1. Introduction

The presence or absence of a fixed point (i.e., a point which remains invariant under a map) is an intrinsic property of a map. However, many necessary or sufficient conditions for the existence of such points involve a mixture of algebraic, topological, or metric properties of the mapping or its domain. By Metric Fixed Point Theory, we understand the branch of Fixed Point Theory concerning those results which depend on a metric and which are not preserved when this metric is replaced by another equivalent metric. The first metric fixed point theorem was given by Banach in 1922.

Theorem 1.1 (Banach Contraction Principle, [1]). Let be a complete metric space and a contractive mapping, that is, there exists such that for every . Then has a (unique) fixed point . Moreover, for every .

Banach Theorem is a basic tool in Functional Analysis, Nonlinear Analysis and Differential Equations. Thus, it is natural to look for some generalizations under weaker assumptions.

For many years Metric Fixed Point Theory just studied some extensions of Banach Theorem relaxing the contractiveness condition, and the extension of this result for multivalued mappings. In the 1960s, Metric Fixed Point Theory received a strong boost when Kirk [2] proved that every (singlevalued) nonexpansive mapping , defined from a convex closed bounded subset of a reflexive Banach space with normal structure, has a fixed point.

The celebrated Kirk's theorem had a profound impact in the development of Fixed Point Theory and iniciated the search of more general conditions for a Banach space and for a subset which assure the existence of fixed points.

The result obtained by Kirk is, in some sense, surprising because it uses geometric properties of Banach spaces (commonly used in Linear Functional Analysis, but rarely considered in Nonlinear Analysis until then). Thus, it is the starting point for a new mathematical field: the application of the Geometric Theory of Banach Spaces to Fixed Point Theory. From that moment on, many researchers have tried to exploit this connection, essentially considering some other geometric properties of Banach spaces which can be applied to prove the existence of fixed points for different types of nonlinear operators (e.g., uniform smoothness, Opial property, nearly uniform convexity, nearly uniform smoothness, etc.).

Fixed Point Theory for multivalued mappings has useful applications in Applied Sciences, in particular, in Game Theory and Mathematical Economics. Thus, it is natural to study the problem of the extension of the known fixed point results for singlevalued mappings to the setting of multivalued mappings.

Some theorems of existence of fixed points of single-valued mappings have already been extended to the multivalued case. For example, in 1969 Nadler [3] extended the Banach Contraction Principle to multivalued contractive mappings in complete metric spaces. However, many other questions remain open, for instance, the possibility of extending the well-known Kirk's Theorem [2], that is, do Banach spaces with weak normal structure have the fixed point property (FPP) for multivalued nonexpansive mappings?

There are many properties of Banach spaces which imply weak normal structure and consequently the FPP for singlevalued mappings (e.g., uniform convexity, nearly uniform convexity, uniform smoothness, …). Thus, it is natural to consider the following problem: Do these properties also imply the FPP for multivalued mappings? As a consequence, some partial answers to the problem of extending Kirk's Theorem have appeared, which are directed to prove that those properties imply the existence of fixed point for multivalued nonexpansive mappings.

Here we present the main known results and current research directions in this subject. This paper can be considered as a survey, but some new results are also included.

2. Preliminaries

In this section we recall the notion of normal structure and some properties of Banach spaces which imply normal structure.

Normal structure plays an essential role in some problems of Metric Fixed Point Theory, especially those concerning nonexpansive mappings. The notion of normal structure was introduced by Brodskiĭ and Mil’man [4] in 1948 in order to study fixed points of isometries. Later, the notion of normal structure was generalized for the weak topology.

Definition 2.1. A Banach space is said to have normal structure (NS) (resp., weak normal structure (-NS)) if for every bounded closed (resp., weakly compact) convex subset of with there exists such that .

In 1965 Kirk [2] obtained a strong connection between normal structure and the FPP for nonexpansive mappings.

Theorem 2.2. Let be a bounded closed (resp., weakly compact) convex subset of a Banach space and let be a nonexpansive mapping (i.e., for every ). If is a reflexive Banach space with normal structure (resp., a Banach space with -NS), then has a fixed point.

Bynum [5] defined two coefficients related to normal structure and weak normal structure.

Definition 2.3. The normal structure coefficient of a Banach space is defined by where denotes the eter of defined by and denotes the Chebyshev radius of defined by .
The weakly convergent sequence coefficient of is defined by where the infimum is taken over all weakly convergent sequences which are not norm convergent, where, denote the asymptotic eter and radius of respectively.

