Does Kirk's Theorem Hold for Multivalued Nonexpansive Mappings?

Abstract

Fixed Point Theory for multivalued mappings has many useful applications in Applied Sciences, in particular, in Game Theory and Mathematical Economics. Thus, it is natural to try of extending the known fixed point results for single-valued mappings to the setting of multivalued mappings. Some theorems of existence of fixed points of single-valued mappings have already been extended to the multivalued case. However, many other questions remain still open, for instance, the possibility of extending the well-known Kirk's Theorem, 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 single-valued mappings (for example, 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? In this way, some partial answers to the problem of extending Kirk's Theorem have appeared, proving 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 shown.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Mohamed A. Khamsi

Facultad de Matemáticas, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain

Received 25 September 2009; Accepted 29 December 2009

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 X be a complete metric space and T:X[arrow right]X a contractive mapping, that is, there exists k∈[0,1) such that d(Tx,Ty)≤kd(x,y) for every x,y∈X . Then T has a (unique) fixed point _x0 . Moreover, _x0 =_{lim n}^Tn x for every x∈X .

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 T:C[arrow right]C , defined from a convex closed bounded subset C 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 C 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 Brodski[ibreve] 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 X is said to have normal structure (NS) (resp., weak normal structure (w -NS)) if for every bounded closed (resp., weakly compact) convex subset C of X with diam (C):=sup {||x-y||:x,y∈C}>0, there exists x∈C such that sup {||x-y||:y∈C}<diam(C) .

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

Theorem 2.2.

Let C be a bounded closed (resp., weakly compact) convex subset of a Banach space X and let T:C[arrow right]C be a nonexpansive mapping (i.e., ||Tx-Ty||≤||x-y|| for every x,y∈C ). If X is a reflexive Banach space with normal structure (resp., a Banach space with w -NS), then T 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 X is defined by [figure omitted; refer to PDF] where diam (A) denotes the diam eter of A defined by diam (A)=sup {||x-y||:x,y∈A} and r(A) denotes the Chebyshev radius of A defined by r(A)=inf {sup {||x-y||:y∈A}:x∈A} .

The weakly convergent sequence coefficient of X is defined by [figure omitted; refer to PDF] where the infimum is taken over all weakly convergent sequences {_xn } which are not norm convergent, where, [figure omitted; refer to PDF] denote the asymptotic diam eter and radius of {_xn }, respectively.

We recall that X is said to have uniform normal structure (UNS) (resp., weak uniform normal structure (w -UNS)) if N(X)>1 (resp., WCS(X)>1 ). 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 X has uniform normal structure, then X 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 X satisfies the Opial property if for every weakly null sequence {_xn } and every x≠0 in X, we have [figure omitted; refer to PDF] We will say that X satisfies the nonstrict Opial property if [figure omitted; refer to PDF] under the same conditions.

The Opial modulus of X is defined for c≥0 as [figure omitted; refer to PDF] where the infimum is taken over all x∈X with ||x||≥c and all weakly null sequences {_xn } in X with _{lim inf n} ||_xn ||≥1 .

We will say that X satisfies the uniform Opial property if _rX (c)>0 for all c>0 .

There are some relationships between the notions of Opial property and normal structure. If X is a Banach space which satisfies the Opial property, then X has w -NS [10]. On the other hand, WCS(X)≥1+_rX (1) [9, Theorem 3.2 ]. Consequently, X has w -UNS if _rX (1)>0 .

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

[figure omitted; refer to PDF] for each ...∈[0,2] , or equivalently

[figure omitted; refer to PDF]

The Clarkson modulus _δX (...) and the coefficient of normal structure N(X) are related by the following inequality: N(X)≥^{(1-_δX (1))-1} . Consequently, the condition _δX (1)>0 implies that X 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 1 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 Sekowski [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 X is nearly uniformly convex (NUC) if and only if

[figure omitted; refer to PDF] for each ...>0 , or equivalently

[figure omitted; refer to PDF] where [varphi] is a measure of noncompactness. Also we are going to use the following equivalent definition: X is NUC if and only if X is reflexive and

[figure omitted; refer to PDF] for each ...>0 , or equivalently

[figure omitted; refer to PDF]

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

[figure omitted; refer to PDF] and consequently,

[figure omitted; refer to PDF] If the space X satisfies the nonstrict Opial property, then _Δ0 (X) coincides with _{[straight epsilon]χ} (X) .

On the other hand, if _{[straight epsilon]β} (X)<1 (in particular, if X is NUC), then X is reflexive and has weak uniform normal structure (see [13, page 125]).

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

[figure omitted; refer to PDF] where _ρX is the modulus of smoothness of X , defined by

[figure omitted; refer to PDF] for t≥0 .

It is known that ^{ρX[variant prime]} (0)<1/2 implies that X is reflexive and has uniform normal structure [15-17]. However, the infinite-dimensional generalization of uniform smoothness, nearly uniform smoothness, does not imply normal structure [13, Example VI.2 ].

