In this paper, we prove the existence of a common fixed point in $C(K)$, the Chebyshev center of K, for a left reversible semigroup of isometry mappings. This existence result improves the results obtained by Lim et al. and Brodskii and Milman.

Fixed points of left reversible semigroup of isometry mappings in Banach spaces

In this paper, we derive a fixed-point theorem for self-mappings. That is, it is shown that every isometric self-mapping on a weakly compact convex subset of a strictly convex Banach space has a fixed point.

/pdf/fixed-point-theorem-for-isometric-self-mappings-31guom8272.pdf

Fixed-Point Theorem for Isometric Self-Mappings

Fixed point theorems of semigroup of isometry mappings and α-nonexpansive mappings on weak⁎ compact convex sets

In this article, we prove that if K is a nonempty weakly compact convex set in a Banach space such that K has the hereditary fixed-point property (FPP) and F is a commuting family of isometry mappings on K, then there exists a point in C(K) which is fixed by every member in F whenever C(K) is a compact set. Also, we give an example to show that C(K), the Chebyshev center of K, need not be invariant under isometry maps. This example answers the question as to whether the Chebyshev center is invariant under isometry maps. Furthermore, we give a simple example to illustrate that Lim’s center, as introduced by Lim, is different from the Chebyshev center.

Lim’s center and fixed-point theorems for isometry mappings

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Lim’s center and fixed-point theorems for isometry mappings