x) = y−x,but the converse is not necessarily true. For more details (see [5],[7]).

Definition 1.6. (see [8]) The function f defined on the invex set K ⊆ Rn is saidto be preinvex with respect η, if for every x, y ∈ K and t ∈ [0, 1], we have that

f (x+ tη(y, x)) ≤ (1− t)f(x) + tf(y).

The concept of preinvexity is more general than convexity since every convexfunction is preinvex with respect to the mapping η(y, x) = y − x, but the converseis not true.

The Gauss-Jacobi type quadrature formula has the following∫ b

a

(x− a)p(b− x)qf(x)dx =

+∞∑k=0

Bm,kf(γk) +R?m|f |, (1.3)

for certain Bm,k, γk and rest R?m|f | (see [18]).


1488 KHAN ET AL 1487-1503

HERMITE-HADAMARD TYPE FRACTIONAL INTEGRAL INEQUALITIES. . . 3

Recently, Liu (see [19]) obtained several integral inequalities for the left-hand sideof (1.3) under the Definition 1.4 of P -function.

Also in (see [20]), Ozdemir et al. established several integral inequalities concerningthe left-hand side of (1.3) via some kinds of convexity.

Motivated by these results, in Section 2, the notion of MT(r;g,m,ϕ)-preinvex func-tion is introduced and some new integral inequalities for the left-hand side of (1.3)involving MT(r;g,m,ϕ)-preinvex functions are given. In Section 3, some generaliza-tions of Hermite-Hadamard type inequalities for nonnegative MT(r;g,m,ϕ)-preinvexfunctions that are twice differentiable via fractional integrals are given. In Section4, some applications to special means are given. In Section 5, some conclusions andfuture research are given. These general inequalities give us some new estimatesfor Hermite-Hadamard type fractional integral inequalities.

2. New integral inequalities for MT(r;g,m,ϕ)-preinvex functions

Definition 2.1. (see [1]) A set K ⊆ Rn is said to be m-invex with respect to themapping η : K×K×(0, 1] −→ Rn for some fixed m ∈ (0, 1], if mx+tη(y, x,m) ∈ Kholds for each x, y ∈ K and any t ∈ [0, 1].

Remark 2.2. In Definition 2.1, under certain conditions, the mapping η(y, x,m)could reduce to η(y, x). For example when m = 1, then the m-invex set degeneratesan invex set on K.

We next give new definition, to be referred as MT(r;g,m,ϕ)-preinvex function.

Definition 2.3. Let K ⊆ Rn be an open m-invex set with respect to η : K ×K ×(0, 1] −→ Rn, g : [0, 1] −→ (0, 1) be a differentiable function and ϕ : I −→ K isa continuous function. The function f : K −→ (0,+∞) is said to be MT(r;g,m,ϕ)-preinvex function with respect to η, if

f(mϕ(y) + g(t)η(ϕ(x), ϕ(y),m)) ≤Mr (f(ϕ(x)), f(ϕ(y)),m; g(t)) (2.1)

holds for any fixed m ∈ (0, 1] and for all x, y ∈ I, t ∈ [0, 1], where

Mr(f(ϕ(x)), f(ϕ(y)),m; g(t))

=

[m√g(t)

2√

1−g(t)fr(ϕ(x)) +

m√

1−g(t)2√g(t)

fr(ϕ(y))

] 1r

, if r 6= 0;

[f(ϕ(x))

] m√g(t)

2√

1−g(t)[f(ϕ(y))

]m√1−g(t)2√g(t) , if r = 0,

is the weighted power mean of order r for positive numbers f(ϕ(x)) and f(ϕ(y)).

Remark 2.4. In Definition 2.3, it is worthwhile to note that the class MT(r;g,m,ϕ)(I)is a generalization of the class MT(I) given in Definition 1.2 for r = m = 1 withrespect to η(ϕ(x), ϕ(y),m) = ϕ(x) − mϕ(y), ϕ(x) = x, ∀x, y ∈ I, g(t) = t, ∀t ∈(0, 1).

Let give below a nontrivial example for motivation of this new interesting classof MT(r;g,m,ϕ)-preinvex functions.


1489 KHAN ET AL 1487-1503


Example 2.5. f1, f2 : (1,+∞) −→ (0,+∞), f1(x) = xp, f2(x) = (1 + x)p, p ∈(0, 1

1000

); h : [1, 3/2] −→ (0,+∞), h(x) = (1 + x2)k, k ∈

(0, 1

100

), are sim-

ple examples of the new class of MT(1;t,m,x)-preinvex functions with respect toη(ϕ(x), ϕ(y),m) = ϕ(x)−mϕ(y), ϕ(x) = x, g(t) = t, r = 1, for any fixedm ∈ (0, 1],but they are not convex.

In this section, in order to prove our main results regarding some new integralinequalities involving MT(r;g,m,ϕ)-preinvex functions, we need the following newinteresting lemma:

Lemma 2.6. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ [0, 1] is adifferentiable function. Assume that f : K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] −→R is a continuous function on K with respect to η : K × K × (0, 1] −→ R, formϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m). Then for any fixed m ∈ (0, 1] and p, q > 0, wehave∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)

(x−mϕ(a))p(mϕ(a) + η(ϕ(b), ϕ(a),m)− x)qf(x)dx

= η(ϕ(b), ϕ(a),m)p+q+1

×∫ 1

0

gp(t)(1− g(t))qf(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)].

Proof. It is easy to observe that∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)


= η(ϕ(b), ϕ(a),m)

∫ 1

0

(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m)−mϕ(a))p

×(mϕ(a) + η(ϕ(b), ϕ(a),m)−mϕ(a)− g(t)η(ϕ(b), ϕ(a),m))q

×f(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)]

= η(ϕ(b), ϕ(a),m)p+q+1

×∫ 1

0

gp(t)(1− g(t))qf(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)].

The following definition will be used in the sequel.

Definition 2.7. The Euler beta function is defined for x, y > 0 as

β(x, y) =

∫ 1

0

tx−1(1− t)y−1dt =Γ(x)Γ(y)

Γ(x+ y).

Theorem 2.8. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ (0, 1) is adifferentiable function. Assume that f : K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] −→(0,+∞) is a continuous function on K with respect to η : K ×K × (0, 1] −→ R,for mϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m). Let k > 1 and 0 < r ≤ 1. If f

kk−1 is

MT(r;g,m,ϕ)-preinvex function on an open m-invex set K for any fixed m ∈ (0, 1],then for any fixed p, q > 0, we have∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)



1490 KHAN ET AL 1487-1503


≤(m

2

) k−1rk |η(ϕ(b), ϕ(a),m)|p+q+1B

1k (g(t); k, p, q)

×[Ar2(g(t); r)f

rkk−1 (ϕ(a)) +Ar1(g(t); r)f

rkk−1 (ϕ(b))

] k−1rk

,

where

B(g(t); k, p, q) =

∫ 1

0

gkp(t)(1− g(t))kqd[g(t)];

A1(g(t); r) =

∫ 1−g(0)

1−g(1)

(√1− tt

) 1r

dt;

A2(g(t); r) =

∫ g(1)

g(0)

(√1− tt

) 1r

dt.

Proof. Let k > 1 and 0 < r ≤ 1. Since fkk−1 is MT(r;g,m,ϕ)-preinvex function on

K, combining with Lemma 2.6, Holder inequality and Minkowski inequality for allt ∈ [0, 1] and for any fixed m ∈ (0, 1], we get∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)


≤ |η(ϕ(b), ϕ(a),m)|p+q+1

[∫ 1

0

gkp(t)(1− g(t))kqd[g(t)]

] 1k

×

[∫ 1

0

fkk−1 (mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)]

] k−1k

≤ |η(ϕ(b), ϕ(a),m)|p+q+1B1k (g(t); k, p, q)

×

[∫ 1

0

(m√g(t)

2√

1− g(t)fr(ϕ(b))

kk−1 +

m√

1− g(t)

2√g(t)

fr(ϕ(a))kk−1

) 1r

d[g(t)]

] k−1k

≤(m

2

) k−1rk |η(ϕ(b), ϕ(a),m)|p+q+1B

1k (g(t); k, p, q)

×

∫ 1

0

( √g(t)√

1− g(t)

) 1r

fkk−1 (ϕ(b))d[g(t)]

r

+

∫ 1

0

(√1− g(t)√g(t)

) 1r

fkk−1 (ϕ(a))d[g(t)]

r k−1rk

=(m

2

) k−1rk |η(ϕ(b), ϕ(a),m)|p+q+1B

1k (g(t); k, p, q)

×[Ar2(g(t); r)f

rkk−1 (ϕ(a)) +Ar1(g(t); r)f

rkk−1 (ϕ(b))

] k−1rk

.


1491 KHAN ET AL 1487-1503


Corollary 2.9. Under the same conditions as in Theorem 2.8 for r = 1 andg(t) = t, we get∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)


≤(mπ

4

) k−1k |η(ϕ(b), ϕ(a),m)|p+q+1β

1k (kp+ 1, kq + 1)

×[f

kk−1 (ϕ(a)) + f

kk−1 (ϕ(b))

] k−1k

.

Theorem 2.10. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ (0, 1) isa differentiable function. Assume that f : K = [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)] −→(0,+∞) is a continuous function on K with respect to η : K×K×(0, 1] −→ R, formϕ(a) < mϕ(a) + η(ϕ(b), ϕ(a),m). Let l ≥ 1 and 0 < r ≤ 1. If f l is MT(r;g,m,ϕ)-preinvex function on an open m-invex set K for any fixed m ∈ (0, 1], then for anyfixed p, q > 0, we have∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)


≤(m

2

) 1rl |η(ϕ(b), ϕ(a),m)|p+q+1B

l−1l (g(t); 1, p, q)

×

[Br(g(t);

1

2r, 2pr − 1, 2qr + 1

)frl(ϕ(a))

+Br(g(t);

1

2r, 2pr + 1, 2qr − 1

)frl(ϕ(b))

] 1rl

.

Proof. Let l ≥ 1 and 0 < r ≤ 1. Since f l is MT(r;g,m,ϕ)-preinvex function on K,combining with Lemma 2.6, the well-known power mean inequality and Minkowskiinequality for all t ∈ [0, 1] and for any fixed m ∈ (0, 1], we get∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)


= η(ϕ(b), ϕ(a),m)p+q+1

∫ 1

0

[gp(t)(1− g(t))q

] l−1l[gp(t)(1− g(t))q

] 1l

×f(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)]

≤ |η(ϕ(b), ϕ(a),m)|p+q+1

[∫ 1

0

gp(t)(1− g(t))qd[g(t)]

] l−1l

×

[∫ 1

0

gp(t)(1− g(t))qf l(mϕ(a) + g(t)η(ϕ(b), ϕ(a),m))d[g(t)]

] 1l

≤ |η(ϕ(b), ϕ(a),m)|p+q+1Bl−1l (g(t); 1, p, q)

×

[∫ 1

0

gp(t)(1−g(t))q

(m√g(t)

2√

1− g(t)fr(ϕ(b))l +

m√

1− g(t)

2√g(t)

fr(ϕ(a))l

) 1r

d[g(t)]

] 1l

≤(m

2

) 1rl |η(ϕ(b), ϕ(a),m)|p+q+1B

l−1l (g(t); 1, p, q)


1492 KHAN ET AL 1487-1503


×

(∫ 1

0

gp+12r (t)(1− g(t))q−

12r f l(ϕ(b))d[g(t)]

)r

+

(∫ 1

0

gp−12r (t)(1− g(t))q+

12r f l(ϕ(a))d[g(t)]

)r 1rl

=(m

2

) 1rl |η(ϕ(b), ϕ(a),m)|p+q+1B

l−1l (g(t); 1, p, q)

×

[Br(g(t);

1

2r, 2pr − 1, 2qr + 1

)frl(ϕ(a))

+Br(g(t);

1

2r, 2pr + 1, 2qr − 1

)frl(ϕ(b))

] 1rl

.

