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On the rate of convergence of Krasnosel’skiĭ-Mann iterations and their connection with sums of Bernoullis

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Abstract

In this paper we establish an estimate for the rate of convergence of the Krasnosel’skiĭ-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.

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Authors and Affiliations

  1. Departamento Ingeniería Industrial, Universidad de Chile, República 701, Santiago, Chile

    R. Cominetti

  2. Departamento Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Blanco Encalada 2120, Santiago, Chile

    J. A. Soto

  3. Departamento de Ingeniería Matemática, Universidad de Chile, Blanco Encalada 2120, Santiago, Chile

    J. Vaisman

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  1. R. Cominetti
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  2. J. A. Soto
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  3. J. Vaisman
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Correspondence to R. Cominetti.

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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 Coordinaci ón en Redes ICM/FIC P10-024F.

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Cominetti, R., Soto, J.A. & Vaisman, J. On the rate of convergence of Krasnosel’skiĭ-Mann iterations and their connection with sums of Bernoullis. Isr. J. Math. 199, 757–772 (2014). https://doi.org/10.1007/s11856-013-0045-4

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  • DOI: https://doi.org/10.1007/s11856-013-0045-4

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