On the Rate of Convergence of Krasnosel’skǐi–mann Iterations and Their Connection with Sums of Bernoullis

In this paper we establish an estimate for the rate of convergence of the Krasnosel’skǐı–Mann iteration for computing fixed points of non-expansive maps. Our main result settles the Baillon–Bruck conjecture [3] on the asymptotic regularity of this iteration. The proof proceeds by establishing a connection between these iterates and a stochastic process involving sums of non-homogeneous Bernoulli trials. We also exploit a new Hoeffdingtype inequality to majorize the expected value of a convex function of these sums using Poisson distributions. ∗ Supported by Fondecyt 1100046 and Núcleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F. ∗∗ Supported by Basal-Conicyt project and Núcleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F. Received July 6, 2012 and in revised form November 27, 2012