On the minimal displacement vector of compositions and convex combinations of nonexpansive mappings

TL;DR

This paper extends the analysis of the minimal displacement vector from firmly nonexpansive mappings to general averaged nonexpansive mappings, providing new insights into their compositions and convex combinations.

Contribution

It introduces a novel proof technique based on the Brezis-Haraux theorem and reflected resolvents to generalize existing results in the field.

Findings

Extended minimal displacement vector results to averaged nonexpansive mappings.

Provided examples demonstrating the tightness of the theoretical bounds.

Connected displacement vectors of composite mappings to those of individual operators.

Abstract

Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. Five years ago, it was shown that if finitely many firmly nonexpansive mappings have or "almost have" fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis-Haraux theorem and reflected resolvents, we extend these results from firmly nonexpansive to general averaged nonexpansive mappings. Various examples illustrate the tightness of our results.