A projection method for solving nonlinear problems in reflexive Banach spaces

Simeon Reich; Shoham Sabach

doi:10.1007/s11784-010-0037-5

A projection method for solving nonlinear problems in reflexive Banach spaces

Simeon Reich, Shoham Sabach

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

We study the convergence of an iterative algorithm for finding common fixed points of finitely many Bregman firmly nonexpansive operators in reflexive Banach spaces. Our algorithm is based on the concept of the so-called shrinking projection method and it takes into account possible computational errors. We establish a strong convergence theorem and then apply it to the solution of convex feasibility and equilibrium problems, and to finding zeroes of two different classes of nonlinear mappings.

Original language	English
Pages (from-to)	101-116
Number of pages	16
Journal	Journal of Fixed Point Theory and Applications
Volume	9
Issue number	1
DOIs	https://doi.org/10.1007/s11784-010-0037-5
State	Published - Mar 2011

Keywords

Banach space
Bregman distance
Bregman firmly nonexpansive operator
Bregman inverse strongly monotone mapping
Bregman projection
Convex feasibility problem
Equilibrium problem
Fixed point
Iterative algorithm
Legendre function
Monotone mapping
Totally convex function

All Science Journal Classification (ASJC) codes

Modelling and Simulation
Geometry and Topology
Applied Mathematics

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10.1007/s11784-010-0037-5

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title = "A projection method for solving nonlinear problems in reflexive Banach spaces",

abstract = "We study the convergence of an iterative algorithm for finding common fixed points of finitely many Bregman firmly nonexpansive operators in reflexive Banach spaces. Our algorithm is based on the concept of the so-called shrinking projection method and it takes into account possible computational errors. We establish a strong convergence theorem and then apply it to the solution of convex feasibility and equilibrium problems, and to finding zeroes of two different classes of nonlinear mappings.",

keywords = "Banach space, Bregman distance, Bregman firmly nonexpansive operator, Bregman inverse strongly monotone mapping, Bregman projection, Convex feasibility problem, Equilibrium problem, Fixed point, Iterative algorithm, Legendre function, Monotone mapping, Totally convex function",

author = "Simeon Reich and Shoham Sabach",

note = "Funding Information: This research was supported by the Israel Science Foundation (Grant 647/07), by the Fund for the Promotion of Research at the Technion and by the Technion President{\textquoteright}s Research Fund. We thank the referee for several helpful comments and corrections.",

year = "2011",

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doi = "10.1007/s11784-010-0037-5",

language = "الإنجليزيّة",

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T1 - A projection method for solving nonlinear problems in reflexive Banach spaces

AU - Reich, Simeon

AU - Sabach, Shoham

N1 - Funding Information: This research was supported by the Israel Science Foundation (Grant 647/07), by the Fund for the Promotion of Research at the Technion and by the Technion President’s Research Fund. We thank the referee for several helpful comments and corrections.

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N2 - We study the convergence of an iterative algorithm for finding common fixed points of finitely many Bregman firmly nonexpansive operators in reflexive Banach spaces. Our algorithm is based on the concept of the so-called shrinking projection method and it takes into account possible computational errors. We establish a strong convergence theorem and then apply it to the solution of convex feasibility and equilibrium problems, and to finding zeroes of two different classes of nonlinear mappings.

AB - We study the convergence of an iterative algorithm for finding common fixed points of finitely many Bregman firmly nonexpansive operators in reflexive Banach spaces. Our algorithm is based on the concept of the so-called shrinking projection method and it takes into account possible computational errors. We establish a strong convergence theorem and then apply it to the solution of convex feasibility and equilibrium problems, and to finding zeroes of two different classes of nonlinear mappings.

KW - Banach space

KW - Bregman distance

KW - Bregman firmly nonexpansive operator

KW - Bregman inverse strongly monotone mapping

KW - Bregman projection

KW - Convex feasibility problem

KW - Equilibrium problem

KW - Fixed point

KW - Iterative algorithm

KW - Legendre function

KW - Monotone mapping

KW - Totally convex function

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DO - 10.1007/s11784-010-0037-5

M3 - مقالة

SN - 1661-7738

VL - 9

SP - 101

EP - 116

JO - Journal of Fixed Point Theory and Applications

JF - Journal of Fixed Point Theory and Applications

IS - 1

ER -

A projection method for solving nonlinear problems in reflexive Banach spaces

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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