Fixed point theorems of various nonexpansive actions of semitopological semigroups on weakly/weak* compact convex sets

TL;DR

This paper establishes fixed point theorems for nonexpansive actions of semitopological semigroups on weakly/weak* compact convex sets, extending existing results under various conditions.

Contribution

It introduces new fixed point results for semigroup actions on convex sets, considering weak and weak* topologies, and explores conditions like RNP and distality.

Findings

Existence of common fixed points under certain conditions.

Extensions to weak* compact convex sets with RNP.

Fixed point results for super asymptotically nonexpansive actions.

Abstract

Let $S$ be a right reversible semitopological semigroup, and let $\operatorname{LUC}(S)$ be the space of left uniformly continuous functions on $S$. Suppose that $\operatorname{LUC}(S)$ has a left invariant mean. Let $K$ be a weakly compact convex subset of a Banach space. We show that there always exists a common fixed point for any jointly weakly continuous and super asymptotically nonexpansive action of $S$ on $K$. Several variances involving the weak* compactness, the RNP, the distality of $K$ and/or the left reversibility of $S$ are also provided.