- Research Article
- Open Access

# Relation between Fixed Point and Asymptotical Center of Nonexpansive Maps

- MohammadReza Haddadi
^{1}Email author, - Hamid Mazaheri
^{1}and - MohammadHusseinLabbaf Ghasemi
^{1}

**2011**:175989

https://doi.org/10.1155/2011/175989

© Mohammad Reza Haddadi et al. 2011

**Received:**19 October 2010**Accepted:**22 November 2010**Published:**2 December 2010

## Abstract

We introduce the concept of asymptotic center of maps and consider relation between asymptotic center and fixed point of nonexpansive maps in a Banach space.

## Keywords

- Banach Space
- Positive Integer
- Continuous Mapping
- Compact Subset
- Convex Subset

## 1. Introduction

Many topics and techniques regarding asymptotic centers and asymptotic radius were studied by Edelstein [1], Bose and Laskar [2], Downing and Kirk [3], Goebel and Kirk [4], and Lan and Webb [5]. Now, We recall that definitions of asymptotic center and asymptotic radius.

The set of all asymptotic centers of with respect to is denoted by .

We present new definitions of asymptotic center and asymptotic radius that is for a mapping and obtain new results.

Definition 1.1.

Definition 1.2.

Let
be a nonempty subset of
*.* We say that
has the fixed-point property for continuous mappings of
with asymptotic center if every continuous mapping
admitting an asymptotic center has a fixed point.

Definition 1.3.

Let
be a nonempty subset of
. We say that
has Property
if for every bounded sequence
, the set
is a nonempty and compact subset of
*.*

Example 1.4.

Let be a normed space and a nonempty subset of . It is clear that

(i)if is a compact set, then in nonempty compact set and so has Property ;

(ii)if is a open set, since , therefore is empty and so fail to have Property .

## 2. Main Results

Our new results are presented in this section.

Proposition 2.1.

Let be a Banach space and let be a nonempty closed bounded and convex subset of . If satisfies Property , then every continuous mapping asymptotically admitting a center in has a fixed point.

Proof.

Theorem 2.2.

Let be a Banach space and let be a nonempty closed bounded and convex subset of . If has the fixed-point property for continuous mappings admitting an asymptotic center, then has Property .

Proof.

that is, is an asymptotic center for . Therefore, by Proposition 2.1, has a fixed point in , . Hence sets a contradiction.

Concerning the first case we proceed as follows.

It is a routine to check that is a continuous mapping from to . Furthermore, for every .

Let be a continuous retraction from into the closed convex subset . We can define by . It is clear that is a asymptotic center for and that is fixed-point free.

Proposition 2.1 (Theorem 2.2) is a generalizations of Theorem 3.1 (Theorem 3.3) in [1]. It can be verified that definition of space is not necessary here.

As an easy consequence of both Proposition 2.1 and Theorem 2.2, we deduce the following result.

Corollary 2.3.

Let be a nonempty closed bounded and convex subset of a Banach space . The following conditions are equivalent.

(1) has the fixed-point property for continuous mappings admitting asymptotic center in .

Theorem 2.4.

Let be a Banach space and let be a nonempty closed convex and bounded subset of satisfying Property . If is a nonexpansive mapping, then has a fixed point.

Proof.

Let be a nonexpansive mapping. The multivalued analog of Banach's Contraction Principle allows us to find a sequence in such that .

For each , the compactness of guarantees that there exists satisfying .

Now we define the mapping by . Since the mapping is upper semicontinuous and for every is a compact convex set we can apply the Kakutani-Bohnenblust-Karlin Theorem in [5] to obtain a fixed point for and hence for .

Let be a metric space and a mapping. Then a sequence in is said to be an approximating fixed-point sequence of if .

Let be a bounded closed and convex subset of a Banach space , a nonexpansive mapping and . Then a mappings define by is always asymptotically regular, that is, for every , .

Proposition 2.5.

Let be a Banach space and a closed bounded convex subset of , and . If is a nonexpansive mapping, then the sequence is an asymptotic center for .

Proof.

Therefore is asymptotic center for .

Theorem 2.6.

Let be a normed space, a nonexpansive mapping with an approximating fixed point sequence and be a nonempty subset of such that is a nonempty star-shaped subset of . Then has an approximating fixed-point sequence in .

Proof.

Therefore is the approximating fixed-point sequence in of .

Corollary 2.7.

Let be a normed space, a nonexpansive mapping with an approximating fixed-point sequence and be a nonempty subset of such that . Suppose is a nonempty weakly compact star-shaped subset of . If is demiclosed, then has a fixed point in .

Proof.

By the last theorem, has an approximating fixed-point sequence . Because is weakly compact, there exists a subsequence of such that . Since is demiclosed on and , it follows that . Therefore, .

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.