Nonexpansive maps with surjective displacement

TL;DR

This paper characterizes when nonexpansive maps on Banach spaces have surjective displacement, providing computable conditions especially in finite-dimensional polyhedral norm spaces, with applications to fixed point theory.

Contribution

It offers new necessary and sufficient computable conditions for surjective displacement and fixed point uniqueness in polyhedral norm Banach spaces.

Findings

Provided a computable condition for surjective displacement in finite-dimensional polyhedral spaces.

Established criteria for the uniqueness of fixed points for nonexpansive maps.

Extended results to applications in nonlinear Perron-Frobenius theory.

Abstract

We investigate necessary and sufficient conditions for a nonexpansive map $f$ on a Banach space $X$ to have surjective displacement, that is, for $f - \mathrm{id}$ to map onto $X$. In particular, we give a computable necessary and sufficient condition when $X$ is a finite dimensional space with a polyhedral norm. We give a similar computable necessary and sufficient condition for a fixed point of a polyhedral norm nonexpansive map to be unique. We also consider applications to nonlinear Perron-Frobenius theory and suggest some additional computable sufficient conditions for surjective displacement and uniqueness of fixed points.