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ON THE CONVERGENCE RATE OF THE KRASNOSEL’SKIĬ–MANN ITERATION

Part of: Nonlinear operators and their properties Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  05 January 2017

SHIN-YA MATSUSHITA
SHIN-YA MATSUSHITA*
Affiliation:
Department of Electronics and Information Systems, Akita Prefectural University, Yuri-Honjo, Akita 015-0055, Japan email matsushita@akita-pu.ac.jp
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Abstract

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The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big-$O$ rate to little-$o$ without any other restrictions. The proof relies on the connection between the KM iteration and a useful technique on the convergence rate of summable sequences. Then we apply the result to give new results on convergence rates for the proximal point algorithm and the Douglas–Rachford method.

Keywords

Krasnosel’skiĭ–Mann iterationconvergence ratenonexpansiveproximal point algorithmDouglas–Rachford method

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Secondary: 47H05: Monotone operators and generalizations 47H10: Fixed-point theorems 47J25: Iterative procedures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 162 - 170
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2017 Australian Mathematical Publishing Association Inc.

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