Corollary 2.11. Under the same conditions as in Theorem 2.10 for r = 1 andg(t) = t, we get∫ mϕ(a)+η(ϕ(b),ϕ(a),m)

mϕ(a)


≤(m

2

) 1l |η(ϕ(b), ϕ(a),m)|p+q+1β

l−1l (p+ 1, q + 1)

×

[β

(p+

1

2, q +

3

2

)f l(ϕ(a)) + β

(p+

3

2, q +

1

2

)f l(ϕ(b))

] 1l

.

3. Hermite-Hadamard type fractional integral inequalities forMT(r;g,m,ϕ)-preinvex functions

In this section, in order to prove our main results regarding some generaliza-tions of Hermite-Hadamard type inequalities for MT(r;g,m,ϕ)-preinvex functions viafractional integrals, we need the following new fractional integral identity:

Lemma 3.1. Let ϕ : I −→ K be a continuous function and g : [0, 1] −→ [0, 1] isa differentiable function. Suppose K ⊆ R be an open m-invex subset with respectto η : K × K × (0, 1] −→ R for any fixed m ∈ (0, 1] and let mϕ(a) < mϕ(a) +η(ϕ(b), ϕ(a),m). Assume that f : K −→ R be a twice differentiable function on K

and f ′′ is integrable on [mϕ(a),mϕ(a)+η(ϕ(b), ϕ(a),m)]. Then for α > 0, we have

ηα+1(ϕ(x), ϕ(a),m)

(α+ 1)η(ϕ(b), ϕ(a),m)

×[(1− gα+1(1))f ′(mϕ(a) + g(1)η(ϕ(x), ϕ(a),m))

−(1− gα+1(0))f ′(mϕ(a) + g(0)η(ϕ(x), ϕ(a),m))]

+ηα+1(ϕ(x), ϕ(b),m)

(α+ 1)η(ϕ(b), ϕ(a),m)

×[(1− gα+1(1))f ′(mϕ(b) + g(1)η(ϕ(x), ϕ(b),m))

−(1− gα+1(0))f ′(mϕ(b) + g(0)η(ϕ(x), ϕ(b),m))]


1493 KHAN ET AL 1487-1503


+ηα(ϕ(x), ϕ(a),m)

η(ϕ(b), ϕ(a),m)

×[gα(1)f(mϕ(a) + g(1)η(ϕ(x), ϕ(a),m))

−gα(0)f(mϕ(a) + g(0)η(ϕ(x), ϕ(a),m))]

+ηα(ϕ(x), ϕ(b),m)

η(ϕ(b), ϕ(a),m)

×[gα(1)f(mϕ(b) + g(1)η(ϕ(x), ϕ(b),m))

−gα(0)f(mϕ(b) + g(0)η(ϕ(x), ϕ(b),m))]

− α

η(ϕ(b), ϕ(a),m)

×

[∫ mϕ(a)+g(1)η(ϕ(x),ϕ(a),m)

mϕ(a)+g(0)η(ϕ(x),ϕ(a),m)

(t−mϕ(a))α−1f(t)dt

+

∫ mϕ(b)+g(1)η(ϕ(x),ϕ(b),m)

mϕ(b)+g(0)η(ϕ(x),ϕ(b),m)

(t−mϕ(b))α−1f(t)dt

]

=ηα+2(ϕ(x), ϕ(a),m)

(α+ 1)η(ϕ(b), ϕ(a),m)

×∫ 1

0

(1− gα+1(t))f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))d[g(t)]

+ηα+2(ϕ(x), ϕ(b),m)

(α+ 1)η(ϕ(b), ϕ(a),m)

×∫ 1

0

(1− gα+1(t))f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))d[g(t)]. (3.1)

Proof. A simple proof of the equality can be done by performing two integrationby parts in the integrals from the right side and changing the variable. The detailsare left to the interested reader.

Let denote

If,g,η,ϕ(x;α,m, a, b) =ηα+2(ϕ(x), ϕ(a),m)

(α+ 1)η(ϕ(b), ϕ(a),m)

×∫ 1

0

(1− gα+1(t))f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))d[g(t)]

+ηα+2(ϕ(x), ϕ(b),m)

(α+ 1)η(ϕ(b), ϕ(a),m)

×∫ 1

0

(1− gα+1(t))f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))d[g(t)]. (3.2)

Using relation (3.2), the following results can be obtained for the correspondingversion for power of the absolute value of the second derivative.


1494 KHAN ET AL 1487-1503


Theorem 3.2. Let ϕ : I −→ A be a continuous function and g : [0, 1] −→ (0, 1)is a differentiable function. Suppose A ⊆ R be an open m-invex subset with respectto η : A × A × (0, 1] −→ R for any fixed m ∈ (0, 1] and let mϕ(a) < mϕ(a) +η(ϕ(b), ϕ(a),m). Assume that f : A −→ (0,+∞) be a twice differentiable functionon A. If f ′′q is nonnegative MT(r;g,m,ϕ)-preinvex function, q > 1, p−1 + q−1 = 1,then for α > 0 and 0 < r ≤ 1, we have

|If,g,η,ϕ(x;α,m, a, b)| ≤(m

2

) 1rq C

1p (g(t); p, α)

(α+ 1)|η(ϕ(b), ϕ(a),m)|

×

|η(ϕ(x), ϕ(a),m)|α+2

[Ar2(g(t); r)f ′′(ϕ(a))rq +Ar1(g(t); r)f ′′(ϕ(x))rq

] 1rq

+ |η(ϕ(x), ϕ(b),m)|α+2[Ar2(g(t); r)f ′′(ϕ(b))rq +Ar1(g(t); r)f ′′(ϕ(x))rq

] 1rq

, (3.3)

where C(g(t); p, α) =

∫ 1

0

(1− gα+1(t))pd[g(t)].

Proof. Suppose that q > 1 and 0 < r ≤ 1. Using relation (3.2), nonnegativeMT(r;g,m,ϕ)-preinvexity of f ′′q, Holder inequality, Minkowski inequality and tak-ing the modulus, we have

|If,g,η,ϕ(x;α,m, a, b)| ≤ |η(ϕ(x), ϕ(a),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

×∫ 1

0

|1− gα+1(t)|∣∣f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))

∣∣d[g(t)]

+|η(ϕ(x), ϕ(b),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

×∫ 1

0

|1− gα+1(t)|∣∣f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))

∣∣d[g(t)]

≤ |η(ϕ(x), ϕ(a),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

(1− gα+1(t))pd[g(t)]

) 1p

×(∫ 1

0

f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))qd[g(t)]

) 1q

+|η(ϕ(x), ϕ(b),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

(1− gα+1(t))pd[g(t)]

) 1p

×(∫ 1

0

f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))qd[g(t)]

) 1q

≤ |η(ϕ(x), ϕ(a),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

(1− gα+1(t))pd[g(t)]

) 1p

×

[∫ 1

0

(m√g(t)

2√

1− g(t)f ′′(ϕ(x))rq +

m√

1− g(t)

2√g(t)

f ′′(ϕ(a))rq

) 1r

d[g(t)]

] 1q

+|η(ϕ(x), ϕ(b),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

(1− gα+1(t))pd[g(t)]

) 1p


1495 KHAN ET AL 1487-1503


×

[∫ 1

0

(m√g(t)

2√

1− g(t)f ′′(ϕ(x))rq +

m√

1− g(t)

2√g(t)

f ′′(ϕ(b))rq

) 1r

d[g(t)]

] 1q

≤(m

2

) 1rq |η(ϕ(x), ϕ(a),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|C

1p (g(t); p, α)

×

∫ 1

0

( √g(t)√

1− g(t)

) 1r

f ′′(ϕ(x))qd[g(t)]

r

+

∫ 1

0

(√1− g(t)√g(t)

) 1r

f ′′(ϕ(a))qd[g(t)]

r 1rq

+(m

2

) 1rq |η(ϕ(x), ϕ(b),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|C

1p (g(t); p, α)

×

∫ 1

0

( √g(t)√

1− g(t)

) 1r

f ′′(ϕ(x))qd[g(t)]

r

+

∫ 1

0

(√1− g(t)√g(t)

) 1r

f ′′(ϕ(b))qd[g(t)]

r 1rq

=(m

2

) 1rq C

1p (g(t); p, α)

(α+ 1)|η(ϕ(b), ϕ(a),m)|

×

|η(ϕ(x), ϕ(a),m)|α+2


] 1rq

+|η(ϕ(x), ϕ(b),m)|α+2[Ar2(g(t); r)f ′′(ϕ(b))rq +Ar1(g(t); r)f ′′(ϕ(x))rq

] 1rq

.

Corollary 3.3. Under the same conditions as in Theorem 3.2 for r = 1, g(t) = tand f ′′ ≤ K, we get∣∣∣∣∣−η(ϕ(x), ϕ(a),m)α+1f ′(mϕ(a))− η(ϕ(x), ϕ(b),m)α+1f ′(mϕ(b))

(α+ 1)η(ϕ(b), ϕ(a),m)

+η(ϕ(x), ϕ(a),m)αf(mϕ(a) + η(ϕ(x), ϕ(a),m)) + η(ϕ(x), ϕ(b),m)αf(mϕ(b) + η(ϕ(x), ϕ(b),m))

η(ϕ(b), ϕ(a),m)

− Γ(α+ 1)

η(ϕ(b), ϕ(a),m)

×[Jα(mϕ(a)+η(ϕ(x),ϕ(a),m))−f(mϕ(a)) + Jα(mϕ(b)+η(ϕ(x),ϕ(b),m))−f(mϕ(b))

]∣∣∣∣∣≤ K

(1 + α)1+1p

(mπ2

) 1q

Γ(p+ 1)Γ(

1α+1

)Γ(p+ 1 + 1

α+1

)

1p


1496 KHAN ET AL 1487-1503


×

[|η(ϕ(x), ϕ(a),m)|α+2 + |η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

].

Theorem 3.4. Let ϕ : I −→ A be a continuous function and g : [0, 1] −→ (0, 1)is a differentiable function. Suppose A ⊆ R be an open m-invex subset with respectto η : A × A × (0, 1] −→ R for any fixed m ∈ (0, 1] and let mϕ(a) < mϕ(a) +η(ϕ(b), ϕ(a),m). Assume that f : A −→ (0,+∞) be a twice differentiable functionon A. If f ′′q is nonnegative MT(r;g,m,ϕ)-preinvex function, q ≥ 1, then for α > 0and 0 < r ≤ 1, we have

|If,g,η,ϕ(x;α,m, a, b)|

≤(m

2

) 1rq |η(ϕ(x), ϕ(a),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

(g(1)− g(0)− gα+2(1)− gα+2(0)

α+ 2

)1− 1q

×[

(Ar2(g(t); r)−Ar4(g(t); r, α)) f ′′(ϕ(a))rq

+ (Ar1(g(t); r)−Ar3(g(t); r, α)) f ′′(ϕ(x))rq] 1rq

+(m

2

) 1rq |η(ϕ(x), ϕ(b),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

(g(1)− g(0)− gα+2(1)− gα+2(0)

α+ 2

)1− 1q

×[

(Ar2(g(t); r)−Ar4(g(t); r, α)) f ′′(ϕ(b))rq


, (3.4)

where

A3(g(t); r, α) =

∫ 1−g(0)

1−g(1)t−

12r (1− t) 1

2r+α+1dt;

A4(g(t); r, α) =

∫ g(1)

g(0)

t−12r+α+1(1− t)

12r dt.

