Fixed Point Theory and Applications

, 2011:175989 | Cite as

Relation between Fixed Point and Asymptotical Center of Nonexpansive Maps

  • MohammadReza Haddadi
  • Hamid Mazaheri
  • MohammadHusseinLabbaf Ghasemi
Open Access
Research Article
Part of the following topical collections:
  1. Equilibrium Problems and Fixed Point Theory

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.

Let Open image in new window be a nonempty subset of a Banach space Open image in new window and Open image in new window a bounded sequence in Open image in new window . Consider the functional Open image in new window defined by
The infimum of Open image in new window over Open image in new window is said to be the asymptotic radius of Open image in new window with respect to Open image in new window and is denoted by Open image in new window . A point Open image in new window is said to be an asymptotic center of the sequence Open image in new window with respect to Open image in new window if

The set of all asymptotic centers of Open image in new window with respect to Open image in new window is denoted by Open image in new window .

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

Definition 1.1.

Let Open image in new window be a bounded closed convex subset of Open image in new window . A sequence Open image in new window is said to be an asymptotic center for a mapping Open image in new window if, for each Open image in new window ,

Definition 1.2.

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

Definition 1.3.

Let Open image in new window be a nonempty subset of Open image in new window . We say that Open image in new window has Property Open image in new window if for every bounded sequence Open image in new window , the set Open image in new window is a nonempty and compact subset of Open image in new window .

Example 1.4.

Let Open image in new window be a normed space and Open image in new window a nonempty subset of Open image in new window . It is clear that

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

(ii)if Open image in new window is a open set, since Open image in new window , therefore Open image in new window is empty and so fail to have Property Open image in new window .

2. Main Results

Our new results are presented in this section.

Proposition 2.1.

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

Proof.

Assume that Open image in new window is a continuous mapping and Open image in new window is a asymptotic center. Let Open image in new window has set of asymptotic center Open image in new window . Since Open image in new window has Property Open image in new window , Open image in new window is nonempty and compact and it is easy to see that it is also convex. In order to obtain the result, it will be enough to show that Open image in new window is Open image in new window -invariant since in this case we may apply Schauder's Fixed-Point Theorem [4, Theorem  18.10]. Indeed, let Open image in new window . Since Open image in new window is a asymptotic center for Open image in new window , we have

Therefore Open image in new window .

Theorem 2.2.

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

Proof.

Suppose that Open image in new window fails to have Property Open image in new window . There exists Open image in new window such that either Open image in new window or Open image in new window is noncompact. In the second case, by Klee's theorem  in [6] there exists a continuous function Open image in new window without fixed points ( Open image in new window ). Since a closed convex subset of a normed space is a retract of the space, there exists a continuous mapping Open image in new window such that Open image in new window for all Open image in new window . Define Open image in new window by Open image in new window . Clearly Open image in new window is a continuous mapping. Moreover,

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

Concerning the first case we proceed as follows.

Let Open image in new window . We take Open image in new window such that Open image in new window . For each positive integer Open image in new window , we consider the following nonempty sets:
Fix an arbitrary Open image in new window and define, by induction, a sequence Open image in new window such that Open image in new window and the segment Open image in new window does not meet Open image in new window . Given Open image in new window , there exists a unique positive integer Open image in new window such that Open image in new window . In this case we define

It is a routine to check that Open image in new window is a continuous mapping from Open image in new window to Open image in new window . Furthermore, Open image in new window for every Open image in new window .

Let Open image in new window be a continuous retraction from Open image in new window into the closed convex subset Open image in new window . We can define Open image in new window by Open image in new window . It is clear that Open image in new window is a asymptotic center for Open image in new window and that Open image in new window 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 Open image in new window 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 Open image in new window be a nonempty closed bounded and convex subset of a Banach space Open image in new window . The following conditions are equivalent.

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

(2) Open image in new window has Property Open image in new window .

Let Open image in new window be a nonempty closed convex bounded subset of a Banach space Open image in new window . By Open image in new window we denote the family of all nonempty compact convex subsets of Open image in new window . On Open image in new window we consider the well-known Hausdorff metric Open image in new window . Recall that a mapping Open image in new window is said to be nonexpansive whenever

Theorem 2.4.

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

Proof.

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

For each Open image in new window , the compactness of Open image in new window guarantees that there exists Open image in new window satisfying Open image in new window .

Now we are going to show that for every Open image in new window ,
Taking any Open image in new window , from the compactness of Open image in new window we can find Open image in new window such that
By compactness again we can assume that Open image in new window converges to a point Open image in new window . From above it follows that

Therefore Open image in new window .

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

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

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

Proposition 2.5.

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

Proof.

The above comments guarantee that Open image in new window is an approximated fixed-point sequence for Open image in new window . Let us see that the sequence Open image in new window an asymptotic center for Open image in new window . Given Open image in new window we have

Therefore Open image in new window is asymptotic center for Open image in new window .

Theorem 2.6.

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

Proof.

and so Open image in new window .

For every Open image in new window , Open image in new window is a contraction, so there exists exactly one fixed point Open image in new window of Open image in new window . Now

Therefore Open image in new window is the approximating fixed-point sequence in Open image in new window of Open image in new window .

Corollary 2.7.

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

Proof.

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

References

  1. 1.
    Edelstein M: The construction of an asymptotic center with a fixed-point property. Bulletin of the American Mathematical Society 1972, 78: 206–208. 10.1090/S0002-9904-1972-12918-5MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bose SC, Laskar SK: Fixed point theorems for certain class of mappings. Journal of Mathematical and Physical Sciences 1985,19(6):503–509.MATHMathSciNetGoogle Scholar
  3. 3.
    Downing D, Kirk WA: Fixed point theorems for set-valued mappings in metric and Banach spaces. Mathematica Japonica 1977,22(1):99–112.MATHMathSciNetGoogle Scholar
  4. 4.
    Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.CrossRefGoogle Scholar
  5. 5.
    Lan KQ, Webb JRL: Open image in new window-properness and fixed point theorems for dissipative type maps. Abstract and Applied Analysis 1999,4(2):83–100. 10.1155/S108533759900010XMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Klee VL Jr.: Some topological properties of convex sets. Transactions of the American Mathematical Society 1955, 78: 30–45. 10.1090/S0002-9947-1955-0069388-5MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Mohammad Reza Haddadi et al. 2011

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.

Authors and Affiliations

  • MohammadReza Haddadi
    • 1
  • Hamid Mazaheri
    • 1
  • MohammadHusseinLabbaf Ghasemi
    • 1
  1. 1.Department of Mathematics, Faculty of MathematicsYazd UniversityYazdIran

Personalised recommendations