Applications of uniform asymptotic regularity to fixed point theorems

TL;DR

This paper investigates the properties of uniformly asymptotically regular mappings and their implications for fixed point theorems, establishing new results on the existence of common fixed points in certain semigroup actions.

Contribution

It demonstrates the nonexistence of nontrivial surjective uniformly asymptotically regular mappings and derives fixed point results for semigroups generated by firmly nonexpansive mappings.

Findings

No nontrivial surjective uniformly asymptotically regular mappings exist.

Semigroups generated by firmly nonexpansive mappings have common fixed points.

Provides a qualitative extension to the Markov-Kakutani theorem.

Abstract

We show that there are no nontrivial surjective uniformly asymptotically regular mappings acting on a metric space and derive some consequences of this fact. In particular, we prove that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings on a bounded $\tau$-compact subset of a Banach space has a common fixed point, and give a qualitative complement to the Markov-Kakutani theorem.