Proof. Suppose that q ≥ 1 and 0 < r ≤ 1. Using relation (3.2), nonnegativeMT(r;g,m,ϕ)-preinvexity of f ′′q, the well-known power mean inequality, Minkowskiinequality and taking the modulus, we have

|If,g,η,ϕ(x;α,m, a, b)| ≤ |η(ϕ(x), ϕ(a),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

×∫ 1

0

|1− gα+1(t)|∣∣f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))

∣∣d[g(t)]

+|η(ϕ(x), ϕ(b),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

×∫ 1

0

|1− gα+1(t)|∣∣f ′′(mϕ(b) + g(t)η(ϕ(x), ϕ(b),m))

∣∣d[g(t)]

≤ |η(ϕ(x), ϕ(a),m)|α+2

(α+ 1)|η(ϕ(b), ϕ(a),m)|

(∫ 1

0

(1− gα+1(t))d[g(t)]

)1− 1q

×(∫ 1

0

(1− gα+1(t))f ′′(mϕ(a) + g(t)η(ϕ(x), ϕ(a),m))qd[g(t)]

) 1q


1497 KHAN ET AL 1487-1503


Corollary 3.5. Under the same conditions as in Theorem 3.4 for r = 1, g(t) = tand f ′′ ≤ K, we get∣∣∣∣∣−η(ϕ(x), ϕ(a),m)α+1f ′(mϕ(a))− η(ϕ(x), ϕ(b),m)α+1f ′(mϕ(b))

(α+ 1)η(ϕ(b), ϕ(a),m)

+η(ϕ(x), ϕ(a),m)αf(mϕ(a) + η(ϕ(x), ϕ(a),m)) + η(ϕ(x), ϕ(b),m)αf(mϕ(b) + η(ϕ(x), ϕ(b),m))

η(ϕ(b), ϕ(a),m)

− Γ(α+ 1)

η(ϕ(b), ϕ(a),m)

×[Jα(mϕ(a)+η(ϕ(x),ϕ(a),m))−f(mϕ(a)) + Jα(mϕ(b)+η(ϕ(x),ϕ(b),m))−f(mϕ(b))

]∣∣∣∣∣≤ K

α+ 1

(α+ 1

α+ 2

)1− 1q (m

2

) 1q

(π −√π(α+ 1)Γ

(α+ 3

2

)Γ(α+ 3)

) 1q

×

[|η(ϕ(x), ϕ(a),m)|α+2 + |η(ϕ(x), ϕ(b),m)|α+2

|η(ϕ(b), ϕ(a),m)|

].

Remark 3.6. For different choices of positive values 0 < r ≤ 1, for any fixed m ∈(0, 1], for a particular choices of a differentiable function g : [0, 1] −→ (0, 1), for

example: e−(t+1), sin(π(t+1)

3

), cos

(π(t+1)

3

), etc, and a particular choices of a

continuous function ϕ(x) = ex for all x ∈ R, xn for all x > 0 and for all n ∈ N,etc, by Theorem 3.2 and Theorem 3.4 we can get some special kinds of Hermite-Hadamard type fractional integral inequalities.

4. Applications to special means

Definition 4.1. (see [23]) A function M : R2+ −→ R+, is called a Mean function if

it has the following internality property:

minx, y ≤M(x, y) ≤ maxx, y.

For a Mean function the following holds, M(x, x) = x, ∀x ∈ R+.

We consider some means for arbitrary positive real numbers α, β (α 6= β).

(1) The arithmetic mean:

A := A(α, β) =α+ β

2

(2) The geometric mean:

G := G(α, β) =√αβ

(3) The harmonic mean:

H := H(α, β) =2

1α + 1

β

(4) The power mean:

Pr := Pr(α, β) =

(αr + βr

2

) 1r

, r ≥ 1.


1499 KHAN ET AL 1487-1503


(5) The identric mean:

I := I(α, β) =

1e

(ββ

αα

), α 6= β;

α, α = β.

(6) The logarithmic mean:

L := L(α, β) =β − α

ln(β)− ln(α).

(7) The generalized log-mean:

Lp := Lp(α, β) =

[βp+1 − αp+1

(p+ 1)(β − α)

] 1p

; p ∈ R \ −1, 0.

(8) The weighted p-power mean:

Mp

(α1, α2, · · · , αnu1, u2, · · · , un

)=

(n∑i=1

αiupi

) 1p

where 0 ≤ αi ≤ 1, ui > 0 (i = 1, 2, . . . , n) with∑ni=1 αi = 1.

It is well known that Lp is monotonic nondecreasing over p ∈ R with L−1 := L andL0 := I. In particular, we have the following inequality H ≤ G ≤ L ≤ I ≤ A. Re-cently, the bivariate means have attracted the attention of many researchers, manyremarkable inequalities can be found in the literature [36]-[46]. Now, let a and b bepositive real numbers such that a < b. Consider the function M := M(ϕ(x), ϕ(y)) :[ϕ(x), ϕ(x) + η(ϕ(y), ϕ(x))] × [ϕ(x), ϕ(x) + η(ϕ(y), ϕ(x))] −→ R+, which is one ofthe above mentioned means, ϕ : I −→ A be a continuous function and g : [0, 1] −→(0, 1) is a differentiable function. Therefore one can obtain various inequalities us-ing the results of Section 3 for these means as follows: Replace η(ϕ(x), ϕ(y),m)with η(ϕ(x), ϕ(y)) and setting η(ϕ(x), ϕ(y)) = M(ϕ(x), ϕ(y)), ∀x, y ∈ I, for val-ue m = 1 in (3.3) and (3.4), one can obtain the following interesting inequalitiesinvolving means:

|If,g,M(·,·),ϕ(x;α, 1, a, b)| ≤(

1

2

) 1rq C

1p (g(t); p, α)

(α+ 1)M(ϕ(a), ϕ(b))

×

Mα+2(ϕ(a), ϕ(x))


] 1rq

+Mα+2(ϕ(b), ϕ(x))[Ar2(g(t); r)f ′′(ϕ(b))rq +Ar1(g(t); r)f ′′(ϕ(x))rq

] 1rq

, (4.1)

|If,g,M(·,·),ϕ(x;α, 1, a, b)|

≤(

1

2

) 1rq Mα+2(ϕ(a), ϕ(x))

(α+ 1)M(ϕ(a), ϕ(b))

(g(1)− g(0)− gα+2(1)− gα+2(0)

α+ 2

)1− 1q

×[



+

(1

2

) 1rq Mα+2(ϕ(b), ϕ(x))

(α+ 1)M(ϕ(a), ϕ(b))

(g(1)− g(0)− gα+2(1)− gα+2(0)

α+ 2

)1− 1q


1500 KHAN ET AL 1487-1503


×[



. (4.2)

Letting M(ϕ(x), ϕ(y)) = A,G,H, Pr, I, L, Lp,Mp, ∀x, y ∈ I in (4.1) and (4.2), weget the inequalities involving means for a particular choices of nonnegative twicedifferentiable MT(r;g,1,ϕ)-preinvex function f. The details are left to the interestedreader.

5. Conclusions

In this paper, we proved some new integral inequalities for the left-hand sideof Gauss-Jacobi type quadrature formula involving MT(r;g,m,ϕ)-preinvex function-s. Also, we established some new Hermite-Hadamard type integral inequalities fornonnegative MT(r;g,m,ϕ)-preinvex functions via Riemann-Liouville fractional inte-grals. These results provide new estimates on these types.

Motivated by this new interesting class of MT(r;g,m,ϕ)-preinvex functions we can in-deed see to be vital for fellow researchers and scientists working in the same domain.

We conclude that our methods considered here may be a stimulant for furtherinvestigations concerning Hermite-Hadamard type integral inequalities for variouskinds of preinvex functions involving classical integrals, Riemann-Liouville frac-tional integrals, k-fractional integrals, local fractional integrals, fractional integraloperators, q-calculus, (p, q)-calculus, time scale calculus and conformable fractionalintegrals.

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Muhammad Adil KhanDepartment of Mathematics, University of Peshawer, Pakistan


Yu-Ming Chu∗ (Corresponding author)

Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China,E-mail address: [email protected]

Artion Kashuri

Department of Mathematics, Faculty of Technical Science, University ”Ismail Qemali”,Vlora, Albania


Rozana Liko

Department of Mathematics, Faculty of Technical Science, University ”Ismail Qemali”,

Vlora, AlbaniaE-mail address: [email protected]


1503 KHAN ET AL 1487-1503

UMBRAL CALCULUS APPROACH TO DEGENERATE POLY-GENOCCHI

POLYNOMIALS

TAEKYUN KIM, DAE SAN KIM, LEE CHAE JANG, AND GWAN-WOO JANG

Abstract. In this paper, we apply umbral calculus techniques in order to derive explicit exprissions,

some properties, recurrence relations and identities for degenerate poly-Genocchi polynomials. Further-more, we derive several explicit expressions of degenerate poly-Genocchi polynomials as linear combina-

tions of some of the well-known families of special polynomials.

1. Review on umbral calculus

The purpose of this paper is to use umbral calculus in order to derive some new and interestingexpressions, recurrence relations and identities for degenerate poly-Genocchi polynomials. To do that wefirst recall the umbral calculus very briefly. For a complete treatment, the reader may refer to [10].Let F be the algebra of all formal power series in the single variable t with the coefficients in the field Cof complex numbers:

F =

f(t) =

∞∑k=0

aktk

k!

∣∣∣∣ ak ∈ C

. (1.1)

Let P = C[x] denote the ring of polynomials in x with the coefficients in C, and let P∗ be the vectorspace of all linear functionals on P. For L ∈ P∗, p(x) ∈ P, < L | p(x) > denotes the action of the linear

functional L on p(x). For f(t) =∑∞k=0 ak

tk

k! ∈ F , the linear functional < f(t) | · > on P is defined by

< f(t) |xn >= an, (n ≥ 0). (1.2)

For L ∈ P∗, let fL(t) =∑∞k=0

⟨L|xk

⟩tk

k! ∈ F . Then we evidently have⟨fL(t)|xn

⟩=⟨L|xn

⟩, and the

map L→ fL(t) is a vector space isomorphism from P∗ to F . Thus F may be viewed as the vector spaceof all linear functionals on P as well as the algebra of formal power series in t. So an element f(t) ∈ Fwill be thought of as both a formal power series and a linear functional on P. F is called the umbralalgebra, the study of which is the umbral calculus.

The order o(f(t)) of 0 6= f(t) ∈ F is the smallest integer k such that the coefficients of tk does notvanish. In particular, for 0 6= f(t) ∈ F , it is called an invertible series if o(f(t)) = 0 and a delta series ifo(f(t)) = 1.

Let f(t), g(t) ∈ F , with o(g(t)) = 0, o(f(t)) = 1. Then there exists a unique sequence of polynomials

Sn(x) (degSn(x) = n) such that⟨g(t)f(t)k|Sn(x)

⟩= n!δn,k, for n, k ≥ 0. Such a sequence is called the

Sheffer sequence for the Sheffer pair (g(t), f(t)), which is concisely denoted by Sn(x) ∼ (g(t), f(t)).

2010 Mathematics Subject Classification. 05A19, 05A40, 11B83.Key words and phrases. degenerate poly-Genocchi polynomials, umbral calculus..

1


1504 T. KIM ET AL 1504-1520

2 Umbral calculus approach to degenerate poly-Genocchi polynomials

It is known that Sn(x) ∼ (g(t), f(t)) if and only if

1

g(f(t))exf(t) =

∞∑n=0

Sn(x)tn

n!, (1.3)

where f(t) is the compositional inverse of f(t) satisfying f(f(t)) = f(f(t)) = t.Let pn(x) ∼ (1, f(t)), qn(x) ∼ (1, l(t)). Then the transfer formula says that

qn(x) = x

(f(t)

l(t)

)nx−1pn(x), (n ≥ 1). (1.4)

Let Sn(x) ∼ (g(t)), f(t)). Then we have the Sheffer identity:

Sn(x+ y) =n∑k=0

(n

k

)Sk(x)pn−k(y), (1.5)

where pn(x) = g(t)Sn(x) ∼ (1, f(t)).For Sn(x) ∼ (g(t), f(t)),

f(t)Sn(x) = nSn−1(x). (1.6)

Also, we have the recurrence formula:

Sn+1(x) =

(x− g′(t)

g(t)

)1

f ′(t)Sn(x). (1.7)

Assume that Sn(x) ∼ (g(t), f(t)), rn(x) ∼ (h(t), l(t)). Then

Sn(x) =n∑k=0

Cn,krk(x), (1.8)

where

Cn,k =1

k!

⟨h(f(t))

g(f(t))l(f(t))k|xn

⟩. (1.9)

Finally, we also need the following: for any h(t) ∈ F , p(x) ∈ P,⟨h(t)|xp(x)

⟩=⟨∂th(t)|p(x)

⟩. (1.10)

For sn(x) ∼ (g(t), f(t)),

d

dxSn(x) =

n−1∑l=0

(n

l

)< f(t)|xn−l > Sl(x). (1.11)

2. Introduction

Let r be any integer, and let 0 6= λ ∈ C. The series Lir(x) defined by

Lir(x) =∞∑n=1

xn

nr, (see [3–9]), (2.1)

is the r-th polylogarithmic function for r ≥ 1, and a rational function for r ≤ 0. One immediate propertyof this is

d

dx(Lir+1(x)) =

1

xLir(x). (2.2)


1505 T. KIM ET AL 1504-1520

T. Kim, D. S. Kim, L. C. Jang, G.-W. Jang 3

The degenerate poly-Genocchi polynomials γ(r)n (λ, x) of index r are given by

2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)xλ =

∞∑n=0

γ(r)n (λ, x)

tn

n!. (2.3)

For x = 0, γ(r)n (λ) = γ

(r)n (λ, 0) are called degenerate poly-Genocchi numbers of index r. In particular for

r = 1, γn(λ, x) = γ(1)n (λ, x) may be called degenerate Genocchi polynomials and are given by

2t

(1 + λt)1λ + 1

(1 + λt)xλ =

∞∑n=0

γn(λ, x)tn

n!. (2.4)

They are a degenerate version of the poly-Genocchi polynomials G(r)n (x) of index r given by

2Lir(1− e−t)et + 1

ext =∞∑n=0

G(r)n (x)

tn

n!. (2.5)

These poly-Genocchi polynomials were first introduced in [3] under the name of poly-Euler polynomials

and with the notation E(r)n (x). However, it seems more appropriate to call them poly-Genocchi polyno-

mials, as Gn(x) = G(1)n (x) are the ’classical’ Genocchi polynomials defined by

2t

et + 1ext =

∞∑n=0

Gn(x)tn

n!. (2.6)

Clearly, limλ→0 γ(r)n (λ, x) = G

(r)n (x). Also, we recall that the degenerate Euler polynomials En(λ, x) were

introduced in [1] by Carlitz and are given by

2

(1 + λt)1λ − 1

(1 + λt)xλ =

∞∑n=0

En(λ, x)tn

n!. (2.7)

The degenerate poly-Bernoulli polynomials β(r)n (λ, x) of index r are defined in [6, 7] as

Lir(1− e−t)(1 + λt)

1λ − 1

(1 + λt)xλ =

∞∑n=0

β(r)n (λ, x)

tn

n!. (2.8)

When x = 0, β(r)n (λ) = β

(r)n (λ, 0) are called degenerate poly-Bernoulli numbers of index r. For r = 1,

βn(λ, x) = β(1)n (λ, x) are called degenerate Bernoulli polynomials and given by

t

(1 + λt)1λ − 1

(1 + λt)xλ =

∞∑n=0

βn(λ, x)tn

n!, (2.9)

which were introduced again by Carlitz in [1]. They are a degenerate version of the poly-Bernoulli

polynomials B(r)n (x) of index r given by

Lir(1− e−t)et − 1

ext =∞∑n=0

B(r)n (x)

tn

n!, (2.10)

We note here that this definition of poly-Bernoulli polynomials is slightly different from its original

definition(cf. [4, 5]). Obviously, limλ→0 β(r)n (λ, x) = B

(r)n (x), and Bn(x) = B(1)(x) are the ’classical’

Bernoulli polynomials defined by

t

et − 1ext =

∞∑n=0

Bn(x)tn

n!, (2.11)


1506 T. KIM ET AL 1504-1520


Write Lir(1− e−t) =∑∞n=1 an

tn

n! = t+∑∞n=2 an

tn

n! . Then from (2.3) and (2.7), we see that

∞∑n=0

γ(r)n (λ, x)

tn

n!=∞∑n=1

( n−1∑l=0

(n

l

)an−lEl(λ, x)

) tnn!. (2.12)

In turn, (2.12) implies that

γ(r)0 (λ, x) = 0, γ

(r)1 (λ, x) = 1, degγ(r)

n (λ, x) = n− 1, (n ≥ 1). (2.13)

In this paper, we would like to apply umbral calculus techniques( [2, 7, 9, 10]) in order to deriveexplicit exprissions, some properties, recurrence relations and identities for degenerate poly-Genocchi

polynomials. However, sometimes we can not apply the umbral calculus techniques directly to γ(r)n (λ, x),

since 2Lir(1−e−t)(1+λt)

1λ+1

is not invertible and hence γ(r)n (λ, x) is not a Sheffer sequence. Nevertheless, from (2.3)

and (2.13), we note that

2Lir(1− e−t)t((1 + λt)

1λ + 1)

(1 + λt)xλ =

∞∑n=0

γ(r)n+1(λ, x)

n+ 1

tn

n!. (2.14)

Thus we see from (2.14) thatγ(r)n+1(λ,x)

n+1 is the Sheffer sequence for the pair

((eλt−1)(et+1)

2λLir(1−e−1λ

(eλt−1)), 1λ (eλt − 1)

),

namely

γ(r)n+1(λ, x)

n+ 1∼(

(eλt − 1)(et + 1)

2λLir(1− e−1λ (eλt−1))

,1

λ(eλt − 1)

). (2.15)

Furthermore, we give some explicit expressions of degenerate poly-Genocchi polynomials as linear com-binations of some of the well-known families of special polynomials.

3. Main results

The following Theorems 2.1 and 2.2 are mentioned in [9] and obtained in [7, 9], and will be useful inderiving several explicit expressions of degenerate poly-Genocchi polynomials as linear combinations ofdegenerate Euler or degenerate Genocchi polynomials.

Theorem 3.1. For all integers r ≥ 2, and n ≥ 0, we have the following identities.⟨Lir(1− e−t)

t| xn

⟩=

1

n+ 1

n+1∑m=1

(−1)m+n−1 m!

mrS2(n+ 1,m)

=n∑

m=0

(n

m

)B(r)m

1

n−m+ 1

=1

n+ 1B(r−1)n

= n!∑

j1,··· ,jr−1≥0,j1+···+jr−1=n

r−1∏i=1

Bjiji!(j1 + · · ·+ ji + 1)

.


1507 T. KIM ET AL 1504-1520


Theorem 3.2. For all integers r ≥ 2, and n ≥ 0, we have the following identities.

⟨Lir(1− e−t) | xn+1

⟩=

n+1∑m=1

(−1)m+n+1 m!

mrS2(n+ 1,m)

=n∑

m=0

(n+ 1

m

)B(r)m

= B(r−1)n

= (n+ 1)!∑

j1,··· ,jr−1≥0,j1+···+jr−1=n

r−1∏i=1

Bjiji!(j1 + · · ·+ ji + 1)

.

The following can be seen also from (2.3), but here we will deduce it by using umbral calculus.

γ(r)n (λ, y) =

⟨ ∞∑m=0

γ(r)m (λ, y)

tm

m!| xn

⟩

=

⟨2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλ | xn

⟩=

⟨2Lir(1− e−t)(1 + λt)

1λ + 1

|∞∑l=0

( yλ

)l (log(1 + λt))l

l!xn

⟩

=

n∑l=0

( yλ

)l⟨2Lir(1− e−t)(1 + λt)

1λ + 1

|∞∑m=l

S1(m, l)λm

m!tmxn

⟩

=n∑l=0

( yλ

)l n∑m=l

(n

m

)λmS1(m, l)

⟨2Lir(1− e−t)(1 + λt)

1λ + 1

| xn−m⟩

=

n∑l=0

( yλ

)l n∑m=l

(n

m

)λmS1(m, l)γ

(r)n−m(λ).

(3.1)

Here (x|λ)n = x(x − λ) · · · (x − (n − 1)λ), for n ≥ 1, and (x|λ)0 = 1. Recalling that (x|λ)n =∑nm=0 λ

n−mS1(n,m)xm, we have the following result.

Theorem 3.3. For all integers n ≥ 0, we have the following expressions.

γ(r)n (λ, x) =

n∑m=0

(n

m

)( m∑l=0

λm−lS1(m, l)xl

)γ

(r)n−m(λ)

=n∑

m=0

(n

m

)γ

(r)n−m(λ)(x|λ)m.


1508 T. KIM ET AL 1504-1520


Now, we would like to express the degenerate poly-Genocchi polynomials in terms of degenerate Eulerpolynomials, for this we need to observe the following.

γ(r)n (λ, y) =

⟨2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλ | xn

⟩=

⟨Lir(1− e−t) |

2

(1 + λt)1λ + 1

(1 + λt)yλxn

⟩=

⟨Lir(1− e−t) |

∞∑l=0

El(λ, y)tl

l!xn

⟩

=n∑l=0

(n

l

)El(λ, y)

⟨Lir(1− e−t) | xn−l

⟩.

(3.2)

From (3.2) and Theorem 2.2, we obtain the following explicit expressions for γ(r)n (λ, x) , as linear combi-

nations of the degenerate Euler polynomials.

Theorem 3.4. For all integers n ≥ 0, we have the following identities.

γ(r)n (λ, x) =

n∑l=1

l∑m=1

(n

l

)(−1)l+m

m!

mrS2(l,m)En−l(λ, x)

=

n∑l=1

l−1∑m=0

(n

l

)(l

m

)B(r)m En−l(λ, x)

=n∑l=1

(n

l

)B

(r−1)l−1 En−l(λ, x)

=n∑l=1

∑j1,··· ,jr−1≥0,j1+···+jr−1=l−1

(n)l

r−1∏i=1

Bjiji!(j1 + · · ·+ ji + 1)

En−l(λ, x).

