Super Asymptotically Nonexpansive Actions of Semitopological Semigroups on Frechet and Locally Convex Spaces

TL;DR

This paper proves the existence of common fixed points for super asymptotically nonexpansive actions of certain semitopological semigroups on Fréchet and locally convex spaces, extending fixed point theory in these settings.

Contribution

It establishes fixed point results for super asymptotically nonexpansive actions of semitopological semigroups on Fréchet and locally convex spaces, generalizing previous fixed point theorems.

Findings

Every jointly continuous, super asymptotically nonexpansive action has a common fixed point.

Results extend fixed point theory to broader classes of spaces and actions.

Applicable to semitopological semigroups with invariant means on LUC functions.

Abstract

Let LUC$(S)$ be the space of left uniformly continuous functions on a semitopological semigroup $S$. Suppose that $S$ is right reversible and $\operatorname{LUC}(S)$ has a left invariant mean. Let $(X,d)$ be a Fr\'echet space. Let $\tau$ be a locally convex topology of $X$ weaker than the $d$-topology such that the metric $d$ is $\tau$-lower semicontinuous. Let $K$ be a $d$--separable and $\tau$--compact convex subset of $X$. We show that every jointly $\tau$-continuous and super asymptotically $d$-nonexpansive action $S\times K\mapsto K$ of $S$ has a common fixed point. Similar results in the locally convex space setting are provided.