Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
Volume 6, Issue 4, Pages 1043-1063Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcdss.2013.6.1043
Keywords
Banach space; Bregman distance; Bregman firmly nonexpansive operator; Bregman strongly nonexpansive operator; Bregman projection; fixed point; iterative algorithm; Legendre function; totally convex function
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Funding
- DGES [MTM2009-13997-C02-01]
- Junta de Andalucia [FQM-127]
- Israel Science Foundation [647/07]
- Graduate School of the Technion
- Fund for the Promotion of Research at the Technion
- Technion President's Research Fund
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Diverse notions of nonexpansive type operators have been extended to the more general framework of Bregman distances in reflexive Banach spaces. We study these classes of operators, mainly with respect to the existence and approximation of their (asymptotic) fixed points. In particular, the asymptotic behavior of Picard and Mann type iterations is discussed for quasi-Bregman nonexpansive operators. We also present parallel algorithms for approximating common fixed points of a finite family of Bregman strongly nonexpansive operators by means of a block operator which preserves the Bregman strong nonexpansivity. All the results hold, in particular, for the smaller class of Bregman firmly nonexpansive operators, a class which contains the generalized resolvents of monotone mappings with respect to the Bregman distance.
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