Right Bregman nonexpansive operators in Banach spaces

Victoria Martín-Márquez; Simeon Reich; Shoham Sabach

doi:10.1016/j.na.2012.04.048

Right Bregman nonexpansive operators in Banach spaces

Victoria Martín-Márquez
, Simeon Reich
, Shoham Sabach

Technion - Israel Institute of Technology

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

Abstract

We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.

Original language	English
Pages (from-to)	5448-5465
Number of pages	18
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	75
Issue number	14
DOIs	https://doi.org/10.1016/j.na.2012.04.048
State	Published - Sep 2012

Keywords

Boltzmann-Shannon entropy
Bregman distance
Bregman firmly nonexpansive operator
Fermi-Dirac entropy
Legendre function
Monotone mapping
Nonexpansive operator
Reflexive Banach space
Resolvent
Retraction
T-monotone mapping
Totally convex function

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2012.04.048

Cite this

@article{c63b935fc7834b1ca5783738513620d4,

title = "Right Bregman nonexpansive operators in Banach spaces",

abstract = "We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.",

keywords = "Boltzmann-Shannon entropy, Bregman distance, Bregman firmly nonexpansive operator, Fermi-Dirac entropy, Legendre function, Monotone mapping, Nonexpansive operator, Reflexive Banach space, Resolvent, Retraction, T-monotone mapping, Totally convex function",

author = "Victoria Mart{\'i}n-M{\'a}rquez and Simeon Reich and Shoham Sabach",

note = "Funding Information: The first author was partially supported by DGES (Grant BFM2009-1096-C02-01 ) and by the Junta de Andaluc{\'i}a (Grant FQM-127 ); the second and third authors were partially supported by the Israel Science Foundation (Grant 647/07 ), the Graduate School of the Technion , the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund . Part of the research pertaining to this paper was carried out when the authors were visiting each other{\textquoteright}s institutions. They thank their colleagues for their kind hospitality. All the authors are very grateful to an anonymous referee for a detailed and insightful report, with many helpful comments and suggestions.",

year = "2012",

month = sep,

doi = "10.1016/j.na.2012.04.048",

language = "English",

volume = "75",

pages = "5448--5465",

journal = "Nonlinear Analysis, Theory, Methods and Applications",

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TY - JOUR

T1 - Right Bregman nonexpansive operators in Banach spaces

AU - Martín-Márquez, Victoria

AU - Reich, Simeon

AU - Sabach, Shoham

N1 - Funding Information: The first author was partially supported by DGES (Grant BFM2009-1096-C02-01 ) and by the Junta de Andalucía (Grant FQM-127 ); the second and third authors were partially supported by the Israel Science Foundation (Grant 647/07 ), the Graduate School of the Technion , the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund . Part of the research pertaining to this paper was carried out when the authors were visiting each other’s institutions. They thank their colleagues for their kind hospitality. All the authors are very grateful to an anonymous referee for a detailed and insightful report, with many helpful comments and suggestions.

PY - 2012/9

Y1 - 2012/9

N2 - We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.

AB - We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.

KW - Boltzmann-Shannon entropy

KW - Bregman distance

KW - Bregman firmly nonexpansive operator

KW - Fermi-Dirac entropy

KW - Legendre function

KW - Monotone mapping

KW - Nonexpansive operator

KW - Reflexive Banach space

KW - Resolvent

KW - Retraction

KW - T-monotone mapping

KW - Totally convex function

UR - https://www.scopus.com/pages/publications/84862981826

U2 - 10.1016/j.na.2012.04.048

DO - 10.1016/j.na.2012.04.048

M3 - Article

SN - 0362-546X

VL - 75

SP - 5448

EP - 5465

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 14

ER -

Right Bregman nonexpansive operators in Banach spaces

Abstract

Keywords

ASJC Scopus subject areas

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