chapter 1 introduction - shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/10518/8/08_chapter...
TRANSCRIPT
1
CHAPTER 1
INTRODUCTION
The work reported in this thesis is based on fixed points, common fixed points, approximate
fixed points, end points and approximate best proximity points of a variety of maps satisfying
different contractive conditions. The stability of different iterative procedures is investigated
in a new setting. Several existing results concerning single valued and multi valued maps are
extended to the theory of iterated function systems (IFS) and iterated multifunction systems
(IMS). Further, some minimax and saddle point theorems are also established.
We first provide a brief historical development of the fixed point theory.
1.1 HISTORICAL OUTLINE
The origin of fixed point theory lies in the method of successive approximations used for
proving existence of solutions of differential equations introduced independently by Joseph
Liouville [1] in 1837 and Charles Emile Picard [2] in 1890. But formally it was started in the
beginning of twentieth century as an important part of analysis. The abstraction of this
classical theory is the pioneering work of the great Polish mathematician Stefan Banach [3]
published in 1922 which provides a constructive method to find the fixed points of a map.
However, on historical point of view, the major classical result in fixed point theory is due to
2
L. E. J. Brouwer [4] given in 1912 (also see Zeidler [5], Kirk and Sims [6] and Granas and
Dugundji [7]).
The celebrated Banach contraction principle (BCP) states that “a contraction mapping
on a complete metric space has a unique fixed point”. Banach used the idea of shrinking map
to obtain this fundamental result. The Brouwer fixed point theorem is of great importance in
the numerical treatment of equations. It exactly states that “a continuous map on a closed unit
ball in Rn has a fixed point”. An important extension of this is the Schauder’s fixed point
theorem [8] of 1930 stating “a continuous map on a convex compact subspace of a Banach
space has a fixed point”. These celebrated results have been used, generalized and extended
in various ways by several mathematicians, scientists, economists for single valued and
multivalued mappings under different contractive conditions in various spaces. Kannan [9]
proved a fixed point theorem for the maps not necessarily continuous. This was another
important development in fixed point theory (see also Chatterjea [10]). Thus various results
pertaining to fixed points, common fixed points, coincidence points, etc. have been
investigated for maps satisfying different contractive conditions in different settings, see
among others, Tychonoff [11], Lefschetz [12], Kakutani [13], Tarski [14], Edelstein [15],
Zamfirescu [16], Ćirić [17]-[18], Mishra [19], Singh et al [20], Singh and Mishra [21],
Browder [22], Göhde [23], Suzuki [24]-[25], Suzuki and Kikkawa [26] and references
therein. For a fundamental comparison and development of various contractive conditions,
one may refer Rhoades [27]-[30], Collaco and Silva [31], Murthy [32], Singh and Tomar [33]
and Pant et al [34].
The study of fixed points of multivalued mappings was initiated by Nadler [35]-[37]
and nonexpansive mapping by Markin [38]-[39] using the concept of Hausdorff metric.
Indeed, Nadler [37] proved that a multivalued contraction has a fixed point in a complete
metric space. Ćirić [40] generalized Nadler’s result to multivalued quasi-contraction maps.
Subsequently, it received great attention in applicable mathematics and was extended and
generalized on various settings. Further, hybrid fixed point theory for nonlinear single-valued
and multivalued maps is a new development in the domain of multivalued analysis (see, for
instance, Krasnoselskii [41], Corley [42], Mishra et al [43], Singh and Arora [44], Singh and
Prasad [45], Dhage [46], Singh and Mishra [47]-[50], Singh et al [51]-[52] and references
thereof). For a historical development of the hybrid fixed point theory, one may refer to Singh
and Mishra [47] and Singh et al [51]-[52].
3
Thus fixed point theory has been extensively studied, generalized and enriched in
different approaches such as, metric, topological and order-theoretic (see Goebel and Kirk
[53], Kirk and Sims [54], Brown [55], Carl [56], Andres and Gorniewicz [57], Wegrzyk [58]
and Smart [59]). This advancement in fixed point theory diversified the applications of
various fixed point results in various areas such as the existence theory of differential and
integral equations, dynamic programming, fractal and chaos theory, discrete dynamics,
population dynamics, differential inclusions, system analysis, interval arithmetic, optimization
and game theory, variational inequalities and control theory, elasticity and plasticity theory
and other diverse disciplines of mathematical sciences (see, for instance, Zeidler [5], Corley
[42], Brown [55], Andres and Górniewicz [57], Wegrzyk [58], Smart [59] and Agarwal et al
[60]).
In the following, we give a brief sketch of the development of the topics studied in this thesis.
1.1.1.
In many situations of practical utility, the mapping under consideration may not have an exact
fixed point due to some tight restrictions on the space or the map. Further, there may arrise
some practical situations where the existence of a fixed point is not strictly required but an
approximate fixed point is more than enough. The theory of approximate fixed points plays an
important role in such situations. Approximate fixed point property for various types of
mappings has been a prominent area of research for the last few decades. A classical best
approximation theorem was introduced by Fan [61] in 1969. Afterward, several authors,
including Reich [62], Prolla [63], Sehgal and Singh [64]-[65], have derived extensions of
Fan’s theorem in many directions. Approximate fixed points of continuous maps have been
studied by Gajek et al [66], Rafi and Salami [67], Hadzic [68]-[69] and many others. Tijs et al
[70] studied approximate fixed point theorems for contraction and non-expansive maps by
weakening the conditions on the spaces. Branzei et al [71] further extended these results to
multifunctions in Banach spaces. Singh and Prasad [72] proved an approximate fixed point
theorem for quasi contraction in metric spaces. Recently Berinde [73] obtained some
approximate fixed point theorems for operators satisfying Kannan, Chatterjea and Zamfirescu
type of conditions on metric spaces. P˘acurar and P˘acurar [74] obtained approximate fixed
point results for almost or weak contraction maps.
4
If T is a non-self-mapping, it is probable that the fixed point equation xTx = has no solution.
