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

Abstract

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 $\mathfrak{F}$ is a commuting family of isometry mappings on $K$, then there exists a point in $C\left(K\right)$ which is fixed by every member in $\mathfrak{F}$ whenever $C\left(K\right)$ is a compact set. Also, we give an example to show that $C\left(K\right)$, 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.