The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings

TL;DR

This paper investigates the long-term behavior of compositions of firmly nonexpansive mappings in curved metric spaces, providing insights into their convergence properties for convex minimization problems.

Contribution

It offers a unified analysis of asymptotic behavior of such mappings in $p$-uniformly convex geodesic spaces, improving understanding of convergence in these settings.

Findings

Analysis of asymptotic regularity of compositions

Convergence results for firmly nonexpansive mappings

Application to convex minimization in curved spaces

Abstract

In this paper we provide a unified treatment of some convex minimization problems, which allows for a better understanding and, in some cases, improvement of results in this direction proved recently in spaces of curvature bounded above. For this purpose, we analyze the asymptotic behavior of compositions of finitely many firmly nonexpansive mappings in the setting of $p$-uniformly convex geodesic spaces focusing on asymptotic regularity and convergence results.