Next, in order to express the degenerate poly-Genocchi polynomials in terms of degenerate Genocchipolynomials, we first observe the following.

γ(r)n (λ, y) =

⟨2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλ | xn

⟩=

⟨Lir(1− e−t)

t| 2t

(1 + λt)1λ + 1

(1 + λt)yλxn

⟩=

⟨Lir(1− e−t)

t|∞∑l=0

γl(λ, y)tl

l!xn

⟩

=n∑l=0

(n

l

)γl(λ, y)

⟨Lir(1− e−t)

t| xn−l

⟩.

(3.3)

From (3.3) and Theorem 2.1, we obtain the following explicit expression for γ(r)n (λ, x) as linear com-

binations of degenerate Genocchi polynomials.


1509 T. KIM ET AL 1504-1520


Theorem 3.5. For all integers n ≥ 0, we have the following identities.

γ(r)n (λ, x) =

n−1∑l=0

l+1∑m=1

1

l + 1

(n

l

)(−1)l+m−1 m!

mrS2(l + 1,m)γn−l(λ, x)

=n−1∑l=0

l∑m=0

1

l −m+ 1

(n

l

)(l

m

)B(r)m γn−l(λ, x)

=n−1∑l=0

1

l + 1

(n

l

)B

(r−1)l γn−l(λ, x)

=n−1∑l=0

∑j1,··· ,jr−1≥0,j1+···+jr−1=l

(n)l

r−1∏i=1

Bjiji!(j1 + · · ·+ ji + 1)

γn−l(λ, x).

Here we apply the transfer formula (1.4) to

xn ∼ (1, t),

1λ

(eλt − 1

)(et + 1)

2Lir

(1− e− 1

λ (eλt−1)) γ(r)

n+1(λ, x)

n+ 1∼(

1,1

λ(eλt − 1)

).

(3.4)

Then, for n ≥ 1 we have

1λ

(eλt − 1

)(et + 1)

2Lir

(1− e− 1


n+1(λ, x)

n+ 1= x

(λt

eλt − 1

)nxn−1

= x∞∑l=0

B(n)l

λl

l!tlxn−1

=n−1∑l=0

(n− 1

l

)λlB

(n)l xn−l.

(3.5)

Thus

γ(r)n+1(λ, x)

n+ 1

=n−1∑l=0

(n− 1

l

)λlB

(n)l

2Lir

(1− e− 1

λ (eλt−1))

1λ (eλt − 1) (et + 1)

xn−l

=n−1∑l=0

(n− 1

l

)λlB

(n)l

∞∑m=0

γ(r)m+1(λ)

m+ 1

(1λ

(eλt − 1

))mm!

xn−l

=n−1∑l=0

n−l∑m=0

(n− 1

l

)λl−mB

(n)l

γ(r)m+1(λ)

m+ 1

∞∑j=m

S2(j,m)λj

j!tjxn−l

=n∑l=0

n−l∑m=0

n−l∑j=m

(n− 1

l

)(n− lj

)λl−m+jS2(j,m)B

(n)l

γ(r)m+1(λ)

m+ 1xn−l−j

=n∑j=0

(n−j∑l=0

n−j−l∑m=0

(n− 1

l

)(n− lj

)λn−m−jS2(n− j − l,m)B

(n)l

γ(r)m+1(λ)

m+ 1

)xj , (n ≥ 1).

(3.6)


1510 T. KIM ET AL 1504-1520


Observing thatn∑l=0

S1(n, l)λn−lxl = (x|λ)n ∼(

1,1

λ

(eλt − 1

)),

1λ

(eλt − 1

)(et + 1)

2Lir

(1− e− 1


n+1(λ, x)

n+ 1∼(

1,1

λ

(eλt − 1

)),

(3.7)

we have

γ(r)n+1(λ, x)

n+ 1

=n∑l=0

S1(n, l)λn−l2Lir

(1− e− 1

λ (eλt−1))

1λ (eλt − 1) (et + 1)

xl

=

n∑l=0

S1(n, l)λn−l∞∑m=0

γ(r)m+1(λ)

m+ 1

(1λ

(eλt − 1

))mm!

xl

=n∑l=0

l∑m=0

S1(n, l)λn−l−mγ

(r)m+1(λ)

m+ 1

∞∑j=m

S2(j,m)λj

j!tjxl

=n∑l=0

l∑m=0

l∑j=m

λn−l−m+j

(l

j

)S1(n, l)S2(j,m)

γ(r)m+1(λ)

m+ 1xl−j

=

n∑j=0

n∑l=j

l−j∑m=0

(l

j

)λn−m−jS1(n, l)S2(l − j,m)

γ(r)m+1(λ)

m+ 1xj .

(3.8)

Hence, from (3.6) and (3.8), we get the following theorem.

Theorem 3.6. We have the following expressions.

γ(r)n+1(λ, x)

n+ 1

=n∑j=0

(n−j∑l=0

n−j−l∑m=0

(n− 1

l

)(n− lj

)λn−m−jS2(n− j − l,m)B

(n)l

γ(r)m+1(λ)

m+ 1

)xj , (n ≥ 1)

=n∑j=0

n∑l=j

l−j∑m=0

(l

j

)λn−m−jS1(n, l)S2(l − j,m)

γ(r)m+1(λ)

m+ 1xj , (n ≥ 0).

The following Sheffer identity follows immediately from (1.5).

γ(r)n (λ, x+ y) =

n∑j=0

(n

j

)γ

(r)j (λ, x)(y|λ)n−j . (3.9)

Applying (1.6) to γ(r)n (λ, x), here we get

1

λ(eλt − 1)γ(r)

n (λ, x) = nγ(r)n−1(λ, x). (3.10)

Thus

γ(r)n (λ, x+ λ)− γ(r)

n (λ, x) = nλγ(r)n−1(λ, x). (3.11)


1511 T. KIM ET AL 1504-1520


By using (1.7), we obtain

γ(r)n+1(λ)

n+ 1=

(x− g′(t)

g(t)

)e−λt

γ(r)n (λ, x)

n

=1

nxγ(r)

n (λ, x− λ)− 1

ne−λt

g′(t)

g(t)γ(r)n (λ, x),

(3.12)

where g(t) = (eλt−1)(et+1)

2λLir(

1−e−1λ

(eλt−1)) . Here,

g′(t)

g(t)= (logg(t))

′

=1

t

λteλt

eλt − 1+

tet

et + 1− teλt

Lir

(1− e− 1

λ (eλt−1)) Lir−1

(1− e− 1

λ (eλt−1))

e1λ (eλt−1) − 1

.

(3.13)

Note that g(t)γ(r)n (λ, x) = (x|λ)n =

∑nm=0 λ

n−mS1(n,m)xm, and that the expression in the curly brackethas order ≥ 1.

e−λtg′(t)

g(t)γ(r)n (λ, x)

=1

t

λt

eλt − 1

2λLir

(1− e− 1

λ (eλt−1))

(eλt − 1)(et + 1)+te(1−λ)t

et + 1


(eλt − 1)(et + 1)

− 2

et + 1

λt

eλt − 1

Lir−1

(1− e− 1

λ (eλt−1))

e1λ (eλt−1) − 1

(eλt − 1)(et + 1)

2λLir

(1− e− 1

λ (eλt−1))γ(r)

n (λ, x)

=n∑

m=0

λn−mS1(n,m)1

t

λt

eλt − 1

2λLir

(1− e− 1

λ (eλt−1))

(eλt − 1)(et + 1)+te(1−λ)t

et + 1


(eλt − 1)(et + 1)

− 2

et + 1

λt

eλt − 1

Lir−1

(1− e− 1

λ (eλt−1))

e1λ (eλt−1) − 1

xm

=n∑

m=0

λn−m

m+ 1S1(n,m)

λt

eλt − 1

2λLir

(1− e− 1

λ (eλt−1))

(eλt − 1)(et + 1)+te(1−λ)t

et + 1


(eλt − 1)(et + 1)

− 2

et + 1

λt

eλt − 1

Lir−1

(1− e− 1

λ (eλt−1))

e1λ (eλt−1) − 1

xm+1

(3.14)

The next theorem follows from (3.12), (3.14), (3.15), (3.16), and (3.17).


1512 T. KIM ET AL 1504-1520


Theorem 3.7. For all integers n ≥ 0, the following holds true.

γ(r)n+1(λ, x)

n+ 1− x

nγ(r)n (λ, x− λ)

= − 1

n

n∑m=0

m+1∑l=0

m+1∑j=l

1

m+ 1

(m+ 1

j

)λn−mS1(n,m)S2(j, l)

1

l + 1λm+1−lγ

(r)l+1(λ)Bm+1−j

(xλ

)

+1

2

m+ 1− jl + 1

λj−lγ(r)l+1(λ)Em−j(x+ 1− λ) +

m+1−j∑k=0

(m+ 1− j

k

)λj+k−lB

(r−1)l Em+1−j−kBk

(xλ

).

For the sake of completeness, we compute the three terms in (3.14) as follows:

λt

eλt − 1

2λLir

(1− e− 1

λ (eλt−1))

(eλt − 1)(et + 1)xm+1

=λt

eλt − 1

∞∑l=0

γ(r)l+1(λ)

l + 1

(1λ (eλt − 1)

)ll!

xm+1

=λt

eλt − 1

m+1∑l=0

γ(r)l+1(λ)λ−l

l + 1

∞∑j=l

S2(j, l)λj

j!tjxm+1

=λt

eλt − 1

m+1∑l=0

m+1∑j=l

1

l + 1

(m+ 1

j

)λj−lS2(j, l)γ

(r)l+1(λ)xm+1−j

=m+1∑l=0

m+1∑j=l

1

l + 1

(m+ 1

j

)λm+1−lS2(j, l)γ

(r)l+1(λ)Bm+1−j

(xλ

),

(3.15)

te(1−λ)t

et + 1

2λLir

(1− e− 1

λ (eλt−1))

(eλt − 1)(et + 1)xm+1

=te(1−λ)t

et + 1

m+1∑l=0

m+1∑j=l

1

l + 1

(m+ 1

j

)λj−lS2(j, l)γ

(r)l+1(λ)xm+1−j

=e(1−λ)t

et + 1

m+1∑l=0

m+1∑j=l

m+ 1− jl + 1

(m+ 1

j

)λj−lS2(j, l)γ

(r)l+1(λ)xm−j

=1

2e(1−λ)t

m+1∑l=0

m+1∑j=l

m+ 1− jl + 1

(m+ 1

j

)λj−lS2(j, l)γ

(r)l+1(λ)

2

et + 1xm−j

=1

2

m+1∑j=0

m+1∑j=l

m+ 1− jl + 1

(m+ 1

j

)λj−lS2(j, l)γ

(r)l+1(λ)Em−j(x+ 1− λ),

(3.16)

and


1513 T. KIM ET AL 1504-1520


2

et + 1

λt

eλt − 1

Lir−1

(1− e− 1

λ (eλt−1))

e1λ (eλt−1) − 1

xm+1

=2

et + 1

λt

eλt − 1

∞∑l=0

B(r−1)l

( 1λ (eλt − 1))l

l!xm+1

=2

et + 1

λt

eλt − 1

m+1∑l=0

λ−lB(r−1)l

∞∑j=l

S2(j, l)λj

j!tjxm+1

=m+1∑l=0

m+1∑j=l

λj−l(m+ 1

j

)S2(j, l)B

(r−1)l

λt

eλt − 1

2

et + 1xm+1−j

=

m+1∑l=0

m+1∑j=l

λj−l(m+ 1

j

)S2(j, l)B

(r−1)l

m+1−j∑k=0

(m+ 1− j

k

)Em+1−j−kλ

kBk

(xλ

)

=

m+1∑l=0

m+1∑j=l

m+1−j∑k=0

(m+ 1

j

)(m+ 1− j

k

)λj+k−lS2(j, l)B

(r−1)l Em+1−j−kBk

(xλ

).

