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

TL;DR

This paper proves that a commuting family of weak* continuous nonexpansive mappings on a weak* compact convex subset of a dual Banach space has a nonempty set of common fixed points, which forms a nonexpansive retract, addressing a long-standing open problem.

Contribution

It establishes the existence of a nonempty fixed point set and a nonexpansive retract for commuting weak* continuous nonexpansive mappings, advancing fixed point theory for semigroups.

Findings

Set of common fixed points is nonempty.

The fixed point set is a nonexpansive retract.

Partially solves an open problem in metric fixed point theory.

Abstract

It is shown that if $S$ is a commuting family of weak$^{\ast }$ continuous nonexpansive mappings acting on a weak$^{\ast }$ compact convex subset $C$ of the dual Banach space $E$, then the set of common fixed points of $S$ is a nonempty nonexpansive retract of $C$. This partially solves a long-standing open problem in metric fixed point theory in the case of commutative semigroups.