Applications of uniform asymptotic regularity to fixed point theorems

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.

11 pages, to appear, Journal of Fixed Point Theory and Applications. Referee suggestions incorporated and some new references added

References in corpus (1)

• A common fixed point theorem for a commuting family of weak${}^{\ast }$ continuous nonexpansive mappings