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Iterative methods for approximating fixed points of Bregman nonexpansive operators. (English) Zbl 1266.26023

Summary: 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.

MSC:

26B25 Convexity of real functions of several variables, generalizations
46T99 Nonlinear functional analysis
47H04 Set-valued operators
47H05 Monotone operators and generalizations
47J25 Iterative procedures involving nonlinear operators
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