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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 247402, 5 pages
http://dx.doi.org/10.1155/2010/247402
Research Article

Remarks on Asymptotic Centers and Fixed Points

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand

Received 4 August 2010; Accepted 4 November 2010

Academic Editor: Mitsuharu Ôtani

Copyright © 2010 A. Kaewkhao and K. Sokhuma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a class of nonlinear continuous mappings defined on a bounded closed convex subset of a Banach space . We characterize the Banach spaces in which every asymptotic center of each bounded sequence in any weakly compact convex subset is compact as those spaces having the weak fixed point property for this type of mappings.

1. Introduction

A mapping on a subset of a Banach space is called a nonexpansive mapping if for all . Although nonexpansive mappings are widely studied, there are many nonlinear mappings which are more general. The study of the existence of fixed points for those mappings is very useful in solving the problems of equations in science and applied science.

The technique of employing the asymptotic centers and their Chebyshev radii in fixed point theory was first discovered by Edelstein [1], and the compactness assumption given on asymptotic centers was introduced by Kirk and Massa [2]. Recently, Dhompongsa et al. proved in [3] a theorem of existence of fixed points for some generalized nonexpansive mappings on a bounded closed convex subset of a Banach space with assumption that every asymptotic center of a bounded sequence relative to is nonempty and compact. However, spaces or sets in which asymptotic centers are compact have not been completely characterized, but partial results are known (see [4, page 93]).

In this paper, we introduce a class of nonlinear continuous mappings in Banach spaces which allows us to characterize the Banach spaces in which every asymptotic center of each bounded sequence in any weakly compact convex subset is compact as those spaces having the weak fixed point property for this type of mappings.

2. Preliminaries

Let be a nonempty closed and convex subset of a Banach space and a bounded sequence in . For , define the asymptotic radius of at as the number Let The number and the set are, respectively, called the asymptotic radius and asymptotic center of relative to . It is known that is nonempty, weakly compact, and convex as is [4, page 90].

Let be a nonexpansive and . Then for , the mapping defined by setting is a contraction mapping. As we have known, Banach contraction mapping theorem assures the existence of a unique fixed point . Since we have the following.

Lemma 2.1. If is a bounded closed and convex subset of a Banach space and if is nonexpansive, then there exists a sequence such that

3. Main Results

Definition 3.1. Let be a bounded closed convex subset of a Banach space . We say that a sequence in is an asymptotic center sequence for a mapping if, for each ,

We say that is a D-type mapping whenever it is continuous and there is an asymptotic center sequence for .

The following observation shows that the concept of D-type mappings is a generalization of nonexpansiveness.

Proposition 3.2. Let be a nonexpansive mapping. Then is a D-type mapping.

Proof. It is easy to see that is continuous. By Lemma 2.1, there exists a sequence such that
For , Then This implies that is an asymptotic center sequence for . Thus is a D-type mapping.

Definition 3.3. We say that a Banach space has the weak fixed point property for D-type mappings if every D-type self-mapping on every weakly compact convex subset of has a fixed point.

Now we are in the position to prove our main theorem.

Theorem 3.4. Let be a Banach space. Then has the weak fixed point property for D-type mappings if and only if the asymptotic center relative to each nonempty weakly compact convex subset of each bounded sequence of is compact.

Proof. Suppose the asymptotic center of any bounded sequence of relative to any nonempty weakly compact convex subset of is compact. Let be a weakly compact convex subset of and a D-type mapping having as an asymptotic center sequence. Let and , respectively, be the asymptotic radius and the asymptotic center of relative to . Since is weakly compact and convex, is nonempty weakly compact and convex. For every , since is an asymptotic center sequence for , we have Hence , which implies that is -invariant. By the assumption, is a compact set. By using Schauder's fixed point theorem, we can conclude that has a fixed point in and hence has a fixed point in .
Now suppose has the weak fixed point property for D-type mappings, and suppose there exists a weakly compact convex subset of and a bounded sequence in whose asymptotic center relative to is not compact. By Klee's theorem (see [4, page 203]), there exists a continuous, fixed point free mapping . We see that is an asymptotic center sequence for . Indeed, since for each , we have Then is a D-type mapping. Thus should have a fixed point which is a contradiction.

In 2007, García-Falset et al. [5] introduced another concept of centers of mappings.

Definition 3.5. Let be a bounded closed convex subset of a Banach space . A point is said to be a center for a mapping if, for each , A mapping is said to be a J-type mapping whenever it is continuous and it has some center .

Definition 3.6. We say that a Banach space has the J-weak fixed point property if every J-type self-mapping of every weakly compact subset of has a fixed point.

Employing the above definitions, the authors proved a characterization of the geometrical property of the Banach spaces introduced in 1973 by Bruck Jr. [6]: a Banach space has property whenever the weakly compact convex subsets of its unit sphere are compact sets.

Theorem 3.7 (see [5, Theorem 16]). Let be a Banach space. Then has property if and only if has the J-weak fixed point property.

It is easy to see that a center of a mapping can be seen as an asymptotic center sequence for the mapping by setting for all . This leads to the following conclusion.

Proposition 3.8. Let be a J-type mapping. Then is a D-type mapping.

Consequently, we have the following proposition.

Proposition 3.9. Let be a Banach space. If has the weak fixed point property for D-type mappings, then has the J-weak fixed point property.

From Theorems 3.4 and 3.7, and Proposition 3.9, we can conclude this paper by the following result.

Theorem 3.10. Let be a Banach space. If the asymptotic center relative to every nonempty weakly compact convex subset of each bounded sequence of is compact, then has property .

Dedication

The authors dedicate this work to Professor Sompong Dhompongsa on the occasion of his 60th birthday.

Acknowledgments

This work was completed with the support of the Commission on Higher Education and The Thailand Research Fund under Grant MRG5180213. The author would like to express his deep gratitude to Professor Dr. Sompong Dhompongsa whose guidance and support were valuable for the completion of the paper.

References

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