3. Some Properties Implying Weak Normal Structure 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 K(X) (resp., KC(X) ) will denote the family of all nonempty compact (resp., compact convex) subsets of X . We recall that a multivalued mapping T:X[arrow right]K(X) is said to be nonexpansive if H(Tx,Ty)≤||x-y|| for every x,y∈X , where H(·,·) denotes the Hausdorff metric given by

[figure omitted; refer to PDF] for every bounded subsets A and B of X .

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

Theorem 3.1 (Lami Dozo [18, Theorem 3.2 ]).

Let X be a Banach space which satisfies the Opial property, let C be a weakly compact convex subset of X, and let T:C[arrow right]K(C) be a nonexpansive mapping. Then T has a fixed point, that is, there exists x∈C such that x∈Tx .

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

Theorem 3.2 (Lim [19]).

Let X be a uniformly convex Banach space, let C be a closed bounded convex subset of X and T:C[arrow right]K(C) be a nonexpansive mapping. Then T has a fixed point.

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

[figure omitted; refer to PDF] where r(C,{_xn }) denotes the asymptotic radius of {_xn } with respect to C defined by

[figure omitted; refer to PDF]

Theorem 3.3 (Kirk and Massa [21]).

Let C be a closed bounded convex subset of a Banach space X and T:C[arrow right]KC(C) a nonexpansive mapping. If the asymptotic center in C of each bounded sequence of X is nonempty and compact, then T 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, k -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 β and χ .

Theorem 3.4 (see [25, Theorem 3.4 ]).

Let C be a closed convex subset of a reflexive Banach space X and {_xn } a bounded sequence in C which is regular with respect to C (i.e., the asymptotic radius is invariant for all subsequences of {_xn } ). Then [figure omitted; refer to PDF] where the Chebyshev radius of a bounded subset D of X relative to C is defined by [figure omitted; refer to PDF] Moreover, if X satisfies the nonstrict Opial property, then [figure omitted; refer to PDF]

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 3.5 ]).

Let C be a nonempty closed bounded convex subset of a Banach space X such that _{[straight epsilon]β} (X)<1 . Let T:C[arrow right]KC(C) be a nonexpansive mapping. Then T has a fixed point.

This result guarantees, in particular, the existence of fixed points in nearly uniformly convex spaces (because _{[straight epsilon]β} (X)=0 if X 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 {_xn } is regular with respect to C if r(C,{_xn })=r(C,{_xni }) for all subsequences {_xni } of {_xn } , and {_xn } is asymptotically uniform with respect to C if A(C,{_xn })=A(C,{_xni }) for all subsequences {_xni } of {_xn } .

Definition 3.6.

A Banach space X is said to satisfy the (DL)-condition if there exists λ∈[0,1) such that for every weakly compact convex subset C of X and for every bounded sequence {_xn } in C which is regular with respect to C [figure omitted; refer to PDF]

A Banach space X is said to satisfy property (D) if there exists λ∈[0,1) such that for any nonempty weakly compact convex subset C of X , any bounded sequence {_xn } in C which is regular and asymptotically uniform with respect to C , and any sequence {_yn }⊂A(C,{_xn }) which is regular and asymptotically uniform with respect to C, we have [figure omitted; refer to PDF]

From the definition it is easy to deduce that property (D) is weaker than the (DL)-condition. Dhompongsa et al. proved in [28, Theorem 3.2 ] and [28, Theorem 3.5 ] that property (D) implies w -NS and the FPP for multivalued nonexpansive mappings.

Theorem 3.7 (see [28, Theorem 3.3 ]).

Let X be a Banach space satisfying property (D). Then X has w -NS.

Theorem 3.8 (see [28, Theorem 3.6 ]).

Let C be a nonempty weakly compact convex subset of a Banach space X which satisfies property (D). Let T:C[arrow right]KC(C) be a nonexpansive mapping. Then T has a fixed point.

From Theorem 3.5 every Banach space with _{[straight epsilon]β} (X)<1 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 w -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 5 ]); 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 3.4 ]).

Let X be a uniformly nonsquare Banach space with property WORTH. Then X satisfies the (DL)-condition.

We recall that a Banach space X is uniformly nonsquare if there exists δ>0 such that ||x+y||...||x-y||≤2-δ for every x,y∈_BX or equivalently J(X)<2, where J(X) denotes the James constant of X defined by [figure omitted; refer to PDF] X is said to satisfy property WORTH if [figure omitted; refer to PDF] for any x∈X and any weakly null sequence {_xn } in X .

Theorem 3.10 (Dhompongsa et al. [28, Theorem 3.7 ]).

Let X be Banach space such that [figure omitted; refer to PDF] where _CNJ (X) denotes the Jordan-von Neumann constant of X defined by [figure omitted; refer to PDF] Then X satisfies property (D).

