Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces

TL;DR

This paper analyzes the convergence rates of an inexact Krasnosel'skii-Mann iteration in Banach spaces, providing new bounds and extending results to continuous-time evolution equations.

Contribution

It introduces a new metric bound for fixed-point residuals, establishing convergence rates for inexact iterations and their variants in Banach spaces.

Findings

Established a new convergence rate bound for inexact Krasnosel'skii-Mann iterations.

Extended convergence results to continuous-time nonautonomous evolution equations.

Applied results to variants like approximate projections, Ishikawa, and diagonal schemes.

Abstract

We study the convergence of an inexact version of the classical Krasnosel'skii-Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point residuals, from which we derive their rate of convergence as well as the convergence of the iterates towards a fixed point. The results are applied to three variants of the basic iteration: infeasible iterations with approximate projections, the Ishikawa iteration, and diagonal Krasnosels'kii-Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.