Fixed point properties and Q-nonexpansive retractions in locally convex spaces

TL;DR

This paper investigates fixed point properties and the existence of Q-nonexpansive retractions in locally convex spaces, extending fixed point results to semigroups and addressing Bruck's problem.

Contribution

It proves the weak fixed point property for commuting semigroups of Q-nonexpansive mappings in locally convex spaces, solving Bruck's problem.

Findings

Weak fixed point property extends to commuting semigroups

Existence of Q-nonexpansive retractions for amenable semigroups

Addresses Bruck's problem in locally convex spaces

Abstract

Suppose that Q is a family of seminorms on a locally convex space E which determines the topology of E. We study the existence of Q-nonexpansive retractions for families of Q-nonexpansive mappings and prove that a separated and sequentially complete locally convex space $E$ that has the weak fixed point property, has the weak fixed point property for commuting separable semitopological semigroups of Q-nonexpansive mappings. This proves the Bruck's problem [5] for locally convex spaces. Moreover, we prove the existence of Q-nonexpansive retractions for the right amenable Q-nonexpansive semigroups.