In such a case best approximation theorems explore the existence of an approximate solution
whereas best proximity point theorems analyze the existence of an optimal approximate
solution. In 2003, Kirk et al [75] introduced 2-cyclic contraction, which becomes a relevant
research area in search of best proximity points. Eldred and Veeramani [76] defined a new
contraction which is more abstract formulation than 2-cyclic contraction. This definition is
more general than the notion of cyclical maps, in the sense that if the sets intersect, then every
point is a best proximity point. It turns out that many of the contractive-type conditions which
have been investigated for fixed points ensure the existence of best proximity points. Some
results of this kind are obtained in Bari et al [77], Al-Thagafi and Shahzad [78], Karpagam
and Agrawal [79], Petric [80], Mishra and Pant [81], etc. Recently, Mohsenalhosseini et al
[82] also defined a stronger concept of approximate best proximity points and obtained some
interesting existence results.
1.1.2.
It is a well known that the celebrated Banach contraction principle (BCP) is one of the
main tools for both the theoretical and the computational aspects in mathematical sciences.
This theorem has witnessed numerous generalizations and extensions in the literature due to
its simplicity and constructive approach. Kannan [9] by taking an entirely different condition,
proved a fixed point theorem for operators, which need not be contraction or contractive.
Indeed, he was the first to propose a fixed point theorem for a discontinuous map. Jungck [83]
obtained an important generalization of (BCP) in the form of common fixed point theorem for
commuting pair of maps. Sessa [84] introduced the concept of weakly commuting maps. This
concept was further improved by Jungck and Rhoades [85] with the notion of weakly
compatible mappings. Pant [86] studied the common fixed point theorem of noncommuting
maps. Singh and Mishra [49] studied coincidence and fixed points of reciprocally continuous
and compatible hybrid maps. Aamri and Moutawakil [87] defined property E. A., Liu et al
[88] generalized it as common property E. A. and obtained interesting results. Branciari [89]
studied contractive conditions of integral type and obtained an integral version of the Banach
contraction principle. A number of papers appeared for the maps satisfying the integral type
conditions in different settings, see for example, Rhoades [90], Vijayaraju et al [91], Aliouche
[92], Djoudi and Aliouche [93], Pathak et al [94], Pathak and Verma [95]-[96], Day et al [97]
and many others.
5
After the initiation of fuzzy sets by Zadeh [98], fuzzy metric spaces are introduced by many
authors such as Kramosil and Michalek [99], George and Veeramani [100], Gregori and
Sapena [101], Kaleva and Seikkala [102], Park [103], Adibi et al [104], Saadati and Park
[105]-[106] in different ways. Grabiec [107] first extended BCP and Edelstein fixed point
theorem to fuzzy metric space Thereafter, may authors worked on this line (see, George and
Veeramani [100], Gregori and Sapena [101], Schweizer et al [108], Song [109], Mihet [110],
Mishra et al [111]-[113], Jha [114] etc). Gogeun [115] generalized fuzzy set to −L fuzzy set
in 1967. Another extension includes intuitionistic fuzzy sets proposed by Atanassov [116] in
1986. Saadati et al [117], Saadati [118] generalized the notions of fuzzy metric spaces due to
George and Veeramani [100] and intuitionistic fuzzy metric spaces due to Saadati and Park
[105] by introducing the notion of −L fuzzy metric spaces with the help of continuous t-
norms. These developments enthused the fuzzyfication of fixed point theory and authors
contributed profusely in this line, see for example, [100]-[101], [107], [119]-[128] and
references thereof.
1.1.3.
Fixed point theory has always been exciting in itself and its applications in new areas.
Currently, it has found new and hot areas of activity. The initiation of fixed point theory in
computer science by Tarski [14] and Scarf [129] enhances its applicability in different
domains. Due to the advent of speedy and fast computational tools, a new horizon has been
provided to fixed point theory. The fixed point equations are solved by means of some
iterative procedures. In view of their concrete applications, it is of great interest to know
whether these iterative procedures are numerically stable or not. The study of stability of
iterative procedures enjoys a celebrated place in applicable mathematics due to chaotic
behavior of functions in discrete dynamics, fractal graphics and various other numerical
computations where computer programming is involved. This kind of problem for real-valued
functions was first discussed by M. Urabe [130] in 1956. The first result on the stability of
iterative procedures on metric spaces is due to Alexander M. Ostrowski [131]. This result was
extended to multivalued operators by Singh and Chadha [132]. Czerwik et al [133]-[134]
extended this to the setting of generalized metric spaces.
The stability of iterative schemes in various settings has thus been studied during the
last three decades by a number of authors, see for instance, Agarwal et al [60], Czerwik et al
6
[133]-[134], Ahmed [135], Berinde [136]-[137], Ćirić and Ume [138], Chidume et al [139],
Goebel and Kirk [140], Harder and Hicks [141]-[142], Osilike [143]-[144], Rhoades [145]-
[146], Singh et al [147]-[149], Olatinwo and Imoru [150], Osilike and Udomene [151], Singh
and Prasad [152], Mishra and Kalinde [153], Mishra et al [154] and several references thereof.
Recently Timis and Berinde [155] and Timis [156] respectively studied weak stability and
weak −2w stability of an operator in metric spaces.
1.1.4.
The study of fixed point problems for multivalued maps was initiated by Kakutani
[13] in the year 1941 in finite dimensional spaces by generalizing the Brouwer’s fixed point
theorem [4]. This was the beginning of the fixed point theory of multimaps having a vital
connection with the minimax theory in games. Kakutani used his results to obtain a simple
proof of von Neumann’s minimax theorem [157]. This was extended to infinite dimensional
Banach spaces by Bohnenblust and Karlin [158]. Further, Sion [159] generalized von
Neumann results on the basis of Knaster, Kuratowski and Mazurkiewicz (KKM) theorem
[160]. At the beginning, the results of the KKM theory were established for convex subsets of
topological vector spaces mainly by Ky Fan [161]-[164]. Horvath [165]-[167] extended most
of the Fan’s results in the KKM theory to H-spaces by replacing the convexity condition by
contractibility. Park [168] in 1992 established new versions of KKM theorems and minimax
inequalities on H-spaces which were extended by Park and Kim [169]-[173]. Thereafter more
general spaces such as abstract convex space and KKM spaces were introduced by Park in
[174]-[178]. In this way, KKM theory has been continuously upgraded and enriched to
enhance its applicability to the wider range of problems, such as minimax theorems, saddle
point theorems in game theory, economics and optimization theory, variational inequalities
etc., (see for instance [5], [13], [158]-[159], [162]-[163], [165], [168]-[195]).
1.1.5.
