Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces

Abstract

In this paper, we introduce a new iterative scheme by a hybrid method and prove a strong convergence theorem of a common element in the set of fixed points of a finite family of closed quasi-Bregman strictly pseudocontractive mappings and common solutions to a system of equilibrium problems in reflexive Banach space. Our results extend important recent results announced by many authors. MSC: 47H09; 47J25

Split common fixed point problem for Bregman demigeneralized mappings in Banach spaces with applications

“…(i). [26] Let C be a nonempty closed convex subset of reflexive Banach space E. Let k be a real number in (0, 1), the map T : C → E is called quasi-Bregman strictly pseudocontractive mapping if F(T ) = ∅, ∀x ∈ C and p ∈ F(T ),…”

“…A map T : C → C is called quasi-Bregman relatively nonexpansive [16] if F(T ) = ∅,F(T ) = F(T ) and D f (T x, p) D f (x, p) for all x ∈ C and p ∈ F(T ). T is said to be quasi-Bregman strictly pseudocontractive [26] if there exists a constant λ ∈ [0, 1) and…”

“…In 1967, Bregman [6] introduced an effective technique using the Bregman distance function D f for designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman's technique is applied in various ways in order to design and analyze iterative algorithm for solving not only feasibility and optimization problems, but also algorithms for solving fixed point problems for nonlinear mappings (see, e.g., [1,8,15,20,21,26] and the references therein).…”

Some results on a finite family of Bregman quasi-strict pseudo-contractions

“…(4) T is said to be Bregman quasi-strictly pseudo-contractive [20] if there exists a constant k ∈ [0, 1) and…”

“…Since then, many authors obtained strong convergence theorems for nonlinear operators based on the generalized projections in Banach spaces; see [14,22] and the references therein. Another way is to use the Bregman distance instead of the norm, Bregman (quasi-)nonexpansive mappings instead of the (quasi-)nonexpansive mappings and the Bregman projection instead of the metric projection; see [18,20,21] and the references therein.…”