Modified Hybrid Block Iterative Algorithm for Uniformly Quasi‐ϕ‐Asymptotically Nonexpansive Mappings

Abstract

A new hybrid Bregman projection method is considered for finding common solutions of the set of common fixed points of an infinite family of closed, uniformly asymptotic regular and uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings, the set of solutions to a variational inequality problem and the set of common solutions to a system of generalized mixed equilibrium problems, strong convergence theorems of common elements are proved by using new analysis techniques and Bregman mappings in the setting of uniformly smooth and 2-uniformly convex real Banach spaces. Our results improve and generalize many important known recent results in the current literature, because Bregman projection mapping generalizes the generalized projection mapping and the metric projection mapping. c 2016 All rights reserved. Keywords: Hybrid Bregman projection method, Bregman totally quasi-D-asymptotically nonexpansive mapping, variational inequality problem, generalized mixed equilibrium problem, uniformly smooth Banach space, 2-uniformly convex Banach space. 2010 MSC: 47J20, 47H09, 47H10.

Inertial hybrid algorithm for generalized mixed equilibrium problems, zero problems, and fixed points of some nonlinear mappings in the intermediate sense

“…Let S ⊆ H be a nonempty convex, closed, and bounded. Let A ∶ S → H be an 𝜂-hemicontinuous map that satisfies (11), 𝜑 ∶ S × S → R be a bifunction satisfying conditions (F1)-(F4) and let g ∶ S → R ∪ {+∞} be a proper, convex, and lower semicontinuous function. For b > 0 and d ∈ H, define a map T b ∶ H → S as follows;…”

“…Based on the above discussion on the different classes of maps, the class of maps described in (7), seems to be the most general, since various classes can be considered as its special subclasses. Definition 1 (see, e.g., Feng et al [11]). A map T ∶ S → S is known as;…”

“…Since the introduction of the monotone map defined in (11), the attentions of many researchers have been attracted toward the studies of nonlinear problems, such as variational inequality and equilibrium problems involving this type of monotone map. For instance, Chen et al [15], studied the following generalized mixed equilibrium problem (shortly written as GMEP) of finding a point d ∈ S such that…”

“…In particular, for p = 2, A is c-strongly monotone map. Furthermore, a mapping described in ( 9) satisfies (11) with…”