Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

TL;DR

This paper proves that compositions and convex combinations of asymptotically regular firmly nonexpansive mappings in Hilbert spaces are also asymptotically regular, extending previous results and using advanced operator theory techniques.

Contribution

It generalizes Bauschke's 2003 result to a broader class of mappings by showing compositions and convex combinations preserve asymptotic regularity.

Findings

Compositions of asymptotically regular firmly nonexpansive mappings are asymptotically regular.

Convex combinations of such mappings also maintain asymptotic regularity.

The proof uses Hilbert product spaces and the Brezis-Haraux range approximation.

Abstract

Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors (nearest point mappings). The proof resides in a Hilbert product space and it relies upon the Brezis-Haraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex combinations.