On the rate of convergence of Krasnoselski-Mann iterations and their connection with sums of Bernoullis
Settles a long-standing open conjecture in fixed point theory, providing a definitive convergence rate for a fundamental iterative method.
The paper proves the Baillon-Bruck conjecture on the asymptotic regularity of the Krasnoselski-Mann iteration for non-expansive maps, establishing an explicit rate of convergence.
In this paper we establish an estimate for the rate of convergence of the Krasnosel'ski\vı-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 Hoeffding-type inequality to majorize the expected value of a convex function of these sums using Poisson distributions.