We recall that is said to have uniform normal structure (UNS) (resp., weak uniform normal structure (-UNS)) if (resp., ). Notice that this is not the common definition of weak uniform normal structure and is often known as Bynum's condition. It is known that if has uniform normal structure, then is reflexive [6].

In the latest fifty years, some geometrical properties implying normal structure have been studied. Here we are going to recall some of these properties and some results which prove that these properties imply the existence of fixed point for multivalued mappings.

First we consider the Opial property. Opial [7] was the first who studied such a property giving applications to Fixed Point Theory. The uniform Opial property was defined in [8] by Prus, and the Opial modulus was introduced in [9] by Lin et al.

Definition 2.4. We will say that a Banach space satisfies the Opial property if for every weakly null sequence and every in we have We will say that satisfies the nonstrict Opial property if under the same conditions.
The Opial modulus of is defined for as where the infimum is taken over all with and all weakly null sequences in with .
We will say that satisfies the uniform Opial property if for all .

There are some relationships between the notions of Opial property and normal structure. If is a Banach space which satisfies the Opial property, then has -NS [10]. On the other hand, [9, Theorem ]. Consequently, has -UNS if .

Next we study the uniform convexity of the space, which is another geometrical property related with normal structure. We recall that a Banach space is uniformly convex (UC) if and only if

for each , or equivalently

The Clarkson modulus and the coefficient of normal structure are related by the following inequality: . Consequently, the condition implies that is reflexive and has uniform normal structure. In particular, notice that not only do uniformly convex spaces have normal structure, but so do all those spaces which do not have segments of length greater than or equal to near the unit sphere.

In 1980 Huff [11] initiated the study of nearly uniform convexity which is an infinite-dimensional generalization of uniform convexity. Independently of Huff, Goebel and Sękowski [12] also introduced a property which is equivalent to nearly uniform convexity under the name of noncompact uniform convexity. It is known that a Banach space is nearly uniformly convex (NUC) if and only if

for each , or equivalently

where is a measure of noncompactness. Also we are going to use the following equivalent definition: is NUC if and only if is reflexive and

for each , or equivalently

When is a reflexive Banach space, is the separation measure and is the Hausdorff measure (for definitions see, for instance, [13] or [14]), we have the following relationships among the different moduli:

and consequently,

If the space satisfies the nonstrict Opial property, then coincides with .

On the other hand, if (in particular, if is NUC), then is reflexive and has weak uniform normal structure (see [13, page 125]).

The dual concept of uniform convexity is uniform smoothness which is also related to normal structure. A Banach space is said to be uniformly smooth (US) if

where is the modulus of smoothness of , defined by

for .

It is known that implies that is reflexive and has uniform normal structure [15–17]. However, the infinite-dimensional generalization of uniform smoothness, nearly uniform smoothness, does not imply normal structure [13, Example ].

3. Some Properties Implying Weak Normal Structure and the FPP for Multivalued Mappings

In this section we are going to show some results which prove that some properties implying weak normal structure also imply the existence of fixed point for multivalued nonexpansive mappings. As a consequence these results give some partial answers to the problem of extending Kirk's Theorem.

Throughout this section (resp., ) will denote the family of all nonempty compact (resp., compact convex) subsets of . We recall that a multivalued mapping is said to be nonexpansive if for every , where denotes the Hausdorff metric given by

for every bounded subsets and of .

In 1973 Lami Dozo gave the following result of existence of fixed point for those spaces which satisfy the Opial property.

Theorem 3.1 (Lami Dozo [18, Theorem ]). Let be a Banach space which satisfies the Opial property, let be a weakly compact convex subset of and let be a nonexpansive mapping. Then has a fixed point, that is, there exists such that .

In 1974 Lim [19] gave a similar result for uniformly convex spaces using Edelstein's method of asymptotic centers [20].

Theorem 3.2 (Lim [19]). Let be a uniformly convex Banach space, let be a closed bounded convex subset of and be a nonexpansive mapping. Then has a fixed point.

In 1990 Kirk and Massa proved the following partial generalization of Lim's Theorem using asymptotic centers of sequences and nets. We recall that, given a bounded sequence in a Banach space and a subset of , the asymptotic center of with respect to is defined by

where denotes the asymptotic radius of with respect to defined by

Theorem 3.3 (Kirk and Massa [21]). Let be a closed bounded convex subset of a Banach space and a nonexpansive mapping. If the asymptotic center in of each bounded sequence of is nonempty and compact, then has a fixed point.

