Remarks on Asymptotic Centers and Fixed Points

Abstract

We introduce a class of nonlinear continuous mappings defined on a bounded closed convex subset of a Banach space 𝑋. 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 𝑇 on a subset 𝐸 of a Banach space 𝑋 is called a nonexpansive mapping if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐸. 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 𝐸 of a Banach space with assumption that every asymptotic center of a bounded sequence relative to 𝐸 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 𝐸 be a nonempty closed and convex subset of a Banach space 𝑋 and {𝑥_𝑛} a bounded sequence in 𝑋. For 𝑥∈𝑋, define the asymptotic radius of {𝑥_𝑛} at 𝑥 as the number𝑟𝑥,𝑥_𝑛=limsup_𝑛→∞𝑥_𝑛−𝑥.(2.1) Let𝑟≡𝑟𝐸,𝑥_𝑛∶=inf𝑟𝑥,𝑥_𝑛,𝑥∶𝑥∈𝐸𝐴≡𝐴𝐸,_𝑛∶=𝑥∈𝐸∶𝑟𝑥,𝑥_𝑛=𝑟.(2.2) The number 𝑟 and the set 𝐴 are, respectively, called the asymptotic radius and asymptotic center of {𝑥_𝑛} relative to 𝐸. It is known that 𝐴(𝐸,{𝑥_𝑛}) is nonempty, weakly compact, and convex as 𝐸 is [4, page 90].

Let 𝑇∶𝐸→𝐸 be a nonexpansive and 𝑧∈𝐸. Then for 𝛼∈(0,1), the mapping 𝑇_𝛼∶𝐸→𝐸 defined by setting𝑇_𝛼𝑥=(1−𝛼)𝑧+𝛼𝑇𝑥(2.3) is a contraction mapping. As we have known, Banach contraction mapping theorem assures the existence of a unique fixed point 𝑥_𝛼∈𝐸. Sincelim_𝛼→1−𝑥_𝛼−𝑇𝑥_𝛼=lim_𝛼→1−(1−𝛼)𝑧−𝑇𝑥_𝛼=0,(2.4) we have the following.

Lemma 2.1. If 𝐸 is a bounded closed and convex subset of a Banach space and if 𝑇∶𝐸→𝐸 is nonexpansive, then there exists a sequence {𝑥_𝑛}⊂𝐸 such that lim_𝑛→∞x_𝑛−𝑇𝑥_𝑛=0.(2.5)

3. Main Results

Definition 3.1. Let 𝐸 be a bounded closed convex subset of a Banach space 𝑋. We say that a sequence {𝑥_𝑛} in 𝑋 is an asymptotic center sequence for a mapping 𝑇∶𝐸→𝑋 if, for each 𝑥∈𝐸, limsup_𝑛→∞𝑥_𝑛−𝑇𝑥≤limsup_𝑛→∞𝑥_𝑛−𝑥.(3.1)

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

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

Proposition 3.2. Let 𝑇∶𝐸→𝐸 be a nonexpansive mapping. Then 𝑇 is a D-type mapping.

Proof. It is easy to see that 𝑇 is continuous. By Lemma 2.1, there exists a sequence {𝑥_𝑛} such that lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=0.(3.2)
For 𝑥∈𝐸, 𝑥_𝑛−𝑇𝑥≤𝑥_𝑛−𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑇𝑥≤𝑥_𝑛−𝑇𝑥_𝑛+𝑥_𝑛−𝑥.(3.3) Then limsup_𝑛→∞𝑥_𝑛−𝑇𝑥≤limsup_𝑛→∞𝑥_𝑛−𝑥.(3.4) This implies that {𝑥_𝑛} is an asymptotic center sequence for 𝑇. Thus 𝑇 is a D-type mapping.

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

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

Theorem 3.4. Let 𝑋 be a Banach space. Then 𝑋 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 𝑋 is compact.

Proof. Suppose the asymptotic center of any bounded sequence of 𝑋 relative to any nonempty weakly compact convex subset of 𝑋 is compact. Let 𝐸 be a weakly compact convex subset of 𝑋 and 𝑇∶𝐸→𝐸 a D-type mapping having {𝑥_𝑛} as an asymptotic center sequence. Let 𝑟 and 𝐴, respectively, be the asymptotic radius and the asymptotic center of {𝑥_𝑛} relative to 𝐸. Since 𝐸 is weakly compact and convex, 𝐴 is nonempty weakly compact and convex. For every 𝑥∈𝐴, since {𝑥_𝑛} is an asymptotic center sequence for 𝑇, we have 𝑟≤limsup_𝑛→∞𝑥_𝑛−𝑇𝑥≤limsup_𝑛→∞𝑥_𝑛−𝑥=𝑟.(3.5) Hence 𝑇(𝑥)∈𝐴, which implies that 𝐴 is 𝑇-invariant. By the assumption, 𝐴 is a compact set. By using Schauder's fixed point theorem, we can conclude that 𝑇 has a fixed point in 𝐴 and hence 𝑇 has a fixed point in 𝐸.
Now suppose 𝑋 has the weak fixed point property for D-type mappings, and suppose there exists a weakly compact convex subset 𝐾 of 𝑋 and a bounded sequence {𝑥_𝑛} in 𝑋 whose asymptotic center 𝐴 relative to 𝐾 is not compact. By Klee's theorem (see [4, page 203]), there exists a continuous, fixed point free mapping 𝑇∶𝐴→𝐴. We see that {𝑥_𝑛} is an asymptotic center sequence for 𝑇. Indeed, since 𝑇𝑥∈𝐴 for each 𝑥∈𝐴, we have limsup_𝑛→∞𝑥_𝑛−𝑇𝑥=𝑟=limsup_𝑛→∞𝑥_𝑛−𝑥.(3.6) Then 𝑇 is a D-type mapping. Thus 𝑇 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 𝐸 be a bounded closed convex subset of a Banach space 𝑋. A point 𝑥₀∈𝑋 is said to be a center for a mapping 𝑇∶𝐸→𝑋 if, for each 𝑥∈𝐸, 𝑇𝑥−𝑥₀≤𝑥−𝑥₀.(3.7) A mapping 𝑇∶𝐸→𝑋 is said to be a J-type mapping whenever it is continuous and it has some center 𝑥₀∈𝑋.

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

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

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

It is easy to see that a center 𝑥₀∈𝑋 of a mapping 𝑇∶𝐸→𝑋 can be seen as an asymptotic center sequence {𝑥_𝑛} for the mapping 𝑇 by setting 𝑥_𝑛≡𝑥₀ for all 𝑛∈ℕ. This leads to the following conclusion.

Proposition 3.8. Let 𝑇∶𝐸→𝑋 be a J-type mapping. Then 𝑇 is a D-type mapping.

Consequently, we have the following proposition.

Proposition 3.9. Let 𝑋 be a Banach space. If 𝑋 has the weak fixed point property for D-type mappings, then 𝑋 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 𝑋 be a Banach space. If the asymptotic center relative to every nonempty weakly compact convex subset of each bounded sequence of 𝑋 is compact, then 𝑋 has property (𝐶).

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.