TY - JOUR

T1 - Rates of convergence for inexact Krasnosel’skii–Mann iterations in Banach spaces

AU - Bravo, Mario

AU - Cominetti, Roberto

AU - Pavez-Signé, Matías

N1 - Funding Information:
This work was partially supported by Núcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003. Mario Bravo was partially funded by FONDECYT 11151003. Roberto Cominetti and Matías Pavez-Signé gratefully acknowledge the support provided by FONDECYT 1130564 and FONDECYT 1171501.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2019/5/1

Y1 - 2019/5/1

N2 - We study the convergence of an inexact version of the classical Krasnosel’skii–Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point 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’kii–Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.

AB - We study the convergence of an inexact version of the classical Krasnosel’skii–Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point 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’kii–Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators.

KW - Evolution equations

KW - Fixed point iterations

KW - Nonexpansive maps

KW - Rates of convergence

UR - http://www.scopus.com/inward/record.url?scp=85041127566&partnerID=8YFLogxK

U2 - 10.1007/s10107-018-1240-1

DO - 10.1007/s10107-018-1240-1

M3 - Article

AN - SCOPUS:85041127566

VL - 175

SP - 241

EP - 262

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -