Abstract
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.
Highlights
It is well known that many nonlinear problems can be reduced to the search for fixed points of nonlinear operators
In the present paper we study a generalization of Algorithm (7) in the more general framework of Bregman distances
In particular, that any Bregman strongly nonexpansive operator is asymptotically regular
Summary
It is well known that many nonlinear problems can be reduced to the search for fixed points of nonlinear operators. Kikkawa and Takahashi [23] have recently applied method (6) to the problem of finding common fixed points of a finite family of nonexpansive mappings in Banach spaces They studied the algorithm xn+1 = wni αni xn + 1 − αni Tixn ,. In the present paper we study a generalization of Algorithm (7) in the more general framework of Bregman distances In this connection we introduce the block operator corresponding to a finite family of Bregman operators of nonexpansive type and prove several results concerning the relations between the common fixed points of the family and the block operator. The third and the fourth sections are devoted to the analysis of Picard and Mann iterations, respectively In these two sections we prove convergence results for Bregman nonexpansive operators. But not least, section (Section 6) we introduce the block operator and prove several results concerning approximating fixed points of such operators
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