Theorem 3.11 (Domínguez Benavides and Gavira [29, Corollary 1 ]).

Let X be a Banach space such that [figure omitted; refer to PDF] Then X satisfies the (DL)-condition. In particular, uniformly smooth Banach spaces (^{ρX[variant prime]} (0)=0 ) satisfy the (DL)-condition.

Theorem 3.12 (Domínguez Benavides and Gavira [29, Corollary 2 ]).

Let X be a Banach space such that one of the following two equivalent conditions is satisfied

(1) _rX (1)>0,

(2) _Δ0 (X)<1 .

Then X satisfies the (DL)-condition.

Theorem 3.13 (Saejung [30, Theorem 3 ]).

A Banach space X has property (D) whenever _{[straight epsilon]0} (X)<WCS(X) .

This result improves Theorem 3.10 because it is easy to see that _CNJ (X)≥1+(1/4)^{(_{[straight epsilon]0} (X))2} .

Theorem 3.14 (Kaewkhao [31, Corollary 3.2 ]).

Let X be a Banach space such that [figure omitted; refer to PDF] where J(X) denotes the James constant of X defined by [figure omitted; refer to PDF] and μ(X) denotes the coefficient of worthwhileness of X defined as the infimum of the set of real numbers r>0 such that [figure omitted; refer to PDF] for all x∈X and all weakly null sequences {_xn } in X . Then X satisfies the (DL)-condition.

Remark 3.15.

This result improves Theorem 3.9 because if X is a uniformly nonsquare Banach space with property WORTH, then [figure omitted; refer to PDF]

Theorem 3.16 (Kaewkhao [31, Theorem 4.1 ]).

Let X be a Banach space such that [figure omitted; refer to PDF] Then X satisfies the (DL)-condition.

Theorem 3.17 (Gavira [32, Theorem 4 ]).

Let X be a Banach space such that [figure omitted; refer to PDF] Then X 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 _{[cursive l]2,1} while Theorem 3.11 does not (see [32]). However, we do not know if ^{ρX[variant prime]} (0)<1/2 implies ^{ρX[variant prime]} (0)<1/μ(X) .

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 X be a Banach space such that [figure omitted; refer to PDF] where R(X) denotes the García-Falset coefficient of X defined by [figure omitted; refer to PDF] Then X satisfies the (DL)-condition.

Proof.

Let C be a nonempty weakly compact convex subset of X . Let {_xn } be a bounded sequence in C which is regular with respect to C . Denote A=A(C,{_xn }) , r=r(C,{_xn }), and R=R(X) . By translation and multiplication we can assume that {_xn } is weakly null and _{lim n} ||_xn ||=1 . Let z∈A , then _{lim sup n} ||_xn -z||=r≤1 . Denote ||z|| by a . By the definition of R, we have [figure omitted; refer to PDF] For every [straight epsilon]>0, there exists N∈... such that

(1) ||_xN -z||<r+...,

(2) ||_xN +z/a||<R+...,

(3) ||(1/(r+...)-1/(R+...))_xN -(1/(r+...)+1/a(R+...))z||>(1/(r+...)+1/a(R+...) )a((r-...)/r),

(4) ||_xN -(1/(r+...)-1/a(R+...))/(1/(r+...)+1/(R+...))z||>r-....

Consider u=(1/(r+...))(_xN -z)∈_BX and v=(1/(R+...))(_xN +z/a)∈_BX . Using the above estimates we obtain [figure omitted; refer to PDF] where o(...) tends to 0 as ...[arrow right]⁰⁺ . Furthermore, [figure omitted; refer to PDF]

Also we have [figure omitted; refer to PDF]

Define f(r)=2/R+1/r-1 and g(r)=1+r/R . Thus, ||u+v||≥max {f(r),g(r)}-o(...) .

Since f(r)=g(r) for r=_r0 =1-R+^R2 +1-R , we obtain [figure omitted; refer to PDF]

Consequently, we have [figure omitted; refer to PDF]

Since the last inequality is true for every ...>0 and every z∈A , letting ...[arrow right]0 and using the continuity of δ(·) , we obtain [figure omitted; refer to PDF]

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

[figure omitted; refer to PDF] It is an open question if this condition implies the (DL)-condition.

Corollary 3.20.

Let X be a uniformly nonsquare Banach space such that R(X)=1 . Then X 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 Ω be a nonempty set and Σ a nontrivial σ -algebra of subsets of Ω . Let P be a δ -ring of subsets of Ω , such that E∩A∈P for any E∈P and A∈Σ . Let us assume that there exists an increasing sequence of sets _Kn ∈P such that Ω=_...Kn (for instance, P can be the class of sets of finite measure in a σ -finite measure space). By ... we denote the linear space of all simple functions with supports from P . By [physics M-matrix] we will denote the space of all measurable functions, that is, all functions f:Ω[arrow right]... such that there exist a sequence {_gn }∈... , |_gn |≤|f| and _gn (ω)[arrow right]f(ω) for all ω∈Ω .