We do not know a complete characterization of those spaces in which asymptotic centers of bounded sequences are compact. Nevertheless, there are some partial answers, for example, -uniformly convex Banach spaces satisfy that condition [22]. However, an example given by Kuczumov and Prus [23] shows that in nearly uniformly convex spaces, the asymptotic center of a bounded sequence with respect to a closed bounded convex subset is not necessarily compact. Therefore, the problem of obtaining fixed point results in nearly uniformly convex spaces remained open. This question (together with the same question for uniformly smooth spaces) explicitly appeared in a survey about Metric Fixed Point Theory for multivalued mappings published by Xu [24] in 2000.

The analysis of the importance of the asymptotic center in Kirk-Massa Theorem led Domínguez Benavides and Lorenzo to study some connections between asymptotic centers and the geometry of certain spaces, including nearly uniformly convex spaces. Thus, in [25] Domínguez and Lorenzo obtained the following relationship between the Chebyshev radius of the asymptotic center of a bounded sequence and the modulus of noncompact convexity with respect to the measures and .

Theorem 3.4 (see [25, Theorem ]). Let be a closed convex subset of a reflexive Banach space and a bounded sequence in which is regular with respect to (i.e., the asymptotic radius is invariant for all subsequences of ). Then where the Chebyshev radius of a bounded subset of relative to is defined by Moreover, if satisfies the nonstrict Opial property, then

The previous inequalities give an iterative method which reduces at each step the value of the Chebyshev radius for a chain of asymptotic centers. Consequently, Domínguez and Lorenzo deduced in [26] the following partial extension of Kirk's Theorem which, in particular, assures that nearly uniformly convex spaces have the fixed point property for multivalued nonexpansive mappings.

Theorem 3.5 (see [26, Theorem ]). Let be a nonempty closed bounded convex subset of a Banach space such that . Let be a nonexpansive mapping. Then has a fixed point.

This result guarantees, in particular, the existence of fixed points in nearly uniformly convex spaces (because if is NUC), giving a positive answer to one of the previous open problems proposed by Xu.

Dhompongsa et al. [27] observed that the main tool used in the proofs in [25, 26], in order to obtain fixed point results for multivalued nonexpansive mappings, is a relationship between the Chebyshev radius of the asymptotic center of a bounded sequence and the asymptotic radius of the sequence. This relationship also gives an iterative method which reduces at each step the value of the Chebyshev radius for a chain of asymptotic centers. Consequently, in [27, 28] they introduced the Domínguez-Lorenzo condition ((DL)-condition, in short) and property (D) in the following way.

We recall that a sequence is regular with respect to if for all subsequences of , and is asymptotically uniform with respect to if for all subsequences of .

Definition 3.6. A Banach space is said to satisfy the (DL)-condition if there exists such that for every weakly compact convex subset of and for every bounded sequence in which is regular with respect to
A Banach space is said to satisfy property (D) if there exists such that for any nonempty weakly compact convex subset of , any bounded sequence in which is regular and asymptotically uniform with respect to , and any sequence which is regular and asymptotically uniform with respect to we have

From the definition it is easy to deduce that property (D) is weaker than the (DL)-condition. Dhompongsa et al. proved in [28, Theorem ] and [28, Theorem ] that property (D) implies -NS and the FPP for multivalued nonexpansive mappings.

Theorem 3.7 (see [28, Theorem ]). Let be a Banach space satisfying property (D). Then has -NS.

Theorem 3.8 (see [28, Theorem ]). Let be a nonempty weakly compact convex subset of a Banach space which satisfies property (D). Let be a nonexpansive mapping. Then has a fixed point.

From Theorem 3.5 every Banach space with satisfies the (DL)-condition. In this paper we present some other properties concerning geometrical constants of Banach spaces which also imply the (DL)-condition or property (D).

Since our goal is to study if properties implying -NS also imply the FPP for multivalued mappings, a possible approach to that problem is to study if these properties imply either the (DL)-condition or property (D). These results will give only partial answers to the problem of extending Kirk's Theorem for multivalued mappings because we know that uniform normal structure does not imply property (D) ([29, Proposition ]); therefore, the problem of extending Kirk's Theorem cannot be fully solved by this approach. In this setting the following results have been obtained.

Theorem 3.9 (Dhompongsa et al. [27, Theorem ]). Let be a uniformly nonsquare Banach space with property WORTH. Then satisfies the (DL)-condition.
We recall that a Banach space is uniformly nonsquare if there exists such that for every or equivalently where denotes the James constant of defined by is said to satisfy property WORTH if for any and any weakly null sequence in .

