Relation between Fixed Point and Asymptotical Center of Nonexpansive Maps
© Mohammad Reza Haddadi et al. 2011
Received: 19 October 2010
Accepted: 22 November 2010
Published: 2 December 2010
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.
Many topics and techniques regarding asymptotic centers and asymptotic radius were studied by Edelstein , Bose and Laskar , Downing and Kirk , Goebel and Kirk , and Lan and Webb . Now, We recall that definitions of asymptotic center and asymptotic radius.
We present new definitions of asymptotic center and asymptotic radius that is for a mapping and obtain new results.
2. Main Results
Our new results are presented in this section.
Concerning the first case we proceed as follows.
Proposition 2.1 (Theorem 2.2) is a generalizations of Theorem 3.1 (Theorem 3.3) in . 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.
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  to obtain a fixed point for and hence for .
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 .
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 .
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