Relation between Fixed Point and Asymptotical Center of Nonexpansive Maps
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Research Article Relation between Fixed Point and Asymptotical Center of Nonexpansive Maps Mohammad Reza Haddadi, Hamid Mazaheri, and Mohammad Hussein Labbaf Ghasemi Department of Mathematics, Faculty of Mathematics, Yazd University, P. O. Box 89195-741, Yazd, Iran Correspondence should be addressed to Mohammad Reza Haddadi, [email protected] Received 19 October 2010; Accepted 22 November 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 Mohammad Reza Haddadi et al. 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. 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.
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 C be a nonempty subset of a Banach space X and {xn } a bounded sequence in X. Consider the functional ra ·, {xn } : X → R defined by ra x, {xn } lim supxn − x, n→∞
x ∈ X.
1.1
The infimum of ra ·, {xn } over C is said to be the asymptotic radius of {xn } with respect to C and is denoted by ra C, {xn }. A point z ∈ C is said to be an asymptotic center of the sequence {xn } with respect to C if ra z, {xn } inf{ra x, {xn } : x ∈ C}. The set of all asymptotic centers of {xn } with respect to C is denoted by Za C, {xn }.
1.2
2
Fixed Point Theory and Applications
We present new definitions of asymptotic center and asymptotic radius that is for a mapping and obtain new results. Definition 1.1. Let C be a bounded closed convex subset of X. A sequence {xn } ⊆ X is said to be an asymptotic center for a mapping T : C → X if, for each x ∈ C, lim supT x − xn ≤ lim supxn − x. n→∞
n→∞
1.3
Definition 1.2. Let C be a nonempty subset of X. We say that C has the fixed-point property for continuous mappings of C with asymptotic center if every continuous mapping T : C → C admitting an asymptotic center has a fixed point. Definition 1.3. Let C be a nonempty subset of X. We say that C has Property Z if for every bounded sequence {xn } ⊂ X \ C, the set Za C, {xn } is a nonempty and compact subset of C. Example 1.4. Let X be a normed space and C a nonempty subset of X. It is clear that i if C is a compact set, then Za C, {xn } in nonempty compact set and so has Property Z; ii if C is a open set, since Za C, {xn } ⊂ ∂C, therefore Za C, {xn } is empty and so fail to have Property Z.
2. Main Results Our new results are presented in this section. Proposition 2.1. Let X be a Banach space and let C be a nonempty closed bounded and convex subset of X. If C satisfies Property Z, then every continuous mapping T : C → C asymptotically admitting a center in
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