One of the recent advancement of the fixed pint theory of single-valued and multivalued
mappings is in the areas of iterated function systems (IFS), iterated multifunction systems
(IMS) and fractals. Fractal theory is a new domain initiated by Mandelbrot [196] in which
Banach [3] and Nadler [37] fixed point theorems and their variants are of vital importance.
7
Michael Barnsley and Steven Demko [197] popularized the theory of IFS after Hutchinson
[198] gave a formal definition of it in 1981. An IFS usually consists of a complete metric
space together with a finite set of contraction mappings. It was born as an application of the
theory of discrete dynamical systems and is a useful tool to build fractal and other similar sets
(see, for instance, [196], [197]-[229] and references thereof). Indeed, self similar sets are
special class of fractals and there are no objects in nature which have exact structures of self
similar sets. These sets are perhaps the simplest and the most basic structures in the theory of
fractals which should give us much information on what would happen in the general case of
fractals (cf. Kigami [209]-[211]).
This basic notion of IFS has been extended and enriched to more general settings by changing
the condition on mappings or the space by various authors (see for instance, [197]-[199],
[212]-[229] ). In Jain and Fiser [212] and Fiser [215] contraction maps are replaced by weakly
contractive or non-expansive maps. Mate [219] and Rus and Triff [229] replaced contraction
constant by a comparison function to obtain their results. In [230]-[232] the formulations of
the contraction due to Meir and Keeler have been used to generalize the IFS theory. Recently
Mihail and Miculescu [221] introduced the notion of generalized iterated function system
(GIFS), which is a family of functions in a complete metric space and obtained GIFS to be a
natural generalization of the notion of IFS (see [220]-[224]). Further, Llorens-Fuster et al
[218] defined mixed iterated function system by taking more general conditions and obtained
a mixed iterated function system theory for contraction and Meir Keeler contraction maps.
Thus there has been a huge development in the metric as well as topological fixed point
theory. The recent advancement of the computational tools has tremendously enhanced the
scope of the applications of it. The entire theory of contractions and multivalued contractions
is expected to revolutionize not only various branches of mathematics but also diverse
dimensions of knowledge in engineering and sciences.
In the following section, we recall the basic concepts required for our study in the subsequent
chapters regarding fixed point theory for single valued, multivalued and hybrid maps.
8
1.2 BASIC CONCEPTS
We start with the basic concepts related to metric spaces, fixed points and different
contraction conditions.
Definition 1.2.1 [233]. Let X be a non-empty set together with a distance function
.: +→× RXXd The function d is said to be a metric iff for all ,,, Xzyx ∈ the following
conditions are satisfied
(i) ( , ) 0d x y ≥ and ,0),( yxiffyxd ==
(ii) ),,(),( xydyxd =
(iii) ).,(),(),( zydyxdzxd +≤
The pair ),( dX is called a metric space.
Definition 1.2.2 [6]. A sequence in a metric space is a Cauchy sequence if for every ,0>ε
there exists Nn ∈0 such that ,),( ε<mn xxd for all ., 0nmn >
Definition 1.2.3 [6]. A metric space ),( dX is called complete, if every Cauchy sequence
converges in it.
Definition 1.2.4 [6]. The diameter of a set A denoted by )(Aδ is defined as
}.,:),({sup)( AbabadA ∈=δ It means that diameter of the set is the least upper bound of
the distances between the points of the set .A If the diameter is finite, i.e., ,)( ∞<Aδ then A
is bounded.
Definition 1.2.5 [6]. Let ),( dX be a metric space and : .T X X→ Then T has a fixed point if
there is an Xx∈ such that .Tx x= The point x is called a fixed point of T.
Definition 1.2.6 [6]. Let ),( dX be a metric space and .:, XXST → Then x is called a
coincidence (respectively, common fixed) point of T and S, if Xx∈ such that Tx = Sx
(respectively, x = Tx = Sx).
9
In some situations an approximate solution of the problem is more than enough, so we have to
find approximate fixed point instead of fixed point.
Definition 1.2.7 [73]. Let ),( dX be a metric space and : .T X X→ An element 0x X∈ is
called an approximate fixed point (or −ε fixed point) of T if 0 0( , ) ,d Tx x ε< when 0.ε >
Approximate fixed point of the function exists if the defined mapping has approximate fixed
point property.
Definition 1.2.8 [73]. A map T is said to satisfy approximate fixed point property (AFPP)
if for every ,0>ε ,)( φε ≠TFix where )(TFixε is the set of all approximate fixed point of T.
The following condition guarantees the existence of approximate fixed points.
Definition 1.2.9 ([234]). A map XXT →: is said to be asymptotically regular if for any
,Xx∈
.0),(lim 1 =+
∞→xTxTd nn
n
Definition 1.2.10 [137]. Let ),( dX be a metric space. A mapping XXT →: is called
(i) Lipschitzian (or −L Lipschitzian) if there exists 0>L such that for all ,, Xyx ∈
),,(),( yxdLTyTxd ⋅≤ (lm)
(ii) Banach contraction if T is −α Lipschitzian, with [0, 1)α∈ and for all ,, Xyx ∈
( , ) ( , ),d Tx Ty d x yα≤ ⋅ (bc)
(iii) nonexpansive if T is 1-Lipschitzian and for all ,, Xyx ∈
( , ) ( , ),d Tx Ty d x y≤ (nm)
(iv) contractive if ),,(),( yxdTyTxd < for all ,, Xyx ∈ ,yx ≠ (cm)
(v) isometry if ),,(),( yxdTyTxd = for all ., Xyx ∈ (im)
Examples 1.2.1 [137]. (i) Let ],2,2/1[]2,2/1[: →T be defined as ,/1)( xxT = then T is
4-Lipschitzian with Fix (T) = {1}, where ( )Fix T denotes fixed point of the mapping T.
(ii) ,: RRT → ,32/)( += xxT .x R∈ Obviously T is a Banach contraction and ( ) {6}.Fix T =
10
(iii) 1 ,Tx x x R= − ∈ is nonexpansive and ( ) {1/ 2}.Fix T =
(iv) ],,1[],1[: ∞→∞T ,/1)( xxxT += is contractive and ( ) .Fix T φ=
(v) ,2)( += xxT then ( )Fix T φ= is isometry.
The celebrated Banach contraction principle (BCP) is the simplest and one of the most
versatile and fundamental results in the fixed point theory. It not only produces approximation
of any desired accuracy but also determines a-priori and a-posteriori error estimates.
