Abstract
We prove a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces. Furthermore, we apply our method to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our theorems complement some recent results that have been proved for this important class of nonlinear mappings.
Highlights
In this paper, without other specifications, let E be a real reflexive Banach space and E∗ as its dual, let R be the set of real numbers, and let C be a nonempty, closed, and convex subset E
We prove a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces
Let C be a nonempty closed convex subset of int(dom f) and let T : C → int(dom f) be a right quasi-Bregman nonexpansive mapping
Summary
Without other specifications, let E be a real reflexive Banach space and E∗ as its dual, let R be the set of real numbers, and let C be a nonempty, closed, and convex subset E. If C is a nonempty and closed subset of int(dom f), where f is a Legendre and Frechet differentiable function, and T : C → int(dom f) is a right Bregman strongly nonexpansive mapping, it is proved that F(T) is closed (see [30]) They have shown that this class of mappings is closed under composition and convex combination and proved weak convergence of the Picard iterative method to a fixed point of a mapping under suitable conditions (see [31]). Our results complements the recent results due to Reich and Sabach [9], Suantai et al [32], and Zhang and Cheng [33] in the sense that our scheme is applicable for right Bregman strongly nonexpansive self-mappings on C ⊆ E
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