Revisiting jointly firmly nonexpansive families of mappings

TL;DR

This paper advances the theoretical understanding of jointly firmly nonexpansive families of mappings by providing new characterizations, improved convergence rates, and analysis of their asymptotic behavior, enhancing the foundation for proximal method convergence analysis.

Contribution

It offers a new characterization via the classical resolvent identity, improves convergence rate results, and studies the asymptotic behavior of these families.

Findings

Characterization in terms of the classical resolvent identity

Improved convergence rate for the uniform case

Analysis of asymptotic behavior at infinity

Abstract

Recently, the author, together with L. Leustean and A. Nicolae, introduced the notion of jointly firmly nonexpansive families of mappings in order to investigate in an abstract manner the convergence of proximal methods. Here, we further the study of this concept, by giving a characterization in terms of the classical resolvent identity, by improving on the rate of convergence previously obtained for the uniform case, and by giving a treatment of the asymptotic behaviour at infinity of such families.