Let us recall that a set function μ:Σ[arrow right][0,∞] is called a σ -subadditive measure if μ(∅)=0 , μ(A)≤μ(B) for any A⊂B and μ(_...An )≤∑μ(_An ) for any sequence of sets {_An }⊂Σ . By _1A , we denote the characteristic function of the set A .

Definition 4.1.

A functional ρ:...×Σ[arrow right][0,∞] is called a function modular if:

(1) ρ(0,E)=0 for any E∈Σ ;

(2) ρ(f,E)≤ρ(g,E) whenever |f(ω)|≤|g(ω)| for any ω∈Ω , f,g∈... and E∈Σ;

(3) ρ(f,·):Σ[arrow right][0,∞] is a σ -subadditive measure for every f∈...;

(4) ρ(α,A)[arrow right]0 as α decreases to 0 for every A∈P , where ρ(α,A)=ρ(α_1A ,A);

(5) if there exists α>0 such that ρ(α,A)=0 , then ρ(β,A)=0 for every β>0;

(6) for any α>0 , ρ(α,·) is order continuous on P , that is, ρ(α,_An )[arrow right]0 if {_An }⊂P and decreases to ∅ .

A σ -subadditive measure ρ is said to be additive if ρ(f,A∪B)=ρ(f,A)+ρ(f,B), whenever A,B∈Σ such that A∩B=∅ and f∈[physics M-matrix] .

The definition of ρ is then extended to f∈[physics M-matrix] by

[figure omitted; refer to PDF]

Definition 4.2.

A set E is said to be ρ -null if ρ(α,E)=0 for every α>0 . A property p(ω) is said to hold ρ -almost everywhere (ρ -a.e.) if the set {ω∈Ω:p(ω) does not hold} is ρ -null. For example, we will say frequently _fn [arrow right]f ρ -a.e.

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

Under the above conditions, we define the function ρ:[physics M-matrix][arrow right][0,∞] by ρ(f)=ρ(f,Ω) . We know from [41] that ρ satisfies the following properties:

(i) ρ(f)=0 if and only if f=0 ρ -a.e.

(ii) ρ(αf)=ρ(f) for every scalar α with |α|=1 and f∈[physics M-matrix] .

(iii): ρ(αf+βg)≤ρ(f)+ρ(g) if α+β=1 , α,β≥0 and f,g∈[physics M-matrix] .

: In addition, if the following property is satisfied

(iii)^{[variant prime]} : ρ(αf+βg)≤αρ(f)+βρ(g) if α+β=1 , α,β≥0 and f,g∈[physics M-matrix] ,

we say that ρ is a convex modular.

A function modular ρ is called σ -finite if there exists an increasing sequence of sets _Kn ∈P such that 0<ρ(_1Kn )<∞ and Ω=_...Kn .

The modular ρ defines a corresponding modular space _Lρ , which is given by

[figure omitted; refer to PDF]

In general the modular ρ is not subadditive and therefore does not behave as a norm or a distance. But one can associate to a modular an F -norm. In fact, when ρ is convex, the formula

[figure omitted; refer to PDF] defines a norm which is frequently called the Luxemburg norm. The formula

[figure omitted; refer to PDF] defines a different norm which is called the Amemiya norm. Moreover, _||·||l and _||·||a are equivalent norms. We can also consider the space

[figure omitted; refer to PDF]

Definition 4.3.

A function modular ρ is said to satisfy the _Δ2 -condition if [figure omitted; refer to PDF]

It is known that the _Δ2 -condition is equivalent to _Eρ =_Lρ .

Definition 4.4.

A function modular ρ is said to satisfy the _Δ2 -type condition if there exists K>0 such that for any f∈_Lρ we have ρ(2f)≤Kρ(f).

In general, the _Δ2 -type condition and _Δ2 -condition are not equivalent, even though it is obvious that the _Δ2 -type condition implies the _Δ2 -condition.

Definition 4.5.

Let _Lρ be a modular space.

(1) The sequence {_fn }⊂_Lρ is said to be ρ -convergent to f∈_Lρ if ρ(_fn -f)[arrow right]0 as n[arrow right]∞ .

(2) The sequence {_fn }⊂_Lρ is said to be ρ -a.e. convergent to f∈_Lρ if the set {ω∈Ω:_fn (ω)...f(ω)} is ρ -null.

(3) A subset C of _Lρ is called ρ -a.e. closed if the ρ -a.e. limit of a ρ -a.e. convergent sequence of C always belongs to C .

(4) A subset C of _Lρ is called ρ -a.e. compact if every sequence in C has a ρ -a.e. convergent subsequence in C .

