Convergence analysis for system of equilibrium problems and left bregman strongly relatively nonexpansive mapping

In this paper, we prove strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces. We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.     

Parallel iterative methods for solving systems of generalized mixed equilibrium problems in reflexive Banach spaces

The purpose of this paper is to introduce three parallel iterative methods which use techniques of Bregman distances, Bregman projections, Bregman strongly nonexpansive operators and hybrid or shrinking projection methods to solve systems of generalized mixed equilibrium problems in a real reflexive Banach space.     

Approximating a Zero of Sum of Two Monotone Operators Which Solves a Fixed Point Problem in Reflexive Banach Spaces

Abstract

The purpose of this paper is to introduce an iterative method for approximating a point in the set of zeros of the sum of two monotone mappings, which is also a solution of a fixed point problem for a Bregman strongly nonexpansive mapping in a real reflexive Banach space. With our iterative technique, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a variational inclusion problem for sum of two monotone mappings and the set of solutions of a fixed point problem for Bregman strongly nonexpansive mapping. We give applications of our result to convex minimization problem, convex feasibility problem, variational inequality problem, and equilibrium problem. Our result complements and extends some recent results in literature.     

可数族弱Bregman相对非扩展映像的收敛性分析

在自反Banach空间中,引入可数族弱Bregman相对非扩张映像概念,构造了两种迭代算法求解可数族弱Bregman相对非扩张映像的公共不动点.在适当条件下,证明了两种迭代算法产生的序列的强收敛性.     

A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance

In this paper, using a hybrid extragradient method, we introduce a new iterative process for approximating a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. For this purpose, we introduce Bregman–Lipschitz-type condition for a pseudomonotone bifunction. It seems that these results for pseudomonotone bifunctions are first in reflexive Banach spaces. This paper concludes with certain applications, where we utilize our results to study the determination of a common point of the solution set of a variational inequality problem and the fixed point set of a finite family of multi-valued relatively nonexpansive mappings. A numerical example to support our main theorem will be exhibited.     

STRONG CONVERGENCE THEOREMS FOR FINITE EQUILIBRIUM PROBLEMS AND BREGMAN TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES

In this paper, we propose a new hybrid iterative scheme for finding a common solution to a finite of equilibrium problems and fixed point of Bregman totally quasiasymptotically nonexpansive mapping in reflexive Banach spaces. Moreover, we prove some strong convergence theorems under suitable control conditions.     

Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces

In this paper, we introduce and study a hybrid iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem for a Bregman relatively nonexpansive mapping in reflexive Banach spaces. We prove that the sequences generated by the hybrid iterative algorithm converge strongly to a common solution of these problems. Further, we give some consequences of the main result. Finally, we discuss a numerical example to demonstrate the applicability of the iterative algorithm.     

Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces

We study the convergence of two iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in reflexive Banach spaces. Both algorithms take into account possible computational errors. We establish two strong convergence theorems and then apply them to the solution of convex feasibility, variational inequality and equilibrium problems.     

Two Projection Algorithms for Solving the Split Common Fixed Point Problem

Journal of Optimization Theory and Applications - We study the split common fixed point problem for Bregman relatively nonexpansive operators in real reflexive Banach spaces. Using Bregman...     

A strong convergence theorem for solving the split feasibility and fixed point problems in Banach spaces

In this paper, we introduce a new parallel iterative method for finding a common solution of the multiple-set split feasibility and fixed point problems concerning left Bregman strongly nonexpansive mappings in Banach spaces.     

Parallel iterative methods for Bregman strongly nonexpansive operators in reflexive Banach spaces

The purpose of this paper is to introduce two parallel iterative methods which are based on the hybrid projection and shrinking projection methods to find a common fixed point of a finite family of Bregman strongly nonexpansive operators in a reflexive Banach space X. And we also give some applications of main results for some related problems.     