Theorem 3.10 (Dhompongsa et al. [28, Theorem ]). Let be Banach space such that where denotes the Jordan-von Neumann constant of defined by Then satisfies property (D).

Theorem 3.11 (Domínguez Benavides and Gavira [29, Corollary ]). Let be a Banach space such that Then satisfies the (DL)-condition. In particular, uniformly smooth Banach spaces () satisfy the (DL)-condition.

Theorem 3.12 (Domínguez Benavides and Gavira [29, Corollary ]). Let be a Banach space such that one of the following two equivalent conditions is satisfied (1)(2). Then satisfies the (DL)-condition.

Theorem 3.13 (Saejung [30, Theorem ]). A Banach space has property (D) whenever .

This result improves Theorem 3.10 because it is easy to see that .

Theorem 3.14 (Kaewkhao [31, Corollary ]). Let be a Banach space such that where denotes the James constant of defined by and denotes the coefficient of worthwhileness of defined as the infimum of the set of real numbers such that for all and all weakly null sequences in . Then satisfies the (DL)-condition.

Remark 3.15. This result improves Theorem 3.9 because if is a uniformly nonsquare Banach space with property WORTH, then

Theorem 3.16 (Kaewkhao [31, Theorem ]). Let be a Banach space such that Then satisfies the (DL)-condition.

Theorem 3.17 (Gavira [32, Theorem ]). Let be a Banach space such that Then satisfies the (DL)-condition.

Remark 3.18 s. (i) This result is a strict generalization of Theorem 3.16 (see [32]).
(ii) Theorem 3.17 applies to the Bynum space while Theorem 3.11 does not (see [32]). However, we do not know if implies .

Finally we show a new result which gives a property implying the (DL)-condition in terms of Clarkson modulus and the García-Falset coefficient.

Theorem 3.19. Let be a Banach space such that where denotes the García-Falset coefficient of defined by Then satisfies the (DL)-condition.

Proof. Let be a nonempty weakly compact convex subset of . Let be a bounded sequence in which is regular with respect to . Denote , and . By translation and multiplication we can assume that is weakly null and . Let , then . Denote by . By the definition of we have For every there exists such that (1)(2)(3)(4)
Consider and . Using the above estimates we obtain where tends to 0 as . Furthermore,
Also we have
Define and . Thus, .
Since for , we obtain
Consequently, we have
Since the last inequality is true for every and every , letting and using the continuity of , we obtain

In [33] it is proved that has normal structure under the slightly weaker condition

It is an open question if this condition implies the (DL)-condition.

Corollary 3.20. Let be a uniformly nonsquare Banach space such that . Then satisfies the (DL)-condition.

4. Fixed Point Results for Multivalued Nonexpansive Mappings in Modular Function Spaces

The theory of modular spaces was initiated by Nakano [34] in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz [35] in 1959. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated and solved in modular spaces (see, for instance, [36–39]). In particular, Dhompongsa et al. [40] have obtained some fixed point results for multivalued mappings in modular functions spaces.

Let us recall some basic concepts about modular function spaces (for more details the reader is referred to [41, 42]).

Let be a nonempty set and a nontrivial -algebra of subsets of . Let be a -ring of subsets of , such that for any and . Let us assume that there exists an increasing sequence of sets such that (for instance, can be the class of sets of finite measure in a -finite measure space). By we denote the linear space of all simple functions with supports from . By we will denote the space of all measurable functions, that is, all functions such that there exist a sequence , and for all .

Let us recall that a set function is called a -subadditive measure if , for any and for any sequence of sets . By we denote the characteristic function of the set .

Definition 4.1. A functional is called a function modular if: (1) for any ;(2) whenever for any , and (3) is a -subadditive measure for every (4) as decreases to for every , where (5)if there exists such that , then for every (6)for any ,   is order continuous on , that is, if and decreases to .

A -subadditive measure is said to be additive if whenever such that and .

The definition of is then extended to by

Definition 4.2. A set is said to be -null if for every . A property is said to hold -almost everywhere (-a.e.) if the set is -null. For example, we will say frequently -a.e.

Note that a countable union of -null sets is still -null. In the sequel we will identify sets and whose symmetric difference is -null, similarly we will identify measurable functions which differ only on a -null set.

Under the above conditions, we define the function by . We know from [41] that satisfies the following properties: