Bravo, M., Cominetti, R., & Pavez-Signe, M. (2019). Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces. Math. Program., 175(1-2), 241–262.
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.