In this paper we study the relation between invariant submean and normal structure in a Banach space. This is used to give an improvement and different proof of a fixed point theorem of Lim (also of Belluce and Kirk for commutative semigroups) for left reversible semigroup of nonexpansive mappings on weakly compact convex subsets of a Banach space with normal structure.
1996 Academic Press, Inc.
0. Introduction
Let S be a semitopological semigroup, i.e., S is a semigroup with Hausdorff topology such that for each a # S, the mappings s ร sa and s ร as from S into S are continuous. Let RUC(S) denote the space of bounded right uniformly continuous real-valued functions on S; S is called left reversible if any two closed right ideals of S have non-void intersection. A closed convex subset K of a Banach space E has normal structure [6, p. 39] if for each bounded closed convex subset H of K which contains more than one point, there is a point x # H which is not a diametral point of H, i.e., sup[&x& y& : y # H]<$(H), where $(H)= the diameter of H.
Belluce and Kirk [3] first proved that if K is a nonempty weakly compact convex subset of a Banach space and if K has complete normal structure, then every family of commuting nonexpansive self-maps on K has a common fixed point. Later Lim [14, Theorem 3] extended this theorem to article no. 0144