{\sigma}-Porosity of the set of strict contractions in a space of non-expansive mappings

TL;DR

This paper proves that within the space of non-expansive mappings on a Banach space, the subset of strict contractions is {}-porous, indicating it is small or negligible in a topological sense.

Contribution

It establishes that the set of strict contractions is {}-porous in the space of non-expansive mappings, highlighting its topological insignificance.

Findings

The set of strict contractions is {}-porous.

Strict contractions form a negligible subset in the space of non-expansive mappings.

The result applies to mappings on bounded, closed, convex subsets of Banach spaces.

Abstract

We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uni- form convergence. We show that the set of strict contractions is a {\sigma}-porous subset.