(3.17)

We note that

< f(t)|xn−l > =

⟨1

λlog(1 + λt) | xn−l

⟩= λ−1

⟨ ∞∑m=1

(−1)m−1λm(m− 1)!tm

m!| xn−l

⟩= (−λ)n−l−1(n− l − 1)!.

(3.18)

Thus, from (1.11), we get

d

dx

(γ

(r)n+1(λ, x)

n+ 1

)= n!

n−1∑l=0

(−λ)n−l−1

l!(n− l)γ

(r)l+1(λ, x)

l + 1. (3.19)

Here we use (1.10) in order to get an expression of γ(r)n (λ, x). For this, assume that n ≥ 1.

γ(r)n (λ, y) =

⟨2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλ | xn

⟩=

⟨(∂t

2Lir(1− e−t)(1 + λt)

1λ + 1

)(1 + λt)

yλ | xn−1

⟩+

⟨2Lir(1− e−t)(1 + λt)

1λ + 1

(∂t(1 + λt)

yλ | xn−1

)⟩.

(3.20)


1514 T. KIM ET AL 1504-1520


The second term of (3.20) is easily seen to be equal to yγ(r)n−1(λ, y − λ). For the first term of (3.20),

we observe that

∂t

(2Lir(1− e−t)(1 + λt)

1λ + 1

)2Lir−1(1−e−t)

1−e−t e−t((1 + λt)1λ + 1)− 2Lir(1− e−t)(1 + λt)

1λ−1(

(1 + λt)1λ + 1

)2

=2

(1 + λt)1λ + 1

Lir−1(1− e−t)et − 1

− 1

1 + λt

2Lir(1− e−t)(1 + λt)

1λ + 1

+1

2(1 + λt)

2

(1 + λt)1λ + 1

2Lir(1− e−t)(1 + λt)

1λ + 1

.

(3.21)

So, the first term can be written as three sums:⟨Lir−1(1− e−t)

et − 1

2

(1 + λt)1λ + 1

(1 + λt)yλ | xn−1

⟩−⟨

1

1 + λt

2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλ | xn−1

⟩+

⟨1

2(1 + λt)

2

(1 + λt)1λ + 1

2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλ | xn−1

⟩.

(3.22)

Now, we compute the three terms in (3.22) as follows⟨Lir−1(1− e−t)

et − 1

2

(1 + λt)1λ + 1

(1 + λt)yλ | xn−1

⟩=

⟨Lir−1(1− e−t)

et − 1| 2

(1 + λt)1λ + 1

(1 + λt)yλxn−1

⟩=

⟨Lir−1(1− e−t)

et − 1|∞∑l=0

El(λ, y)tl

l!xn−1

⟩

=

n−1∑l=0

(n− 1

l

)El(λ, y)

⟨Lir−1(1− e−t)

et − 1| xn−1−l

⟩

=

n−1∑l=0

(n− 1

l

)El(λ, y)B

(r−1)n−1−l,

(3.23)

⟨1

1 + λt

2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλ | xn−1

⟩=

⟨1

1 + λt| 2Lir(1− e−t)

(1 + λt)1λ + 1

(1 + λt)yλxn−1

⟩=

⟨1

1 + λt|∞∑l=0

γ(r)l (λ, y)

tl

l!xn−1

⟩

=n−1∑l=0

(n− 1

l

)γ

(r)l (λ, y)

⟨1

1 + λt| xn−1−l

⟩

=n−1∑l=0

(n− 1

l

)γ

(r)l (λ, y)(−λ)n−1−l(n− 1− l)!,

(3.24)


1515 T. KIM ET AL 1504-1520


and ⟨1

2(1 + λt)

2

(1 + λt)1λ + 1

2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλ | xn−1

⟩=

⟨1

2(1 + λt)

2

(1 + λt)1λ + 1

| 2Lir(1− e−t)(1 + λt)

1λ + 1

(1 + λt)yλxn−1

⟩=

⟨1

2(1 + λt)

2

(1 + λt)1λ + 1

|∞∑l=0

γ(r)l (λ, y)

tl

l!xn−1

⟩

=n−1∑l=0

(n− 1

l

)γ

(r)l (λ, y)

⟨1

2(1 + λt)| 2

(1 + λt)1λ + 1

xn−1−l⟩

=n−1∑l=0

(n− 1

l

)γ

(r)l (λ, y)

⟨1

2(1 + λt)|∞∑m=0

Em(λ)tm

m!xn−1−l

⟩

=n−1∑l=0

(n− 1

l

)γ

(r)l (λ, y)

n−1−l∑m=0

(n− 1− l

m

)Em(λ)

⟨1

2

1

1 + λt| xn−1−l−m

⟩

=1

2

n−1∑l=0

n−1−l∑m=0

(n− 1

l

)(n− 1− l

m

)γ

(r)l (λ, y) Em(λ)(−λ)n−1−l−m(n− 1− l −m)!

(3.25)

Putting (3.20), (3.22), (3.23), (3.24), (3.25) altogether, we obtain the following result.


γ(r)n (λ, x) = xγ

(r)n−1(λ, x− λ) +

n−1∑l=0

(n− 1)!

l!

1

(n− 1− l)!B

(r−1)n−1−lEl(λ, x)

−(−λ)n−1−lγ(r)l (λ, x) +

1

2

n−1−l∑m=0

1

m!Em(λ)(−λ)n−1−l−mγ

(r)l (λ, x)

.

From now on, we will exploit (1.9) in order to express degenerate poly-Genocchi polynomials as linearcombinations of well known families of polynomials. For this, we remind the reader that

γ(r)n+1(λ, x)

n+ 1∼(

(eλt − 1)(et + 1)


,1

λ(eλt − 1)

). (3.26)

We letγ(r)n+1(λ,x)

n+1 =∑nk=0 Cn,kEk(λ, x), with noting that

En(λ, x) ∼(et + 1

2,

1

λ(eλt − 1)

). (3.27)


1516 T. KIM ET AL 1504-1520


Then

Cn,k =1

k!

⟨(1 + λt)

1λ + 1

2

2λLir(1− e−t)λt((1 + λt)

1λ + 1)

tk | xn⟩

=1

k!

⟨Lir(1− e−t)

t| tkxn

⟩=

(n

k

)⟨Lir(1− e−t)

t| xn−k

⟩=

1

n− k + 1

(n

k

)B

(r−1)n−k .

(3.28)

Thus we have shown

γ(r)n+1(λ, x) =

n∑k=0

(n+ 1

k

)B

(r−1)n−k Ek(λ, x). (3.29)

To express degenerate poly-Genocchi polynomials as linear combinations of Euler polynomials, writeγ(r)n+1(λ,x)

n+1 =∑nk=0 Cn,kEk(x), with noting that

En(λ, x) ∼(et + 1

2, t

). (3.30)

Then

Cn,k =1

k!

⟨(1 + λt)

1λ + 1

2


1λ + 1)

(1

λlog(1 + λt)

)k| xn

⟩

= λ−k⟨Lir(1− e−t)

t| 1

k!(log(1 + λt))kxn

⟩= λ−k

⟨Lir(1− e−t)

t|∞∑l=k

S1(l, k)λl

l!tlxn

⟩

= λ−kn∑l=k

(n

l

)λlS1(l, k)

⟨Lir(1− e−t)

t| xn−l

⟩

= λ−kn∑l=k

1

n− l + 1

(n

l

)λlS1(l, k)B

(r−1)n−l .

(3.31)

Thus we have derived the following theorem.

Theorem 3.9. For all n ≥ 0, we have the following.

γ(r)n+1(λ, x) =

n∑k=0

n∑l=k

(n+ 1

l

)λl−kS1(l, k)B

(r−1)n−l Ek(x).

Letγ(r)n+1(λ,x)

n+1 =∑nk=0 Cn,kβ

(r)k (λ, x), with

β(r)n (λ, x) ∼

(et − 1

Lir(1− e−1λ (eλt−1)

,1

λ(eλt − 1)

). (3.32)


1517 T. KIM ET AL 1504-1520


Then

Cn,k =1

k!

⟨(1 + λt)

1λ − 1

Lir(1− e−t)2λLir(1− e−t)λt((1 + λt)

1λ + 1)

| tkxn⟩

=

(n

k

)⟨2

(1 + λt)1λ + 1

| (1 + λt)1λ − 1

txn−k

⟩

=

(n

k

)⟨2

(1 + λt)1λ + 1

|∞∑l=0

(1|λ)l+1

l + 1

tl

l!xn−k

⟩

=

(n

k

) n−k∑l=0

(n− kl

)(1|λ)l+1

l + 1

⟨2

(1 + λt)1λ + 1

| xn−k−l⟩

=

(n

k

) n−k∑l=0

(n− kl

)(1|λ)l+1

l + 1En−k−l(λ).

(3.33)

Thus we have shown the following result.

Theorem 3.10. For n ≥ 0, we have the following.

γ(r)n+1(λ, x) =

n∑k=0

n−k∑l=0

(n+ 1

l + 1

)(n− lk

)(1|λ)l+1En−k−l(λ)β

(r)k (λ, x).

Writeγ(r)n+1(λ,x)

n+1 =∑nk=0 Cn,kβk(λ, x), with noting that

βn(λ, x) ∼(λ(et − 1)

eλt − 1,

1

λ(eλt − 1)

). (3.34)

Then

Cn,k =1

k!

⟨λ((1 + λt)

1λ − 1)

λt


1λ + 1)

| tkxn⟩

=

(n

k

)⟨2Lir(1− e−t)t((1 + λt)

1λ + 1)

| (1 + λt)1λ − 1

txn−k

⟩

=

(n

k

) n−k∑l=0

(n− kl

)(1|λ)l+1

l + 1

⟨2Lir(1− e−t)t((1 + λt)

1λ + 1)

| xn−k−l⟩

=

(n

k

) n−k∑l=0

(n− kl

)(1|λ)l+1

l + 1

γ(r)n−k−l+1(λ)

n− k − l + 1.

(3.35)

Thus we obtain the following theorem.

Theorem 3.11. For all n ≥ 0, the following holds true.

γ(r)n+1(λ, x) =

n∑k=0

n−k∑l=0

1

l + 1

(n+ 1

l

)(n− l + 1

k

)(1|λ)l+1γ

(r)n−k−l+1(λ)βk(λ, x).

Lastly, we would like to express the degenerate poly-Genocchi polynomials in terms of Bernoulli poly-nomials, with noting that

Bn(x) ∼(et − 1

t, t

), (3.36)


1518 T. KIM ET AL 1504-1520


we letγ(r)n+1(λ,x)

n+1 =∑nk=0 Cn,kBk(x). Then

Cn,k =1

k!

