An iterative process for a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems in reflexive Banach spaces

The equilibrium problem and its generalizations had a great influence in the development of some branches of pure and applied sciences. The equilibrium problems theory provides a natural and novel approach for some problems arising in nonlinear analysis, physics and engineering, image reconstruction, economics, finance, game theory and optimization. In recent times, there were many methods in order to solve the equilibrium problem and its generalizations. Some authors proposed many iterative methods and studied the convergence of such iterative methods for equilibrium problems and nonexpansive mappings in the setting of Hilbert spaces and Banach spaces. Note that a generalized mixed equilibrium problem is a generalization of an equilibrium problem and a Bregman totally quasi-asymptotically nonexpansive mapping is a generalization of a nonexpansive mapping in reflexive Banach spaces. The purpose of this paper is to combine the parallel method with the Bregman distance and the Bregman projection in order to introduce a new parallel hybrid iterative process which is to find common solutions of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems. After that, we prove that the proposed iteration strongly converges to the Bregman projection of initial element on the intersection of common fixed point set of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and the solution set of a system of generalized mixed equilibrium problems in reflexive Banach spaces. As application, we obatin some strong convergence results for a Bregman totally quasi-asymptotically nonexpansive mapping and a generalized mixed equilibrium problem in reflexive Banach spaces. These results are extensions and improvements to the main results in [7, 8]. In addition, a numerical example is provided to illustrate for the obtained result.

An inertial iterative algorithm for generalized equilibrium problems and Bregman relatively nonexpansive mappings in Banach spaces

The aim of this paper is to introduce and study an inertial hybrid iterative method for solving generalized equilibrium problems involving Bregman relatively nonexpansive mappings in Banach spaces. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational example to emphasize the efficiency and relevancy of main result.

Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems

<p style='text-indent:20px;'>In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.</p>