Bravo, M., & Cominetti, R. (2018). Sharp convergence rates for averaged nonexpansive maps. Isr. J. Math., 227(1), 163–188.
Abstract: We establish sharp estimates for the convergence rate of the Kranosel'skiAMann fixed point iteration in general normed spaces, and we use them to show that the optimal constant of asymptotic regularity is exactly . To this end we consider a nested family of optimal transport problems that provide a recursive bound for the distance between the iterates. We show that these bounds are tight by building a nonexpansive map T: [0, 1](N) > [0, 1](N) that attains them with equality, settling a conjecture by Baillon and Bruck. The recursive bounds are in turn reinterpreted as absorption probabilities for an underlying Markov chain which is used to establish the tightness of the constant 1/root pi.

Bravo, M., Cominetti, R., & PavezSigne, M. (2019). Rates of convergence for inexact Krasnosel'skiiMann iterations in Banach spaces. Math. Program., 175(12), 241–262.
Abstract: We study the convergence of an inexact version of the classical Krasnosel'skiiMann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixedpoint 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'kiiMann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.
