Nonexpansive maps with surjective displacement

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 − 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.