Theorem 1.2.1 [3]. Let ),( dX be a complete metric space and XXT →: satisfies (bc).
Then T has a unique fixed point ,p i.e., Tp = p and lim .n
nT x p
→∞=
Moreover, we have a-priori error estimate
( , ) ( , )1
nnd T x p d Tx xα
α≤
−
and the a-posteriori error estimate
1( , ) ( , ).1
n n nd T x p d T x T xαα
−≤−
)).(),((1
)),(( 1 xTxTdpxTd nnn −
−≤
αα
There are various generalizations of the celebrated contraction mapping principle. These
generalizations are obtained either by weakening the contraction condition of the map by
giving a sufficiently rich structure to the space in order to compensate the relaxation of the
contraction condition or by extending the structure of the space or sometime combining both
the approaches.
Kannan [9] was the first to propose a fixed point theorem for a discontinuous map. He exactly
proved the following.
Theorem 1.2.2 [9] (Kannan Contraction Theorem). Let T be a self map on a complete
metric space (X, d) satisfying the following condition (usually called Kannan contraction),
for all Xyx ∈, and some [0,1/ 2),β ∈
11
[ ].),(),(),( TyydTxxdTyTxd +≤ β (kc)
Then T has a unique fixed point.
Kannan’s theorem motivated numerous extensions and generalizations of the BCP and his
own fixed point theorem on various settings. Chatterjea [10] proved a fixed point theorem for
discontinuous mapping satisfying a condition which is actually a kind of dual of Kannan
mapping.
Definition 1.2.11 [10]. Let ),( dX be a metric space. A mapping XXT →: is called
Chatterjea contraction if for all Xyx ∈, and some [0,1/ 2),γ ∈
[ ].),(),(),( TxydTyxdTyTxd +≤ γ (cc)
Zamfirescu [16] obtained some interesting results by combining Banach, Kannan and
Chatterjea contraction conditions.
Definition 1.2.12 [16]. Let ),( dX be a metric space. A mapping XXT →: is called
Zamfirescu contraction if for all Xyx ∈, and some [0, 1),α ∈ 1, [0, ),2
β γ ∈ satisfies at
least one of the following conditions.
(i) ),(),( yxdTyTxd α≤
(ii) [ ]),(),(),( TyydTxxdTyTxd +≤ β (zc)
(iii) [ ].),(),(),( TxydTyxdTyTxd +≤ γ
Among various generalizations of the Banach contraction, the Ćirić [17]-[18] contraction
(also called as quasi-contraction) is considered to be the most general one.
Definition 1.2.13. Let ),( dX be a metric space. A mapping XXT →: is called quasi-
contraction (Ćirić [17]-[18]) if
)},,(),,(),,(),,(),,(max{.),( TxydTyxdTyydTxxdyxdkTyTxd ≤ (qc)
for some 10 <≤ k and all yx, in .X
12
Definition 1.2.14 [235]. A mapping XXT →: is called weak or almost contraction if there
exist )1,0(∈α and 0≥L such that for all ,, Xyx ∈
).,(),(),( TxydLyxdTyTxd +≤α (ac)
It is important to note that any mapping satisfying Banach, Kannan, Chatterjea, Zamfirescu, or
Ciric (with constant k in )2/1,0( ) type conditions are a weak or almost contraction, see [71].
Jungck [83] extended Banach contraction in the following manner.
Definition 1.2.15 [83]. Let ( , )X d be a metric space and , : .T S X X→ The following
condition is known as Jungck contraction.
),,(),( SySxdTyTxd ⋅≤α (jc)
for some [0, 1)α∈ and all ., Xyx ∈
Although this condition was known to Singh and Kulshrestha [236] but Jungck was credited
for presenting constructive proof regarding the existence of a common fixed point of
commuting maps. Further various fixed point and common fixed point results are investigated
in the literature for the maps satisfying different conditions. We recall some of them as
follows.
Definition 1.2.16. Let ( , )X d be a metric space and .:, XXST → The mappings T and S are
(i) commuting [83], if TSx = STx for all ,Xx∈
(ii) weakly commuting [84], if ),(),( SxTxdSTxTSxd ≤ for all ,Xx∈
(iii) compatible [237, 238], if ,0),(lim =∞→ nnn
STxTSxd whenever }{ nx is a sequence in X such
that
,limlim tSxTx nnnn==
∞→∞→ for some ,Xt∈
(iv) weakly compatible [85], if they commute at their coincidence points; i.e., if Tu = Su for
some u in X, then TSu = STu,
(v) satisfying the property (E. A.) [87], if there exists a sequence }{ nx such that
,limlim tSxTx nnnn==
∞→∞→ for some .Xt∈
13
Notice that weakly commuting mappings are compatible and compatible mappings are weakly
compatible but the converse need not be true (see [87], [237]-[239]). It is easy to see that two
noncompatible mappings satisfy the property (E. A.).
Fixed point theory for multivalued mappings is a natural generalization of the theory of single
valued mappings. Most of the results including the well known fixed point theorems of
Banach and others have been extensively studied for the multivalued cases, see, for instance
[35]-[41], [55], [57]-[59], [63], [85] and several references therein. These multivalued fixed
point theorems have applications in control theory, convex optimization, differential
equations, economics, etc. (see also [178], [180]).
In the following, we discuss the basic concepts and preliminaries regarding multivalued fixed
point theory.
Definition 1.2.17 [59]. If T is a multivalued map, i.e., from X to the collection of nonempty
subsets of .X Then a point p in X is called a fixed point of T if .Tpp∈
Definition 1.2.18 [59]. Let XXS →: and ).(: XCBXT → A point Xp∈ is a coincidence
(respectively, common fixed) point of S and T if TpSp∈ (respectively, TpSpp ∈= ).
Example 1.2.2. Consider ),0[ ∞=X with the usual metric. Define XXS →: and
)(: XCBXT → as
∞∈∈
=),1[3)1,0[0
xifxxif
Sx and
∞∈+∈
=),1[]31,1[)1,0[}{
xifxxifx
Tx
We have ,1]4,1[31 TS =∈= that is 1=x is a coincidence point of S and T and 0 is the
common fixed point of S and T.
Definition 1.2.19 [240]. Let A and B be the nonempty compact subsets of ,X then the
distance between a point x and set A is defined as ),(min),( yxdAxdAy∈
= .
The distance from set A to set B is given as ),(max),( BxdBAdAx∈
= .