⟨λ((1 + λt)

1λ − 1)

log(1 + λt)


1λ + 1)

|(

1

λlog(1 + λt)

)kxn

⟩

= λ−k

⟨2Lir(1− e−t)t((1 + λt)

1λ + 1)

λt

log(1 + λt)

(1 + λt)1λ − 1

t| 1

k!(log(1 + λt))kxn

⟩

= λ−k

⟨2Lir(1− e−t)t((1 + λt)

1λ + 1)

λt

log(1 + λt)

(1 + λt)1λ − 1

t|∞∑l=k

S1(l, k)λl

l!tlxn

⟩

= λ−kn∑l=0

(n

l

)λlS1(l, k)

⟨2Lir(1− e−t)t((1 + λt)

1λ + 1)

λt

log(1 + λt)| (1 + λt)

1λ − 1

txn−l

⟩

= λ−kn∑l=0

(n

l

)λlS1(l, k)

⟨2Lir(1− e−t)t((1 + λt)

1λ + 1)

λt

log(1 + λt)|∞∑m=0

(1|λ)m+1

m+ 1

tm

m!xn−l

⟩

= λ−kn∑l=0

(n

l

)λlS1(l, k)

n−l∑m=0

(n− lm

)(1|λ)m+1

m+ 1

⟨2Lir(1− e−t)t((1 + λt)

1λ + 1)

| λt

log(1 + λt)xn−l−m

⟩

= λ−kn∑l=k

(n

l

)λlS1(l, k)

n−l∑m=0

(n− lm

)(1|λ)m+1

m+ 1

⟨2Lir(1− e−t)t((1 + λt)

1λ + 1)

|∞∑j=0

bjλj

j!tjxn−l−m

⟩

= λ−kn∑l=k

(n

l

)λlS1(l, k)

n−l∑m=0

(n− lm

)(1|λ)m+1

m+ 1

n−l−m∑j=0

(n− l −m

j

)λjbj

γ(r)n−l−m−j+1

n− l −m− j + 1.

(3.37)

Here bj are the Bernoulli numbers of the second kind given by tlog(1+t) =

∑∞j=0 bj

tj

j! . Thus we have

derived the following result.


γ(r)n+1(λ, x)

=n∑k=0

n∑l=k

n−l∑m=0

n−l−m∑j=0

1

m+ 1

(n+ 1

j

)(n− j + 1

m

)(n− j −m+ 1

l

)λl+j−kS1(l, k)

× (1|λ)m+1bjγ(r)n−l−m−j+1Bk(x).

References

1. L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., 15(1979),51–88. 2, 22. R. Dere, Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. (Kyung-

shang), 22 (2013), no.3, 433-438. 23. H. Jolany, M. Aliabadi, R.B. Corcino, M.R. Darafsheh, A note on multi poly-Euler numbers and Bernoulli polynomials,

Gen. Math., 20(2012), no. 2-3, 122–134. 2.1, 24. D. S. Kim, T. Kim, Higher-order Bernoulli and poly-Bernoulli mixed type polynomials, Georgian Math. J., 22(2015),

265-272. 2.1, 25. D. S. Kim, T. Kim, A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials, Russ. J. Math. Phys., 22

(2015), no. 1, 26–33. 2.1, 2

6. D. S. Kim, T. Kim, T. Mansour, J.-J. Seo, Fully degenerate poly-Bernoulli polynomials with a q parameter, Filomat,

30(2016), no. 4, 1029-1035. 2.1, 27. D.S. Kim, T. Kim, H.I. Kwon , T. Mansour, Degenerate poly-Bernoulli polynomials with umbral calculus viewpoint, J.

Inequal. Appl., 215 (2015), 215–228. 2.1, 2, 2, 3


1519 T. KIM ET AL 1504-1520


8. D. S. Kim, T. Kim, Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials,

Adv. Stud. Contemp. Math. (Kyungshang), 23 (2013), no.4, 621-636. 2.1

9. T. Kim, D.S. Kim, Poly-Genocchi polynomials with umbral calculus viewpoint, Preprint. 2.1, 2, 310. S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc.[Harcourt Brace Jo-

vanovich, Dublishers], New York, 1984. 1, 2

Department of Mathematics, College of Science Tianjin Polytechnic University, Tianjin 300160, China,

Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of KoreaE-mail address: [email protected]

Department of Mathematics, Sogang University, Seoul 121-742, Republic of KoreaE-mail address: [email protected]

Graduate School of Education, Konkuk University, Seoul 143-701, Republic of KoreaE-mail address: [email protected]

Department Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea



1520 T. KIM ET AL 1504-1520

QUADRATIC (ρ1, ρ2)-FUNCTIONAL INEQUALITY IN FUZZY NORMED

SPACES

YOUNGHUN JO, JUNHA PARK, TAEKSEUNG KIM, JAEMIN KIM∗ AND CHOONKIL PARK∗

Abstract. In this paper, we introduce and solve the following quadratic (ρ1, ρ2)-functionalinequality

N(f(x+ y) + f(x− y) − 2f(x) − 2f(y), t) (0.1)

≤ min

(N

(ρ1

(2f

(x+ y

2

)+ 2f

(x− y

2

)− f(x) − f(y)

), t),

N(ρ2

(4f

(x+ y

2

)+ f(x− y) − 2f(x) − 2f(y)

), t))

in fuzzy normed spaces, where ρ1 and ρ2 are fixed nonzero real numbers with 1|ρ1|

+ 12|ρ2|

< 1,

and f(0) = 0.Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic (ρ1, ρ2)-

functional inequality (0.1) in fuzzy Banach spaces.

1. Introduction and preliminaries

Katsaras [16] defined a fuzzy norm on a vector space to construct a fuzzy vector topological

structure on the space. Some mathematicians have defined fuzzy norms on a vector space from

various points of view [10, 20, 43]. In particular, Bag and Samanta [2], following Cheng and

Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric

is of Kramosil and Michalek type [19]. They established a decomposition theorem of a fuzzy

norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3].

We use the definition of fuzzy normed spaces given in [2, 24, 25] to investigate the Hyers-Ulam

stability of quadratic (ρ1, ρ2)-functional inequality in fuzzy Banach spaces.

Definition 1.1. [2, 24, 25, 26] Let X be a real vector space. A function N : X × R→ [0, 1] is

called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R,

(N1) N(x, t) = 0 for t ≤ 0;

(N2) x = 0 if and only if N(x, t) = 1 for all t > 0;

(N3) N(cx, t) = N(x, t|c|) if c 6= 0;

(N4) N(x+ y, s+ t) ≥ minN(x, s), N(y, t);(N5) N(x, ·) is a non-decreasing function of R and limt→∞N(x, t) = 1.

(N6) for x 6= 0, N(x, ·) is continuous on R.

2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40.Key words and phrases. fuzzy Banach space; quadratic (ρ1, ρ2)-functional inequality; fixed point method;

Hyers-Ulam stability.∗Corresponding author.


1521 YOUNGHUN JO ET AL 1521-1529

Y. JO, J. PARK, T. KIM, J. KIM AND C. PARK

The pair (X,N) is called a fuzzy normed vector space.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [23, 24].

Definition 1.2. [2, 24, 25, 26] Let (X,N) be a fuzzy normed vector space. A sequence xn in X

is said to be convergent or converge if there exists an x ∈ X such that limn→∞N(xn − x, t) = 1

for all t > 0. In this case, x is called the limit of the sequence xn and we denote it by

N -limn→∞ xn = x.

Definition 1.3. [2, 24, 25, 26] Let (X,N) be a fuzzy normed vector space. A sequence xnin X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all

n ≥ n0 and all p > 0, we have N(xn+p − xn, t) > 1− ε.

It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If

each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy

normed vector space is called a fuzzy Banach space.

We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous

at a point x0 ∈ X if for each sequence xn converging to x0 in X, then the sequence f(xn)converges to f(x0). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be

continuous on X (see [3]).

The stability problem of functional equations originated from a question of Ulam [42] con-

cerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer

to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for

additive mappings and by Th.M. Rassias [35] for linear mappings by considering an unbounded

Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by Gavruta

[11] by replacing the unbounded Cauchy difference by a general control function in the spirit of

Th.M. Rassias’ approach. The stability problems of several functional equations have been ex-

tensively investigated by a number of authors and there are many interesting results concerning

this problem (see [7, 13, 15, 17, 18, 21, 31, 32, 33, 36, 37, 38, 39, 40, 41]).

Park [29, 30] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability

of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces.

We recall a fundamental result in fixed point theory.

Let X be a set. A function d : X × X → [0,∞] is called a generalized metric on X if d

satisfies

(1) d(x, y) = 0 if and only if x = y;

(2) d(x, y) = d(y, x) for all x, y ∈ X;

(3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.

Theorem 1.4. [4, 9] Let (X, d) be a complete generalized metric space and let J : X → X

be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element

x ∈ X, either

d(Jnx, Jn+1x) =∞



QUADRATIC (ρ1, ρ2)-FUNCTIONAL INEQUALITY IN FUZZY NORMED SPACES

for all nonnegative integers n or there exists a positive integer n0 such that

(1) d(Jnx, Jn+1x) <∞, ∀n ≥ n0;(2) the sequence Jnx converges to a fixed point y∗ of J ;

(3) y∗ is the unique fixed point of J in the set Y = y ∈ X | d(Jn0x, y) <∞;(4) d(y, y∗) ≤ 1

1−Ld(y, Jy) for all y ∈ Y .

In 1996, G. Isac and Th.M. Rassias [14] were the first to provide applications of stability theory

of functional equations for the proof of new fixed point theorems with applications. By using

fixed point methods, the stability problems of several functional equations have been extensively

investigated by a number of authors (see [5, 6, 23, 27, 28, 33, 34]).

In Section 2, we solve the quadratic (ρ1, ρ2)-functional inequality (0.1) and prove the Hyers-

Ulam stability of the quadratic (ρ1, ρ2)-functional inequality (0.1) in fuzzy Banach spaces by

using the fixed point method.

Throughout this paper, assume that ρ1 and ρ2 are fixed nonzero real numbers with 1|ρ1|+

12|ρ2| <

1.

2. Quadratic (ρ1, ρ2)-functional inequality (0.1)

In this section, we solve and investigate the quadratic (ρ1, ρ2)-functional inequality in fuzzy

normed spaces.

Lemma 2.1. Let X be a real vector space and (Y,N) be a fuzzy normed vector space. If a

mapping f : X → Y satisfies f(0) = 0 and

N(f(x+ y) + f(x− y)− 2f(x)− 2f(y), t) (2.1)

≤ min(N

(ρ1

(2f

(x+ y

2

)+ 2f

(x− y

2

)− f(x)− f(y)

), t

),

N

(ρ2

(4f

(x+ y

2

)+ f(x− y)− 2f(x)− 2f(y)

), t

))for all x, y ∈ X and all t > 0. Then f is quadratic.

Proof. Assume that f : X → Y satisfies (2.1).

Letting y = 0 in (2.1), we get

1 ≤ min(N(ρ1

(4f(x

2

)− f(x)

), t),(ρ2

(4f(x

2

)− f(x)

), t))

≤ N(

(ρ1 + ρ2)(

4f(x

2

)− f(x)

), 2t)

Thus

f(x

2

)=

1

4f(x) (2.2)

for all x ∈ X.

Now we consider P : X → Y that

P (x, y) = f(x+ y) + f(x− y)− 2f(x)− 2f(y)




and we consider

α =1

|ρ1|+

1

2|ρ2|.