(5) A subset C of _Lρ is called ρ -bounded if [figure omitted; refer to PDF]

We know by [41] that under the _Δ2 -condition the norm convergence and modular convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with the _Δ2 -type condition. In the sequel we will assume that the modular function ρ is convex and satisfies the _Δ2 -type condition. Hence, the ρ -convergence defines a topology which is identical to the norm topology.

In the same way as the Hausdorff distance defined on the family of bounded closed subsets of a metric space, we can define the analogue to the Hausdorff distance for modular function spaces. We will speak of ρ -Hausdorff distance even though it is not a metric.

Definition 4.6.

Let C be a nonempty subset of _Lρ . We will denote by _Fρ (C) the family of nonempty ρ -closed subsets of C and by _Kρ (C) the family of nonempty ρ -compact subsets of C . Let _Hρ (·,·) be the ρ -Hausdorff distance on _Fρ (_Lρ ) , that is, [figure omitted; refer to PDF] where dis_tρ (f,B)=inf {ρ(f-g):g∈B} is the ρ -distance between f and B . A multivalued mapping T:C[arrow right]_Fρ (_Lρ ) is said to be a ρ -contraction if there exists a constant k∈[0,1) such that [figure omitted; refer to PDF] If it is valid when k=1 , then T is called ρ -nonexpansive.

A function f∈C is called a fixed point for a multivalued mapping T if f∈TF .

Dhompongsa et al. [40] stated the Banach Contraction Principle for multivalued mappings in modular function spaces.

Theorem 4.7 (see [40, Theorem 3.1 ]).

Let ρ be a convex function modular satisfying the _Δ2 -type condition, C a nonempty ρ -bounded ρ -closed subset of _Lρ , and T:C[arrow right]_Fρ (C) a ρ -contraction mapping, that is, there exists a constant k∈[0,1) such that [figure omitted; refer to PDF] Then T has a fixed point.

By using that result, they proved the existence of fixed points for multivalued ρ -nonexpansive mappings.

Theorem 4.8 (see [40, Theorem 3.4 ]).

Let ρ be a convex function modular satisfying the _Δ2 -type condition, C a nonempty ρ -a.e. compact ρ -bounded convex subset of _Lρ , and T:C[arrow right]_Kρ (C) a ρ -nonexpansive mapping. Then T has a fixed point.

They also applied the above theorem to obtain fixed point results in the Banach space _L1 (resp., _{[cursive l]1} ) for multivalued mappings whose domains are compact in the topology of the convergence locally in measure (resp., ^w* -topology).

Consider the space _Lp (Ω,μ) for a σ -finite measure μ with the usual norm. Let C be a bounded closed convex subset of _Lp for 1<p<∞ and T:C[arrow right]K(C) a multivalued nonexpansive mapping. Because of uniform convexity of _Lp , it is known that T has a fixed point. For p=1 , T can fail to have a fixed point even in the singlevalued case for a weakly compact convex set C (see [43]). However, since _L1 is a modular space where ρ(f)=_∫Ω |f|dμ=||f|| for all f∈_L1 , Theorem 4.8 implies the existence of a fixed point when we define mappings on a ρ -a.e. compact ρ -bounded convex subset of _L1 . Thus the following can be stated.

Corollary 4.9 (see [40, Corollary 3.5 ]).

Let (Ω,μ) be as above, C⊂_L1 (Ω,μ) a nonempty bounded convex set which is compact for the topology of the convergence locally in measure, and T:C[arrow right]K(C) a nonexpansive mapping. Then T has a fixed point.

In the case of the space _{[cursive l]1} , we also can obtain a bounded closed convex set C and a nonexpansive mapping T:C[arrow right]C which is fixed point free. Indeed, consider the following easy and well-known example.

Let

[figure omitted; refer to PDF] Define a nonexpansive mapping T:C[arrow right]C by

[figure omitted; refer to PDF] then T is a fixed point free map. However, if we consider _Lρ =_{[cursive l]1} , where ρ(x)=||x|| , for all x∈_{[cursive l]1} , then ρ -a.e. convergence and ^ω* -convergence are identical on bounded subsets of _{[cursive l]1} (see [36]). This fact leads to the following corollary.

Corollary 4.10 (see [40, Corollary 3.6 ]).

Let C be a nonempty ^ω* -compact convex subset of _{[cursive l]1} and T:C[arrow right]K(C) a nonexpansive mapping. Then T has a fixed point.

Next we will give a property of closed convex bounded subsets of _{[cursive l]1} more general than weak star compactness which implies the fixed point property for nonexpansive mappings.

Domínguez et al. introduced in [44] some compactness conditions concerning proximinal subsets called Property (P). Following this idea we will use the following similar notion for modular function spaces.

Definition 4.11.

Let C be a nonempty ρ -closed convex ρ -bounded subset of _Lρ . It is said that C has Property (_Pρ ) if for every f∈_Lρ , which is the ρ -a.e. limit of a sequence in C , the set _Pρ,C (f) is a nonempty and ρ -compact subset of C , where _Pρ,C (f)={g∈C:ρ(g-f)=dis_tρ (f,C)}.

Using that notion and the following two lemmas, we obtain a new fixed point result for multivalued ρ -nonexpansive mappings.

Lemma 4.12 (see [40, Lemma 3.3 ]).

Let ρ be a convex function modular satisfying the _Δ2 -type condition, f∈_Lρ , and K a nonempty ρ -compact subset of _Lρ . Then there exists _g0 ∈K such that [figure omitted; refer to PDF]

Lemma 4.13 (see [37, Lemma 1.3 ]).

Let ρ be a function modular satisfying the _Δ2 -type condition, and {_fn}n be a sequence in _Lρ such that _fn [arrow right]ρ-a.e.f∈_Lρ and there exists k>1 such that _{sup n} ρ(k(_fn -f))<∞ . Then, [figure omitted; refer to PDF]

Theorem 4.14.

Let ρ be a convex function modular satisfying the _Δ2 -type condition, C a nonempty ρ -closed ρ -bounded convex subset of _Lρ satisfying Property (_Pρ ) such that every sequence in C has a ρ -a.e. convergent subsequence in _Lρ , and T:C[arrow right]_Kρ C(C) a ρ -nonexpansive mapping. Then T has a fixed point.

Proof.

Fix _f0 ∈C . For each n∈... , the ρ -contraction _Tn :C[arrow right]_Fρ (C) is defined by [figure omitted; refer to PDF] By Theorem 4.7, we can conclude that _Tn has a fixed point, say _fn . It is easy to see that [figure omitted; refer to PDF] By our assumptions, we can assume, by passing through a subsequence, that _fn [arrow right]ρ-a.e.f for some f∈_Lρ . By Lemma 4.12, for each n∈... there exists _gn ∈T_fn such that [figure omitted; refer to PDF] Now we are going to show that _Pρ,C (f)∩Th≠∅ for each h∈_Pρ,C (f) . Taking any h∈_Pρ,C (f) , from the ρ -compactness of Th and Lemma 4.12, we can find _hn ∈Th such that [figure omitted; refer to PDF] and we can assume, by passing through a subsequence, that _hn [arrow right]ρ_h0 for some _h0 ∈Th . From above and using Lemma 4.13, it follows that [figure omitted; refer to PDF] On the other hand, by Lemma 4.13 we also have [figure omitted; refer to PDF] Thus, we deduce ρ(f-_h0 )≤ρ(f-h) , which implies that _h0 ∈_Pρ,C (f) and so _Pρ,C (f)∩Th≠∅ .

Now we define the mapping T...:_Pρ,C (f)[arrow right]KC(_Pρ,C (f)) by T...(h)=_Pρ,C (f)∩Th . From [45, Proposition 2.45 ] we know that the mapping T... is upper semicontinuous. Since _Pρ,C (f)∩Th is a nonempty ρ -compact convex set and the ρ -topology is a norm-topology, we can apply the Kakutani-Bohnenblust-Karlin Theorem (see [14]) to obtain a fixed point for T... and hence for T .

If we apply the previous theorem in the particular case of the space _L1 (Ω,μ) for a σ -finite measure μ with the usual norm, we obtain the following result, which can be also deduced from [44, Theorem 4.9 ].

Corollary 4.15.

Let (Ω,μ) be as above, C⊂_L1 (Ω,μ) a nonempty closed bounded convex set which satisfies Property (P). Suppose, in addition, that every sequence in C has a convergent locally in measure subsequence in _L1 . If T:C[arrow right]KC(C) is a nonexpansive mapping, then T has a fixed point.

If we consider now the space _{[cursive l]1} , then the assumption of existence of a ^w* -convergent subsequence for every sequence in C can be removed and we can state the following result.

Corollary 4.16.

Let C be a nonempty closed bounded convex subset of _{[cursive l]1} which satisfies Property (P). If T:C[arrow right]KC(C) is a nonexpansive mapping, then T has a fixed point.

Notice that in _{[cursive l]1} there exists a subset with Property (P) which is not ^w* -compact.

Example 4.17 (see [44, Example 4.8 ]).

Let (_an ) be a bounded sequence of nonnegative real numbers and let (_en ) be the standard Schauder basis of _{[cursive l]1} . It is clear that the set C:=conv¯(_xn ) , where _xn :=(1+_an )_en , is never weakly star compact. Nevertheless, by using [46, Example 1 ] it is easy to show that C has Property (P) if and only if _N0 :={n∈...:_an =_{inf m∈...am} } is nonempty and finite.

Acknowledgments

The authors are very grateful to the anonymous referee for some useful suggestions to improve the presentation of this paper. This research was partially supported by DGES Grant no.BFM2006-13997-C02-01 and Junta de Andalucía Grant no.FQM-127. This research is dedicated to W. A. Kirk celebrating his wide and deep contribution in Metric Fixed Point Theory.

References

[1] S. Banach, "Sur les opérations dans les ensembles abstraits et leurs applications," Fundamenta Mathematicae , vol. 3, pp. 133-181, 1922.

[2] W. A. Kirk, "A fixed point theorem for mappings which do not increase distances," The American Mathematical Monthly , vol. 72, pp. 1004-1006, 1965.

[3] S. B. Nadler Jr., "Multi-valued contraction mappings," Pacific Journal of Mathematics , vol. 30, pp. 475-488, 1969.

[4] M. S. Brodski[ibreve], D. P. Mil'man, "On the center of a convex set," Doklady Akademii Nauk SSSR , vol. 59, pp. 837-840, 1948.

[5] W. L. Bynum, "Normal structure coefficients for Banach spaces," Pacific Journal of Mathematics , vol. 86, no. 2, pp. 427-436, 1980.

[6] E. Maluta, "Uniformly normal structure and related coefficients," Pacific Journal of Mathematics , vol. 111, no. 2, pp. 357-369, 1984.

[7] Z. Opial, "Weak convergence of the sequence of successive approximations for nonexpansive mappings," Bulletin of the American Mathematical Society , vol. 73, pp. 591-597, 1967.

[8] S. Prus, "Banach spaces with the uniform Opial property," Nonlinear Analysis: Theory, Methods & Applications , vol. 18, no. 8, pp. 697-704, 1992.

[9] P.-K. Lin, K.-K. Tan, H. K. Xu, "Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings," Nonlinear Analysis: Theory, Methods & Applications , vol. 24, no. 6, pp. 929-946, 1995.

[10] J.-P. Gossez, E. Lami Dozo, "Some geometric properties related to the fixed point theory for nonexpansive mappings," Pacific Journal of Mathematics , vol. 40, pp. 565-573, 1972.

[11] R. Huff, "Banach spaces which are nearly uniformly convex," The Rocky Mountain Journal of Mathematics , vol. 10, no. 4, pp. 743-749, 1980.

[12] K. Goebel, T. Sekowski, "The modulus of noncompact convexity," Annales Universitatis Mariae Curie-Sklodowska. Sectio A , vol. 38, pp. 41-48, 1984.

[13] J. M. Ayerbe Toledano, T. Domínguez Benavides, G. López Acedo Measures of Noncompactness in Metric Fixed Point Theory , vol. 99, of Operator Theory: Advances and Applications, pp. viii+211, Birkhäuser, Basel, Switzerland, 1997.

[14] K. Goebel, W. A. Kirk Topics in Metric Fixed Point Theory , vol. 28, of Cambridge Studies in Advanced Mathematics, pp. viii+244, Cambridge University Press, Cambridge, UK, 1990.

[15] B. Turett, "A dual view of a theorem of Baillon," Nonlinear Analysis and Applications (St. Johns, Nfld., 1981) , vol. 80, of Lecture Notes in Pure and Applied Mathematics, pp. 279-286, Marcel Dekker, New York, NY, USA, 1982.

[16] M. A. Khamsi, "On metric spaces with uniform normal structure," Proceedings of the American Mathematical Society , vol. 106, no. 3, pp. 723-726, 1989.

[17] S. Prus, "Some estimates for the normal structure coefficient in Banach spaces," Rendiconti del Circolo Matematico di Palermo , vol. 40, no. 1, pp. 128-135, 1991.

[18] E. Lami Dozo, "Multivalued nonexpansive mappings and Opial's condition," Proceedings of the American Mathematical Society , vol. 38, pp. 286-292, 1973.

[19] T. C. Lim, "A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space," Bulletin of the American Mathematical Society , vol. 80, pp. 1123-1126, 1974.

[20] M. Edelstein, "The construction of an asymptotic center with a fixed-point property," Bulletin of the American Mathematical Society , vol. 78, pp. 206-208, 1972.

[21] W. A. Kirk, S. Massa, "Remarks on asymptotic and Chebyshev centers," Houston Journal of Mathematics , vol. 16, no. 3, pp. 357-364, 1990.

[22] W. A. Kirk, "Nonexpansive mappings in product spaces, set-valued mappings and k -uniform rotundity," Nonlinear Functional Analysis and Its Applications, Part 2 (Berkeley, Calif., 1983) , vol. 45, of Proc. Sympos. Pure Math., pp. 51-64, American Mathematical Society, Providence, RI, USA, 1986.

[23] T. Kuczumow, S. Prus, "Compact asymptotic centers and fixed points of multivalued nonexpansive mappings," Houston Journal of Mathematics , vol. 16, no. 4, pp. 465-468, 1990.

[24] H.-K. Xu, "Metric fixed point theory for multivalued mappings," Dissertationes Mathematicae , vol. 389, pp. 39, 2000.