14
Note that d is not a metric, since d is not symmetric. That is, ),(),( ABdBAd ≠ in general.
The Hausdorff distance ),,( BAh between Aand B is ( , ) max{ ( , ), ( , )}.h A B d A B d B A=
Definition 1.2.20 [240]. Let ),( dX be a metric space and K(X) be the nonempty compact
subsets of X together with a distance function h. Then for all
, , ( ),A B C K X∈ ( , )h A B satisfies following properties.
(i) ( , ) 0h A B ≥ and ( , ) 0 ,h A B iff A B= =
(iii) ( , ) ( , ),h A B h B A=
(iv) ( , ) ( , ) ( , ).h A B h A C h C B≤ +
The pair ( , )K h is called a Hausdorff metric space.
Definition 1.2.21 [137]. Let ),( dX be a metric space and ( )P X is the family of nonempty
subsets of .X Let : ( )T X P X→ satisfies the following condition.
( , ) ( , ),h Tx Ty a d x y≤ ⋅ (mc)
for some )1,0[∈a and all ., Xyx ∈ Then T is called a multivalued contraction.
Further, T is called
(i) nonexpansive, if it is 1-Lipschitzian,
(ii) contractive if ( , ) ( , ),h Tx Ty d x y< for all ,, Xyx ∈ ,yx ≠
(iii) isometry if ( , ) ( , ),h Tx Ty d x y= for all ., Xyx ∈
Definition 1.2.22. Let :T X X→ and : ( ).S X CB X→ Then mappings T and S are
(i) weakly commuting [84], if ( )TSx CB X∈ for all x X∈ and ( , ) ( , ),h TSx STx d Tx Sx≤
(iii) compatible [237, 238], if lim ( , ) 0,n nnh TSx STx
→∞= whenever }{ nx is a sequence in X such
that
lim ( )nnSx A CB X
→∞= ∈ and lim ,nn
Tx t A→∞
= ∈
(iv) weakly compatible [85], if they commute at their coincidence points; i.e., if Tu Su∈ for
some u in X, then TSu = STu,
(v) R-weakly commutativity [86], if ( )TSx CB X∈ for all x X∈ and there exists a real
number 0>R such that
( , ) ( , ),h STx TSx R d Tx Sx≤
15
(v) property (E . A.) [87], if there exists a sequence }{ nx such that
lim ( )nnSx A CB X
→∞= ∈ and .lim AtTxnn
∈=∞→
Liu et al [86] defined a more general condition then property (E. A.) as follows.
Definition 1.2.23 [88]. Let XXgf →:, and ).(:, XCBXGF → The pairs ),( Ff and
),( Gg are said to satisfy the common property (E. A.) if there exists two sequences
}{},{ nn yx in X, some t in X, and A, B in CB(X) such that
,lim AFxnn=
∞→ ,lim BGynn
=∞→
.limlim BAtgxfx nnnnI∈==
∞→∞→
Nadler [37] extended the BCP to multivalued maps in the following manner.
Theorem 1.2.3 [37]. Let ),( dX be a complete metric space and ( )CB X denotes the
collection of all nonempty closed and bounded subsets of X with the Hausdorff metric h.
Suppose : ( , ) ( ( ), )T X d CB X h→ satisfies (mc). Then T has a fixed point in .X
For approximating fixed point of the corresponding contraction operator of a given
nonlinear equation, we need some iterative methods. There are many iterative procedures
given in the literature which can be used in different conditions (see, [2], [147], [241]-[244],
etc.), i.e. if one iterative scheme fails in a given situation other may be applied or some time
we may choose them according to their speed of convergence.
The most popular iterative procedure, called Picard iteration is defined as follows.
Definition 1.2.24 [2]. Let ( , )X d be a metric space and : .T X X→ Choose 0x X∈ and
define 1 0 2 1, ,...x Tx x Tx= = and obtain a relation
...,2,1,0,1 ==+ nTxx nn . (PI)
Here nx is the thn Picard iterate of 0x in X.
16
Banach fixed point theorem uses Picard iteration for the convergence towards a fixed point.
But in many cases Picard iterations may not converge to the fixed point of the map, some
other iterative scheme may be used in such cases.
In 1953, Mann [242] defined an iteration scheme in the following manner.
Definition 1.2.25 [242]. Choose an initial point 0x in a given set and let }{ nα with
,10 <≤ nα be a sequence of real numbers. Then Mann iteration is defined by
...,2,1,0,)1(1 =+−=+ nTxxx nnnnn αα . (MI)
If T is a continuous map and the Mann iterative process converges, then it converges to a
fixed point of .T But if T is not continuous, then there is no guarantee that it will converge to
a fixed point of ,T even if the Mann process converges. This is shown by the following
example.
Example 1.2.3 [137]. Let ]1,0[]1,0[: →T be given by 010 == TT and .10,1 <<= xTx
Then ( ) {0},Fix T = where ( )Fix T is the set of all fixed points of T and the Mann iteration
with 1,1≥= n
nnα converges to 1, which is not a fixed point of .T
Some other scheme may be used in such a case. Ishikawa [243] defined the following iterative
scheme.
Definition 1.2.26 [243]. Choose an initial point 0x in a given set and let }{},{ nn βα be the
sequences of real numbers with .10 <≤≤ nn βα Then
...,2,1,0,)1(
)1(1
=+−=+−=+
nTxxyTyxx
nnnnn
nnnnn
ββαα
(II)
In the similar way, Noor [244] defined a three-step iteration scheme as follows.
17
Definition 1.2.27 [244]. Choose an initial point 0x in a given set and let
)1,0[}{},{},{ ⊂nnn γβα be the sequences of real numbers. Then
...,2,1,0,)1(
)1()1(1
=+−=+−=+−=+
nTxxzTzxyTyxx
nnnnn
nnnnn
nnnnn
γγββαα
(NI)
For a detailed development of different iteration scheme, one may refer Berinde [137].
Jungck [237, 238] generalized BCP, by replacing the identity map with two continuous maps
and obtained a common fixed point theorem. He defined a new iterative scheme using two
mappings in the following way.
Definition 1.2.28 [238]. Let XYTS →:, and ).()( YSYT ⊆ For any ,0 Yx ∈ consider
...,1,0),,(1 ==+ nxTfSx nn .
This procedure was essentially introduced by Jungck, and it becomes the Picard iterative
procedure when XY = and =S identity map.