It follows from (2.1) and (2.2) that

N(P (x, y), t) ≤ min(N(ρ1

2P (x, y), t

), N (ρ2P (x, y), t)

)= min

(N

(1

2P (x, y),

t

|ρ1|

), N

(1

2P (x, y),

t

2|ρ2|

))≤ N

(P (x, y),

(1

|ρ1|+

1

2|ρ2|

)t

)= N (P (x, y), αt)

for all t > 0. By (N5) and (N6),

P (x, y) = f(x+ y) + f(x− y)− 2f(x)− 2f(y) = 0

for all x, y ∈ X, since α < 1. So f : X → Y is quadratic.

We prove the Hyers-Ulam stability of the quadratic (ρ1, ρ2)-functional inequality (2.1) in fuzzy

Banach spaces.

Theorem 2.2. Let X be a real vector space and (Y,N) be a fuzzy normed vector space. Let

ϕ : X2 → [0,∞) be a function such that there exists an L < 1 with

ϕ(x, y) ≤ L

4ϕ(2x, 2y)

for all x, y ∈ X. Let f : X → Y be a mapping satisfying f(0) = 0 and

min

(N(f(x+ y) + f(x− y)− 2f(x)− 2f(y), t),

t

t+ ϕ(x, y)

)(2.3)

≤ min(N

(ρ1

(2f

(x+ y

2

)+ 2f

(x− y

2

)− f(x)− f(y)

), t

),

N

(ρ2

(4f

(x+ y

2

)+ f(x− y)− 2f(x)− 2f(y)

), t

))for all x, y ∈ X and all t > 0. Then Q(x) := N − limn→∞ 4nf

(x2n

)exists for each x ∈ X and

defines a quadratic mapping C : X → Y such that

N(f(x)−Q(x), t) ≥ (2− 2L)t

(2− 2L)t+ βϕ(x, 0)(2.4)

for all x ∈ X and all t > 0 while β = 1|ρ1| + 1

|ρ2| .

Proof. Letting y = 0 in (2.3), we get

t

t+ ϕ(x, 0)≤ min

(N(ρ1

(4f(x

2

)− f(x)

), t), N(ρ2

(4f(x

2

)− f(x)

), t))

(2.5)

≤ N

(4f(x

2

)− f(x),

(1

|ρ1|+

1

|ρ2|

)t

2

)= N

(f(x)− 4f

(x2

),β

2t

)




Consider the set

S := g : X → Y

and introduce the generalized metric on S:

d(g, h) = inf

µ ∈ R+ : N(g(x)− h(x), µt) ≥ t

t+ ϕ(x, 0), ∀x ∈ X,∀t > 0

,

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [22, Lemma 2.1]).

Now we consider the linear mapping J : S → S such that

Jg(x) := 4g(x

2

)for all x ∈ X.

Let g, h ∈ S be given such that d(g, h) = ε. Then

N(g(x)− h(x), εt) ≥ t

t+ ϕ(x, 0)

for all x ∈ X and all t > 0. Hence

N(Jg(x)− Jh(x), Lεt) = N(

4g(x

2

)− 4h

(x2

), Lεt

)= N

(g(x

2

)− h

(x2

),Lεt

4

)≥

Lt4

Lt4 + ϕ

(x2 , 0) ≥ Lt

4Lt4 + L

4ϕ(x, 0)=

t

t+ ϕ(x, 0)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that

d(Jg, Jh) ≤ Ld(g, h)

for all g, h ∈ S.

It follows from (2.5) that

N

(f(x)− 4f

(x2

),β

2t

)≥ t

t+ ϕ(x, 0)

for all x ∈ X and all t > 0. So d(f, Jf) ≤ β2 .

By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following:

(1) Q is a fixed point of J , i.e.,

Q(x

2

)=

1

4Q(x) (2.6)

for all x ∈ X. The mapping Q is a unique fixed point of J in the set

M = g ∈ S : d(f, g) <∞.

This implies that Q is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0,∞)

satisfying

N(f(x)−Q(x), µt) ≥ t

t+ ϕ(x, 0)

for all x ∈ X;

(2) d(Jnf,Q)→ 0 as n→∞. This implies the equality

N - limn→∞

4nf( x

2n

)= Q(x)




for all x ∈ X;

(3) d(f,Q) ≤ 11−Ld(f, Jf), which implies the inequality

d(f,Q) ≤ β

2− 2L.

This implies that the inequality (2.4) holds.

By (2.3),

min

(N

(4n(f

(x+ y

2n

)+ f

(x− y

2n

)− f

( x2n

)− 2f

( y2n

)), 4nt

),

t

t+ ϕ(x2n ,

y2n

))

≤ min(N

(4nρ1

(2f

(x+ y

2n+1

)+ 2f

(x− y2n+1

)− f

( x2n

)− f

( y2n

)), 4nt

),

N

(4nρ2

(4f

(x+ y

2n+1

)+ f

(x− y

2n

)− 2f

( x2n

)− 2f

( y2n

)), 4nt

))for all x, y ∈ X, all t > 0 and all n ∈ N. So

min

(N

(4n(f

(x+ y

2n

)+ f

(x− y

2n

)− f

( x2n

)− 2f

( y2n

)), t

),

t4n

t4n + Ln

4n ϕ (x, y)

)

≤ min(N

(4nρ1

(2f

(x+ y

2n+1

)+ 2f

(x− y2n+1

)− f

( x2n

)− f

( y2n

)), t

),

N

(4nρ2

(4f

(x+ y

2n+1

)+ f

(x− y

2n

)− 2f

( x2n

)− 2f

( y2n

)), t

))Since limn→∞

t4n

t4n

+Ln

4nϕ(x,y)

= 1 for all x, y ∈ X and all t > 0,

N(Q(x+ y) +Q(x− y)− 2Q(x)− 2Q(y), t)

≤ min(N

(ρ1

(2Q

(x+ y

2

)+ 2Q

(x− y

2

)−Q(x)−Q(y)

), t

),

N

(ρ2

(4Q

(x+ y

2

)+Q(x− y)− 2Q(x)− 2Q(y)

), t

))for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping Q : X → Y is quadratic, as

desired.

Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector

space with norm ‖ · ‖ and (Y,N) be a fuzzy normed vector space. Let f : X → Y be a mapping

satisfying f(0) = 0 and

min

(N(f(x+ y) + f(x− y)− 2f(x)− 2f(y), t),

t

t+ θ(‖x‖p + ‖y‖p)

)(2.7)

≤ min(N

(ρ1

(2f

(x+ y

2

)+ 2f

(x− y

2

)− f(x)− f(y)

), t

),

N

(ρ2

(4f

(x+ y

2

)+ f(x− y)− 2f(x)− 2f(y)

), t

))




for all x, y ∈ X and all t > 0. Then Q(x) := N -limn→∞ 4nf( x2n ) exists for each x ∈ X and

defines a quadratic mapping Q : X → Y such that

N (f(x)−Q(x), t) ≥ 2(2p − 4)t

2(2p − 4)t+ βθ‖2x‖p


|ρ2| .

Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(‖x‖p + ‖y‖p) for all x, y ∈ X.

Then we can choose L = 22−p, and we get the desired result.

Theorem 2.4. Let X be a real vector space and (Y,N) be a fuzzy normed vector space. Let

ϕ : X2 → [0,∞) be a function such that there exists an L < 1 with

ϕ(x, y) ≤ 4Lϕ(x

2,y

2

)for all x, y ∈ X. Let f : X → Y be a mapping satisfying (2.3). Then Q(x) := N -limn→∞

14n f (2nx)

exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that

N (f(x)−Q(x), t) ≥ (8− 8L)t

(8− 8L)t+ βϕ(x, 0)(2.8)

for all x ∈ X and all t > 0.

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (2.5) that

N

(f(x)− 1

4f (2x) ,

β

8t

)≥ t

t+ ϕ(x, 0)

for all x ∈ X and all t > 0. Now we consider the linear mapping J : S → S such that

Jg(x) :=1

4g (2x)

for all x ∈ X. Then d(f, Jf) ≤ β8 . Hence

d(f,Q) ≤ β

8− 8L,

which implies that the inequality (2.8) holds.

The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 2. Let X be a normed

vector space with norm ‖ · ‖ and (Y,N) be a fuzzy normed vector space. Let f : X → Y be a

mapping satisfying (2.7). Then Q(x) := N -limn→∞14n f(2nx) exists for each x ∈ X and defines

a quadratic mapping Q : X → Y such that

N (f(x)−Q(x), t) ≥ 2(4− 2p)t

2(4− 2p)t+ βθ‖x‖p


|ρ2| .

Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(‖x‖p + ‖y‖p) for all x, y ∈ X.

Then we can choose L = 2p−2, and we get the desired result.




Acknowledgments

This work was supported by the Seoul Science High School R&E Program in 2017. C. Park

was supported by Basic Science Research Program through the National Research Foundation of

Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937).

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Younghun Jo, Junha Park, Taekseung Kim, Jaemin KimMathematics Branch, Seoul Science High School, Seoul 03066, Korea

E-mail address: [email protected]; [email protected]; [email protected]; [email protected]

Choonkil ParkResearch Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea




TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 26, NO. 8, 2019

Existence and global attractiveness of pseudo almost periodic solutions to impulsive partial stochastic neutral functional differential equations, Zuomao Yan and Fangxia Lu,…1343

Fourier series of functions associated with poly-Genocchi polynomials, Taekyun Kim, Dae San Kim, Dmitry V. Dolgy, and Jin-Woo Park,……………………………………………1387

Tracy-Singh Products and Classes of Operators, Arnon Ploymukda, Pattrawut Chansangiam, Wicharn Lewkeeratiyutkul,…………………………………………………………….1401

On unicity theorems of difference of entire or meromorphic functions, Yong Liu,….1414

On the existence and behavior of the solutions for some difference equations systems, M. M. El-Dessoky and Aatef Hobiny,………………………………………………………….1428

Applications of soft sets to BCC-ideals in BCC-algebras, Sun Shin Ahn,………….1440

Fixed point theorems for various contraction conditions in digital metric spaces, Choonkil Park, Ozgur Ege, Sanjay Kumar, Deepak Jain, and Jung Rye Lee,……………………….1451

Fixed points of Ćirić type ordered F-contractions on partial metric spaces, Muhammad Nazam, Muhammad Arshad, Choonkil Park, and Sungsik Yun,…………………………….1459

The Theoretical Analysis of 𝑙𝑙1-TV Compressive Sensing Model for MRI Image Reconstruction, Xiaoman Liu,………………………………………………………………………..1471

On Fibonacci Z-sequences and their logarithm functions, Hee Sik Kim, J. Neggers, and Keum Sook So,………………………………………………………………………………1479

Hermite-Hadamard type fractional integral inequalities for 𝑀𝑀𝑇𝑇(𝑟𝑟;𝑔𝑔,𝑚𝑚,𝜑𝜑)-preinvex functions, Muhammad Adil Khan, Yu-Ming Chu, Artion Kashuri, and Rozana Liko,………….1487

Umbral calculus approach to degenerate poly-Genocchi polynomials, Taekyun Kim, Dae San Kim, Lee Chae Jang, and Gwan-Woo Jang,………………………………………….1504

Quadratic (𝜌𝜌1,𝜌𝜌2)-functional inequality in fuzzy normed spaces, Younghun Jo, Junha Park, Taekseung Kim, Jaemin Kim, and Choonkil Park,……………………………………1521

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