[25] T. Domínguez Benavides, P. Lorenzo Ramírez, "Fixed-point theorems for multivalued non-expansive mappings without uniform convexity," Abstract and Applied Analysis , vol. 2003, no. 6, pp. 375-386, 2003.

[26] T. Domínguez Benavides, P. Lorenzo Ramírez, "Asymptotic centers and fixed points for multivalued nonexpansive mappings," Annales Universitatis Mariae Curie-Sklodowska. Sectio A , vol. 58, pp. 37-45, 2004.

[27] S. Dhompongsa, A. Kaewcharoen, A. Kaewkhao, "The Domínguez-Lorenzo condition and multivalued nonexpansive mappings," Nonlinear Analysis: Theory, Methods & Applications and Methods , vol. 64, no. 5, pp. 958-970, 2006.

[28] S. Dhompongsa, T. Domínguez Benavides, A. Kaewcharoen, A. Kaewkhao, B. Panyanak, "The Jordan-von Neumann constants and fixed points for multivalued nonexpansive mappings," Journal of Mathematical Analysis and Applications , vol. 320, no. 2, pp. 916-927, 2006.

[29] T. Domínguez Benavides, B. Gavira, "The fixed point property for multivalued nonexpansive mappings," Journal of Mathematical Analysis and Applications , vol. 328, no. 2, pp. 1471-1483, 2007.

[30] S. Saejung, "Remarks on sufficient conditions for fixed points of multivalued nonexpansive mappings," Nonlinear Analysis: Theory, Methods & Applications , vol. 67, no. 5, pp. 1649-1653, 2007.

[31] A. Kaewkhao, "The James constant, the Jordan-von Neumann constant, weak orthogonality, and fixed points for multivalued mappings," Journal of Mathematical Analysis and Applications , vol. 333, no. 2, pp. 950-958, 2007.

[32] B. Gavira, "Some geometric conditions which imply the fixed point property for multivalued nonexpansive mappings," Journal of Mathematical Analysis and Applications , vol. 339, no. 1, pp. 680-690, 2008.

[33] H. Jiao, Y. Guo, F. Wang, "Modulus of convexity, the coefficient R(1,X) , and normal structure in Banach spaces," Abstract and Applied Analysis , vol. 2008, 2008.

[34] H. Nakano Modulared Semi-Ordered Linear Spaces , Maruzen, Tokyo, Japan, 1950.

[35] J. Musielak, W. Orlicz, "On modular spaces," Studia Mathematica , vol. 18, pp. 49-65, 1959.

[36] T. Domínguez Benavides, M. A. Khamsi, S. Samadi, "Asymptotically regular mappings in modular function spaces," Scientiae Mathematicae Japonicae , vol. 53, no. 2, pp. 295-304, 2001.

[37] T. Domínguez Benavides, M. A. Khamsi, S. Samadi, "Asymptotically nonexpansive mappings in modular function spaces," Journal of Mathematical Analysis and Applications , vol. 265, no. 2, pp. 249-263, 2002.

[38] M. A. Khamsi, "Fixed point theory in modular function spaces," Recent Advances on Metric Fixed Point Theory (Seville, 1995) , vol. 48, of Ciencias, pp. 31-57, Universidad de Sevilla, Seville, Spain, 1996.

[39] M. A. Khamsi, W. M. Kozlowski, S. Reich, "Fixed point theory in modular function spaces," Nonlinear Analysis: Theory, Methods & Applications , vol. 14, no. 11, pp. 935-953, 1990.

[40] S. Dhompongsa, T. Domínguez Benavides, A. Kaewcharoen, B. Panyanak, "Fixed point theorems for multivalued mappings in modular function spaces," Scientiae Mathematicae Japonicae , vol. 63, no. 2, pp. 161-169, 2006.

[41] W. M. Kozlowski Modular Function Spaces , vol. 122, of Monographs and Textbooks in Pure and Applied Mathematics, pp. x+252, Marcel Dekker, New York, NY, USA, 1988.

[42] J. Musielak Orlicz Spaces and Modular Spaces , vol. 1034, of Lecture Notes in Mathematics, pp. iii+222, Springer, Berlin, Germany, 1983.

[43] D. E. Alspach, "A fixed point free nonexpansive map," Proceedings of the American Mathematical Society , vol. 82, no. 3, pp. 423-424, 1981.

[44] T. Domínguez Benavides, J. García Falset, E. Llorens-Fuster, P. Lorenzo Ramírez, "Fixed point properties and proximinality in Banach spaces," Nonlinear Analysis: Theory, Methods & Applications , vol. 71, no. 5-6, pp. 1562-1571, 2009.

[45] S. Hu, N. S. Papageorgiou Handbook of Multivalued Analysis. Vol. I. Theory , vol. 419, of Mathematics and Its Applications, pp. xvi+964, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

[46] K. Goebel, T. Kuczumov, "Irregular convex sets with fixed point property for nonexpansive mappings," Colloqium Mathematicum , pp. 259-264, 1979.