Notice that, If we take ,),( nn TxxTf = then the method is called Jungck-Picard iteration
(JPI) and if ,)1(),( nnnnn TxSxxTf αα +−= where ,10 <≤ nα then it is called Jungck-Mann
iteration (JMI) (see [147] & [245]).
Definition 1.2.29 [150]. Let XXS →: and ).()( XSXT ⊆ Define
...,1,0,)1(
)1(1 =
+−=+−=+ n
TxSxSzTzSxSx
nnnnn
nnnnn
ββαα
(JII)
where }{ nα and }{ nβ satisfies
∑ ∏∑+==
+−∞=
≥≤≤=n
jiii
n
jn
nn
aiviii
niii
10
0
}1{)(,)(
,0,1,0)(,1)(
ααα
βαα
converges. It is called Jungck-Ishikawa iteration.
Notice that when =S identity map, it is called Ishikawa iteration.
18
These fixed point iterative procedures have tremendous applications in the problems of
solving nonlinear equations. An iterative scheme is said to be stable if small perturbations
during computations, produce small changes in approximate value of the fixed point
computed by means of these itearions. Formally, Harder and Hicks [142] defined the stability
of an iteration procedure as follows.
Definition 1.2.30 [142]. Let ),( dX be a complete metric space and : .T X X→ Let
Xx nn ⊂∞=0}{ be the sequence generated by an iteration procedure involving T which is
defined by ,...,1,0),,(1 ==+ nxTfx nn where Xx ∈0 is the initial approximation and f is
some function. Suppose ∞=0}{ nnx converges to a fixed point p of .T Let Xy nn ⊂∞
=0}{ and set
...,2,1,0)),,(,( 1 == + nyTfyd nnnε Then, the iteration procedure is said to be −T stable or
stable with respect to T if and only if 0lim =∞→ nnε implies .lim pynn
=∞→
Singh et al [147] first defined stability for Jungck type iterative procedures in the following
manner.
Definition 1.2.31 [147]. Let ,:, XYTS → )()( YSYT ⊆ and z be a coincidence point of T
and ,S that is, ,pTzSz == say. For any ,0 Yx ∈ let the sequence },{ nSx generated by the
iterative procedure ...,1,0),,(1 ==+ nxTfSx nn , converges to .p Let XSyn ⊂}{ be an arbitrary
sequence, and set ...,2,1,0)),,(,( 1 == + nyTfSyd nnnε . Then the iterative procedure
),( nxTf will be called −),( TS stable if and only if 0lim =∞→ nnε implies that .lim pSynn
=∞→
Timis and Berinde [155] defined weak stability using the concept of approximate sequence,
which is defined as follows.
Definition 1.2.32 [137]. Let ),( dX be a metric space and Xx nn ⊂∞=1}{ be a given sequence.
We shall say that Xy nn ∈∞=0}{ is an approximate sequence of }{ nx if, for any ,Nk ∈ there
exists )(kηη = such that
,),( η≤nn yxd for all .kn ≥
19
Definition 1.2.33 [137]. Let ),( dX be a metric space and : .T X X→ Let }{ nx be an iteration
procedure defined by Xx ∈0 and
0),,(1 ≥=+ nxTfx nn .
Suppose that }{ nx converges to a fixed point p of .T If for any approximate sequence
Xyn ⊂}{ of },{ nx 0)),(,(lim 1 =+∞→ nnnyTfyd implies ,lim pynn
=∞→
then we shall say that
iterative procedure is weakly −T stable or weakly stable with respect to .T
Further Timis [156] defined weak −2w stability using the following concept of equivalent
sequences
Definition 1.2.34 [156]. Two sequences { }∞=0nnx and { }∞=0nny are equivalent sequences if
0),( →nn yxd as .∞→n
Remark 1.2.1. Any equivalent sequence is an approximate sequence but the converse may
not be true.
The following example illustrates it.
Example 1.2.4. Let ∞=0}{ nnx be a sequence with .2nxn = First, we take an equivalent sequence
of ∞=0}{ nnx to be ∞
=0}{ nny where .12
nnyn += In this case, we have ,01),( →=
nxyd nn .∞→n
Now, take an approximate sequence of ∞=0}{ nnx to be ,}{ 0
∞=nny where .
122
++=
nnnyn Then,
.,021
12),( ∞→>→
+= n
nnxyd nn
Definition 1.2.35 [156]. Let ),( dX be a metric space and XXT →: be a map. Let }{ nx be an
iteration procedure defined by Xx ∈0 and
0),,(1 ≥=+ nxTfx nn .
20
Suppose that }{ nx converges to a fixed point p of .T If for any equivalent sequence }{ ny of
},{ nx 0)),(,(lim 1 =+∞→ nnnyTfyd implies ,lim pynn
=∞→
then we shall say that iterative scheme is
weak −2w stable with respect to .T
1.3 SOME GENERAL SPACES
In this section we define some general spaces used in the subsequent work reported in this
thesis.
1.3.1 b - METRIC SPACE
First we define a b- metric space introduced by Czerwik [246]. The basic concepts of
convergence, compactness, closedness and completeness are also defined in such space.
Definition 1.3.1.1 [246]. Let X be a non empty set and 1≥b be a given real number. A
function +ℜ→× XXd : is said to be a −b metric iff for all ,,, Xzyx ∈ the following
conditions are satisfied
(i) ,0),( yxiffyxd ==
(ii) ),,(),( xydyxd =
(iii) )].,(),([),( zydyxdbzxd +≤
The pair ),( dX is called a b-metric space.
As noted in [246], mathematical problems such as the problem of metrization of convergence
with respect to measure, lead to the generalization of metric, so called b-metric. The class of
b-metric spaces is effectively larger than that of metric spaces, since a b-metric space is a
metric space when s = 1 in the above condition (iii). The following example of Singh and
Prasad [152] shows that a b-metric on X need not be a metric on X (see also [246, p. 264]).
Example 1.3.1.1 [152]. Let },,,{ 4321 xxxxX = and ,2),( 21 ≥= kxxd
),(),( 4131 xxdxxd = ),(),( 4232 xxdxxd == ,1),( 43 == xxd ),(),( ijji xxdxxd = for all
4,3,2,1, =ji and .4,3,2,1,0),( == ixxd ii Then
21
[ ]),(),(2
),( jnniji xxdxxdkxxd +≤ for 4,3,2,1,, =jin
and if ,2>k the ordinary triangle inequality does not hold.
Definition 1.3.1.2 [214]. Let ),( dX be a −b metric space. Then a sequence Nnnx ∈}{ in X is
called
(a) convergent if and only if there exists Xx∈ such that 0),( →xxd n as .∞→n In this
case, we write ,lim xxnn=
∞→
(b) Cauchy if and only if 0),( →mn xxd as ., ∞→nm
Remark 1.3.1.1 [214].
In a b-metric space ),( dX the following assertions hold
(i) a convergent sequence has a unique limit,
(ii) each convergent sequence is Cauchy,
(iii) in general, a −b metric is not continuous.
Definition 1.3.1.3 [214]. Let ),( dX be a −b metric space. If Y is a nonempty subset of .X
Then the closure Y of Y is the set of limits of all convergent sequences of points in ,Y i.e.,
}.limthatsuch}{sequenceaexiststhere:{ xxxXxY nnNnn =∈=∞→∈
Definition 1.3.1.4 [214]. Let ),( dX be a −b metric space. Then a subset XY ⊂ is called:
(a) closed if and only if for each sequence Nnnx ∈}{ in Y which converges to an element ,x we
have Yx∈ ,
(b) compact if and only if for every sequence of elements of Y there exists a subsequence
that converges to an element of Y,
(c) bounded if and only if .},:),(sup{)( ∞<∈= YbabadYδ
Definition 1.3.1.5 [214]. The −b metric space ),( dX is complete iff every Cauchy sequence
in X converges.
22
In the following section we recall some definitions and preliminaries regarding fuzzy metric
spaces required for our results. We follow R. P. Pant [86], George and Veeramani [100],
Grabiec [107], Mishra et al [111], Schweizer and Sklar [120], Subramanyam [247], Vasuki
[248], V. Pant [249], Singh and Jain [250] and Pathak et al [251] for notations and
preliminaries.
1.3.2 FUZZY METRIC SPACE
Definition 1.3.2.1 [120]. A binary operation ]1,0[]1,0[]1,0[: →×T is a −t norm if T
satisfies following conditions for all ].1,0[,,, ∈dcba
(i) ),,(),( abba TT =
(ii) )),,(,()),,(( cbacba TT =
(iii) ),(),( dcba TT ≤ whenever ca ≤ and ,db ≤ for all ],1,0[,,, ∈dcba
(iv) ,)1,( aa =T for all ].1,0[∈a
Definition 1.3.2.2 [100]. The triplet ( , , )X M T is said to be fuzzy metric space, if X is an
arbitrary set, T is a continuous −t norm and M is a fuzzy set on ),0(2 ∞×X satisfying the
following conditions.
(i) ( , , ) 0,x y t >M
(ii) ( , , ) 1 ,x y t iff x y= =M
(iii) ( ( , , ), ( , , ) ( , , ), , 0, , , ,x y t y z s x z t s t s x y z X≤ + > ∀ ∈T M M M
(iv) ( , , .) : (0, ) [0, 1]x y ∞ →M is continuous.
Definition 1.3.2.3 [107]. Let ( , , )X M T be a fuzzy metric space. A sequence }{ nx in X is
said to be
(i) convergent to a point Xx∈ if 1),,(lim =∞→
txxnnM , for all ,0>t
(ii) a Cauchy sequence if 1),,(lim =+∞→txx npnn
M , for all ,0>t ,0>p
(iii) a complete fuzzy metric space in which every Cauchy sequence converges to a point in it.
23
The concept of commuting mappings in the setting of fuzzy metric space is given by
Subramanyam [247] as follows.
Definition 1.3.2.4 [247]. Mappings A and S of a fuzzy metric space ( , , )X TM into itself are
said to be commuting if ( , , ) 1ASx SAx t =M for all .Xx∈
After this several other weaker conditions of commutativity are defined in fuzzy metric space,
some of them are described as: weakly commuting mappings, compatible mappings, R-weakly
commuting maps, R-weakly commutativity of type )( gA , R-weakly commutativity of type
)( fA , etc., (see [111], [247]-[248]).
Definition 1.3.2.5 [86]. Mappings A and S of a fuzzy metric space ( , , )X TM into itself is
said to be weakly commuting if ( , , ) ( , , )ASx SAx t Ax Sx t≥M M for each Xx∈ and .0>t
In 1994, Mishra et al [111] introduced the concept of compatible mapping as follows:
Definition 1.3.2.6 [111]. Mappings A and S of a fuzzy metric space ( , , )X TM into itself is
said to be compatible if 1),,( =∞→
tSAxASx nn for all ,0>t whenever }{ nx is a sequence in
X such that uSxAx nnnn==
∞→∞→limlim
for some .Xu∈
Vasuki [248] extended the notion of R-weakly commuting maps introduced by Pant [86] in
metric space to fuzzy metric spaces.
Definition 1.3.2.7 [248]. Mappings A and S of a fuzzy metric space ( , , )X M T into itself are
−R weakly commuting provided there exists some positive real number R such that
( , , ) ( , , / )ASx SAx t Ax Sx t R≥M M for each Xx∈ and .0>t
Pathak et al [251] improved the notion of R-weakly commuting mappings in metric spaces to
the notions of R-weakly commutativity of type )( gA and type ).( fA V. Pant [249] extended it
to fuzzy metric space.
24
Definition 1.3.2.8 [249]. Mappings A and S of a fuzzy metric space ( , , )X M T into itself are
−R weakly commuting of type )( fA (or of type )( gA ) provided there exists some positive
real number R, for each Xx∈ and 0>t such that
( , , ) ( , , / )SSx ASx t Sx Ax t R≥M M (or ( , , ) ( , , / )AAx SAx t Ax Sx t R≥M M ).
Jungck and Rhoades [85] introduced the concept of weakly compatible maps which were
found to be more generalized than compatible maps. Weak compatibility in fuzzy metric
space is given by Singh and Jain [250].
Definition 1.3.2.9 [250]. Mappings A and S of a fuzzy metric space ( , , )X M T into itself are
said to be weakly compatible if they commute at the coincidence points, i.e., if Au = Su for
some ,Xu∈ then ASu = SAu.