Strong Convergence Theorems for a Countable Family of Multi-Valued Bregman Quasi-Nonexpansive Mappings in Reflexive Banach Spaces

This article uses the shrinking projection method introduced by Takahashi, Kubota and Takeuchi to propose an iteration algorithm for a countable family of Bregman multi-valued quasi-nonexpansive mappings in order to have the strong convergence under a limit condition in the framework of reflexive Banach spaces. We apply our results to a zero point problem of maximal monotone mappings and equilibrium problems in reflexive Banach spaces. The results presented in the article improve and extend the corresponding results of that found in the literature.     

STRONG CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEM AND BREGMAN TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES

In this paper, we propose a new hybrid iterative scheme for finding a common solution of an equilibrium problem and fixed point of Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces. Moreover, we prove some strong convergence theorems under suitable control conditions. Finally, the application to zero point problem of maximal monotone operators is given by the result.     

A proximal augmented Lagrangian method for equilibrium problems

Considering a recently proposed proximal point method for equilibrium problems, we construct an augmented Lagrangian method for solving the same problem in reflexive Banach spaces with cone constraints generating a strongly convergent sequence to a certain solution of the problem. This is an inexact hybrid method meaning that at a certain iterate, a solution of an unconstrained equilibrium problem is found, allowing a proper error bound, followed by a Bregman projection of the initial iterate onto the intersection of two appropriate halfspaces. Assuming a set of reasonable hypotheses, we provide a full convergence analysis.     

Convergence theorem for equilibrium problem and Bregman strongly nonexpansive mappings in Banach spaces

In this paper, we present an iterative scheme for Bregman strongly nonexpansive mappings in the framework of Banach spaces. Furthermore, we prove the strong convergence theorem for finding common fixed points with the set of solutions of an equilibrium problem.     

Right Bregman nonexpansive operators in Banach spaces

We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.     

A projection method for solving nonlinear problems in reflexive Banach spaces

We study the convergence of an iterative algorithm for finding common fixed points of finitely many Bregman firmly nonexpansive operators in reflexive Banach spaces. Our algorithm is based on the concept of the so-called shrinking projection method and it takes into account possible computational errors. We establish a strong convergence theorem and then apply it to the solution of convex feasibility and equilibrium problems, and to finding zeroes of two different classes of nonlinear mappings.     

An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces

The purpose of this paper is to study split feasibility problems and fixed point problems concerning left Bregman strongly relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We suggest an iterative scheme for the problem and prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Finally, we give numerical examples of our result to study its efficiency and implementation. Our result complements many recent and important results in this direction.     

Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces

In this paper, a new system of parametric generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings in real Banach spaces is introduced and studied. We first generalize the notion of the Yosida approximation in Hilbert spaces introduced by Moudafi to reflexive Banach spaces. Further, by using the notion of the Yosida approximation, we consider a system of parametric generalized Wiener-Hopf equation problems and show its equivalence to the system of parametric generalized mixed implicit equilibrium problems. By using a fixed point formulation of the system of parametric generalized Wiener-Hopf equation problems, we study the behavior and sensitivity analysis of a solution set of the system of parametric generalized mixed implicit equilibrium problems. We prove that, under suitable assumptions, the solution set of the system of parametric generalized mixed implicit equilibrium problems is nonempty, closed and Lipschitz continuous with respect to the parameters. Our results are new, and improve and generalize some known results in this field.     

A new iterative scheme for a countable family of relatively nonexpansive mappings and an equilibrium problem in Banach spaces

In this paper, we construct a new iterative scheme and prove strong convergence theorem for approximation of a common fixed point of a countable family of relatively nonexpansive mappings, which is also a solution to an equilibrium problem in a uniformly convex and uniformly smooth real Banach space. We apply our results to approximate fixed point of a nonexpansive mapping, which is also solution to an equilibrium problem in a real Hilbert space and prove strong convergence of general H-monotone mappings in a uniformly convex and uniformly smooth real Banach space. Our results extend many known recent results in the literature.