# On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings

- Jingxin Zhang
^{1}Email author and - Yunan Cui
^{2}

**2010**:596952

https://doi.org/10.1155/2010/596952

© J. Zhang and Y. Cui. 2010

**Received: **30 July 2010

**Accepted: **5 October 2010

**Published: **11 October 2010

## Abstract

We show some geometric conditions on a Banach space *X* concerning the Jordan-von Neumann constant, Zbăganu constant, characteristic of (separation) noncompact convexity, and the coefficient *R*(1, *X*), the weakly convergent sequence coefficient, which imply the existence of fixed points for multivalued nonexpansive mappings.

## Keywords

## 1. Introduction

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.

In 1969, Nadler [1] established the multivalued version of Banach's contraction principle. One of the most celebrated results about multivalued mappings was given by Lim [2] in 1974. Using Edelstein's method of asymptotic centers, he proved the existence of a fixed point for a multivalued nonexpansive self-mapping where is a nonempty bounded closed convex subset of a uniformly convex Banach space. Since then the metric fixed point theory of multivalued mappings has been rapidly developed. Some other classical fixed point theorems for single-valued mappings have been extended to multivalued mappings. However, many questions remain 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, in short) for multivalued nonexpansive mappings?

implies property (D).

(3)Satit Saejung [5] proved that the condition implies property (D).

(4)Gavira [6] showed that the condition

implies (DL) condition.

In 2007, Domínguez Benavides and Gavira [7] have established FFP for multivalued nonexpansive mappings in terms of the modulus of squareness, universal infinite-dimensional modulus, and Opia modulus. Attapol Kaewkhao [8] has established FFP for multivalued nonexpansive mappings in terms of the James constant, the Jordan-von Neumann Constants, weak orthogonality.

Besides, In 2010, Domínguez Benavides and Gavira [9] have given a survey of this subject and presented the main known results and current research directions.

In this paper, in terms of the Jordan-von Neumann constant, Zb ganu constant, and the coefficient , the weakly convergent sequence coefficient, we show some geometrical properties which imply the property (D) or (DL) condition and so the FPP for multivalued nonexpansive mappings.

## 2. Preliminaries

Let be a Banach space and be a nonempty subset of ; we denote all nonempty bounded closed subsets of by and all nonempty compact convex subsets of by .

It is known that is a nonempty weakly compact convex as is.

Let and be as above; then is called regular relative to if for all subsequence of ; further, is called asymptotically uniform relative to if for all subsequence of . In Banach spaces, we have the following results:

- (2)
(Kirk [11]) if is separable, then contains a subsequence which is asymptotically uniform relative to .

In 2006, Dhompongsa et al. [3] introduced the Domnguez-Lorenzo condition ((DL) condition, in short) in the following way.

Definition 2.1 (see [3]).

The (DL) condition implies weak normal structure [3]. We recll that a Banach space is said to have a weak normal structure (w-NS) if for every weakly compact convex subset of with there exist such that .

The (DL) condition also implies the existence of fixed points for multivalued nonexpansive mappings.

Theorem 2.2 (see [3]).

Let be a weakly compact convex subset of Banach space ; if satisfies (DL) condition, then multivalued nonexpansive mapping has a fixed point.

Definition 2.3 (see [4]).

It was observed that property (D) is weaker than the (DL) condition and stronger than weak normal structure, and Dhompongsa et al. [4] proved that property (D) implies the w-MFPP.

Theorem 2.4 (see [4]).

Let be a weakly compact convex subset of Banach space ; if satisfies property (D), then multivalued nonexpansive mapping has a fixed point.

Before going to the results, let us recall some more definitions. Let be a Banach space.

where the infimum is taken over all weakly (not strongly) null sequences with existing.

The ultrapower of a Banach space has proved to be useful in many branches of mathematics. Many results can be seen more easily when treated in this setting.

*convergers to*

*with respect to*, denoted by , if for each neighborhood of , . A filter on is called an ultrafilter if it is maximal with respect to the set inclusion. An ultrafilter is called trivial if it is of the form for some fixed ; otherwise, it is called nontrivial. Let denote the subspace of the product space equipped with the norm

Note that if is nontrivial, then can be embedded into isometrically. For more details see [14].

## 3. Main Results

Theorem 3.1.

Proof.

Let ; taking a subsequence if necessary, we can assume that for all .

Hence we deduce the desired inequality.

By Theorems 2.2 and 3.1, we have the following result.

Corollary 3.2.

Let be a nonempty bounded closed convex subset of a Banach space such that and a nonexpansive mapping. Then has a fixed point.

Proof.

since , if , then we have which implies that is uniformly nonsquare; hence is reflexive. Thus by Theorems 2.2 and 3.1, the result follows.

Remark 3.3.

Note that ; it is easy to see that Theorem 3.1 includes [6, Theorem ] and Corollary 3.2 includes [6, Corollary ].

We first give the following tool.

Proposition 3.4.

Proof.

Clearly, . To show , suppose are not all zero. Without loss of generality, we assume .

Hence, the inequality follows from the arbitrariness of .

Theorem 3.5.

Proof.

Hence we deduce the desired inequality.

Using Theorem 2.2, we obtain the following corollary.

Corollary 3.6.

Let be a nonempty weakly compact convex subset of a Banach space such that and let be a nonexpansive mapping. Then has a fixed point.

In the following, we present some properties concerning geometrical constants of Banach spaces which also imply the property (D).

Theorem 3.7.

Let be a Banach space. If ; then has property (D).

Proof.

Let be a weakly compact convex subset of ; suppose that and are regular and asymptotically uniform relative to . Passing to a subsequence of , still denoted by , we may assume that and exists.

Hence and the assertion follows by the definition of property (D).

Using Theorems 2.4 and 3.7, we obtain the follwing corollary.

Corollary 3.8.

Let be a nonempty bounded closed convex subset of a reflexive Banach space such that and let be a nonexpansive mapping. Then has a fixed point.

It is known that is NUC if and only if . The above-mentioned definitions and properties can be found in [17].

Theorem 3.9.

Let be a reflexive Banach space. If , then has property (D).

Proof.

Let be a weakly compact convex subset of ; suppose that and are regular and asymptotically uniform relative to . Passing to a subsequence of , still denoted by , we may assume that and exists. Let .

So for any , there exists such that and for all .

Remark 3.10.

Since , Theorem 3.9 implies the [5, Theorem ]. Furthermore, it is easy to see ; then Theorem 3.9 also includes [4, Theorem ].

By Theorem 3.9, we obtain the following Corollary.

Corollary 3.11.

Let be a nonempty bounded closed convex subset of a reflexive Banach space such that and let be a nonexpansive mapping. Then has a fixed point.

Noticing , obviously, Corollary 3.11 extends the following well-known result.

Theorem 3.12 (see [18, Theorem ]).

Let be a nonempty bounded closed convex subset of a reflexive Banach space such that and let be a nonexpansive mapping. Then has a fixed point.

## Declarations

### Acknowledgments

The authors would like to thank the anonymous referee for providing some suggestions to improve the manuscript. This work was supported by China Natural Science Fund under grant 10571037.

## Authors’ Affiliations

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