Many authors defined fuzzy metric spaces in different ways (see for instance [99]-[106],
[252]-[253]). Using the idea of L -fuzzy set [112], Saadati et al [115] introduced the notion
of L -fuzzy metric spaces with the help of continuous t-norms as a generalization of fuzzy
metric space due to George and Veeramani [100] and intuitionistic fuzzy metric space due to
Saadati and Park [105]. The following section devoted to basic concepts related to L -fuzzy
metric spaces.
1.3.3 L -FUZZY METRIC SPACE
Definition 1.3.3.1 [115]. Let ( , )LL= ≤L be a complete lattice, and U be a nonempty set
called a universe. An L -fuzzy set on U is defined as a mapping .: LU → For each
u in ,U )(u represents the degree ( in L ) to which u satisfies .
Lemma 1.3.3.1 [254, 255]. Consider the set *L and the operation *L≤ defined by
},1and]1,0[),(:),{( 212
2121* ≤+∈= xxxxxxL
25
1121*21 ),(),( yxyyxxL
≤⇔≤ and ,22 yx ≥ for every .),(),,( *2121 Lyyxx ∈ Then ),( *
*
LL ≤
is a complete lattice.
Classically, a triangular norm on )],1,0([ ≤ is defined as an increasing, commutative,
associative mapping ]1,0[]1,0[: 2 → satisfying ,),1( xx = for all ].1,0[∈x These
definitions can be straightforwardly extended to any lattice ( , ).LL= ≤L Also 0 inf L=L and
1 sup L=L
Definition 1.3.3.2 [120]. A triangular norm ( −t norm) on L is a mapping 2: L L→T
satisfying the following conditions.
(i) ( )( ( , 1 ) );x L x x∀ ∈ =LT
(ii) 2( ( , ) )( ( , ) ( , ));x y L x y y x∀ ∈ =T T
(iii) 3( ( , , ) ) ( ( , ( , )) ( ( , ), );x y z L x y z x y z∀ ∈ =T T T T
(iv) 4( ( , , , ) ) ( ( , ) ( , )).L L Lx x y y L x x and y y x y x y′ ′ ′ ′ ′ ′∀ ∈ ≤ ≤ ⇒ ≤T T
A −t norm T on L is said to be continuous if for any ,x y∈L and any sequences }{ nx and
}{ ny which converge to x and y, we have
lim ( , ) ( , )n nnx y x y
→∞=T T .
Definition 1.3.3.3 [254]. A negation on L is any decreasing mapping : L L→N satisfying
(0 ) 1=L LN and (1 ) 0 .=L LN If ( ( )) ,x x=N N for all ,Lx∈ then N is called an involutive
negation.
Definition 1.3.3.4 [118]. The 3-triplet ( , , )X M T is said to be an −L fuzzy metric space, if
X is an arbitrary (non-empty) set, is a continuous t-norm on L and M is an −L fuzzy set
on ),0(2 ∞×X satisfying the following conditions for every zyx ,, in X and st, in ),0( ∞ .
(i) ( , , ) 0Lx y t >M L ,
(ii) ( , , ) 1 0,x y t for all t iff x y= > =M L ,
(iii) ( , , ) ( , , ),x y t y x t=M M
(iv) ( ( , , ), ( , , ) ( , , ),Lx y t y z s x z t s≤ +T M M M
26
(v) ( , , .) : (0, )x y L∞ →M is continuous and lim ( , , ) 1 .t
x y t→∞
=M L
In this case M is called an −L fuzzy metric. If ,M N=M M is an intuitionistic fuzzy set,
then the 3-tuple ,( , , )M NX M T is said to be an intuitionistic fuzzy metric space, defined by
Park [103] in 2004, using the concept of intuitionistic fuzzy set [116].
Definition 1.3.3.5 A sequence Nnnx ∈}{ in an −L fuzzy metric space ( , , )X M T is called a
Cauchy sequence, if for each \{0 }Lε ∈ L and ,0>t there exists Nn ∈0 such that for all
,0nnm ≥≥
( , , ) ( )m n Lx x t ε>M N .
The sequence Nnnx ∈}{ is said to be convergent to Xx∈ in −L fuzzy metric space
( , , ),X M T if ( , , ) ( , , ) 1 ,n nx x t x x t= → LM M whenever ,∞→n for every .0>t −L fuzzy metric
space is said to be complete if and only if every Cauchy sequence converges to a point in it.
1.4 THESIS OUTLINE
This thesis is organized as follows.
In Chapter 2, we obtain some basic approximate fixed point results in generalized
metric spaces. Further some existence results concerning approximate fixed points, endpoints
and approximate endpoints of multivalued contractions are also derived. Some quantitative
estimates of the sets of approximate fixed points and approximate endpoints for set-valued
contractions in generalized metric space are developed. These results extend some recent
results in the literature. When the mapping under consideration is a nonself mapping, the fixed
point equation in such cases may not have a solution. Best proximity pair theorems deal with
these problems and provide not only approximate solution but also an optimal approximate
solution. Some approximate best proximity pair theorems are also obtained with application to
Hammerstein integral equation.
The intent of Chapter 3 is to obtain some common fixed point theorems in general
settings. We study the common fixed points of hybrid pair of maps consisting of single and
27
multivalued maps satisfying an integral type contractive condition in b-metric spaces. An
application of the results to functional equation of dynamic programming problem is also
given. Further we define the notion of θ − −L fuzzy metric space and obtain some common
fixed point theorems for maps satisfying integral type conditions in such spaces.
In Chapter 4, we study the stability results of various iterative schemes of maps
satisfying some general condition. Weak and −2w weak stability results are also provided for
these iterative schemes. Our results extend and improve several recent results. A comparative
study of the different iterative schemes is also presented for some examples reported in the
literature.
A generalized version of the KKM theorems by using the concept of Chang and Zhang
[182] is studied in Chapter 5. Our results generalize some of the recent result of Chang and
Zhang [182] and Ansari et al [179]. As applications of the results to game theory, minimax
theorem and saddle point theorem for two-person-zero-sum game are established.
Consequently, a saddle point theorem for two person zero sum parametric game is also
proved.
In Chapter 6, we first define some contraction conditions of more general nature and
establish a generalized iterated function and multi function system theory using those
contraction conditions. Correspondingly some existence and uniqueness results are also
obtained. This theory extends several recent results and enhances the scope of IFS and IMS.
28
ONE SHOULD STUDY MATHEMATICS SIMPLY
BECAUSE IT HELPS TO ARRANGE ONE’S IDEAS
M. W. Lomonossow