Applications of uniform asymptotic regularity to fixed point theorems

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 $τ$-compact subset of a Banach space has a common fixed point, and give a qualitative complement to the Markov-Kakutani theorem.

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