Subinvariant metric functionals for nonexpansive mappings

TL;DR

This paper explores the existence and properties of subinvariant metric functionals for commuting nonexpansive mappings in Banach spaces, highlighting their role in fixed point theory and identifying conditions for unique minimizers.

Contribution

It introduces the concept of subinvariant metric functionals for nonexpansive mappings and demonstrates their significance in fixed point existence in Banach spaces.

Findings

Metric functionals are practical tools for fixed point searches.

In certain Banach subsets, every metric functional has a unique minimizer.

Subinvariance of metric functionals implies the existence of fixed points.

Abstract

We investigate the existence of subinvariant metric functionals for commuting families of nonexpansive mappings in noncompact subsets of Banach spaces. Our findings underscore the practicality of metric functionals when searching for fixed points of nonexpansive mappings. To demonstrate this, we additionally investigate subsets of Banach spaces that have only nontrivial metric functionals. We particularly show that in certain cases every metric functional has a unique minimizer; thus, subinvariance implies the existence of a fixed point.