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.     

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.     

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.     

Bregman distances and Chebyshev sets

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.     

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.     

A Bregman projection method for approximating fixed points of quasi-Bregman nonexpansive mappings

We introduce an abstract algorithm that aims to find the Bregman projection onto a closed convex set. As an application, the asymptotic behavior of an iterative method for finding a fixed point of a quasi-Bregman nonexpansive mapping with the fixed-point closedness property is analyzed. We also show that our result is applicable to Bregman subgradient projectors.     

Inexact Iterative Bregman Method for Optimal Control Problems

In this article, we investigate an inexact iterative regularization method based on generalized Bregman distances of an optimal control problem with control constraints. We show robustness and convergence of the inexact Bregman method under a regularity assumption, which is a combination of a source condition and a regularity assumption on the active sets. We also take the discretization error into account. Numerical results are presented to demonstrate the algorithm.     

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.     

Strong convergence theorem by hybrid method for equilibrium problems,variational inequality problems and maximal monotone operators

We introduce an iterative scheme for finding a common element of the solution set of the equilibrium problem, the solution set of the variational inequality problem for an inverse-strongly-monotone operators and the solution set of a maximal monotone operator in a 2-uniformly convex and uniformly smooth Banach space, and then we present strong convergence theorems which generalize the results of many others.     

Banach空间中两个极大单调算子公共零点的迭代格式

令E为实光滑、一致凸Banach空间,E~*为其对偶空间.令A,B(?)E×E~*为极大单调算子且A~(-1)∩B~(-1)0≠(?).本文将引入新的迭代格式,利用Lyapunov泛函与广义投影算子等技巧,证明迭代序列弱收敛于极大单调算子A和B的公共零点.     

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.     

Convergence theorems of a hybrid algorithm for equilibrium problems

In this paper, we introduce a hybrid iterative scheme for finding a common element of the set of common fixed points of two hemi-relatively non-expansive mappings and the set of solutions of an equilibrium problem by the CQ hybrid method in Banach spaces. Our results improve and extend the corresponding results announced by Cheng and Tian [Y. Cheng, M. Tian, Strong convergence theorem by monotone hybrid algorithm for equilibrium problems, hemi-relatively nonexpansive mappings and maximal monotone operators, Fixed Point Theory Appl. 2008 (2008) 12 pages, doi:10.1155/2008/617248], Takahashi and Zembayashi [W. Takahashi, K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively non-expansive mappings, Fixed Point Theory Appl. (2008) doi:10.1155/2008/528476] and some others.     

极大单调算子和相对非扩展映射的修正杂交迭代算法

在实一致光滑、一致凸Banach空间中提出了两种修正杂交迭代算法,证明了迭代序列既强收敛到极大单调算子的零点, 又强收敛到非扩展映射的不动点的结论. 推广和补充了以往的研究工作.     

On split common fixed point problems

Based on the convergence theorem recently proved by the second author, we modify the iterative scheme studied by Moudafi for quasi-nonexpansive operators to obtain strong convergence to a solution of the split common fixed point problem. It is noted that Moudafi's original scheme can conclude only weak convergence. As a consequence, we obtain strong convergence theorems for split variational inequality problems for Lipschitz continuous and monotone operators, split common null point problems for maximal monotone operators, and Moudafi's split feasibility problem.     

Fixed point solutions of variational inequality and generalized equilibrium problems with applications

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solution of generalized equilibrium problem and the set of solutions of the variational inequality problem for a co-coercive mapping in a real Hilbert space. Then strong convergence of the scheme to a common element of the three sets is proved. Furthermore, new convergence results are deduced and finally we apply our results to solving optimization problems and obtaining zeroes of maximal monotone operators and co-coercive mappings.     

Halpern's Iteration for Bregman Relatively Nonexpansive Mappings in Banach Spaces

In this article, using Bregman functions, we first introduce new modified Mann and Halpern's iterations for finding common fixed points of an infinite family of Bregman relatively nonexpansive mappings in the framework of Banach spaces. Furthermore, we prove the strong convergence theorems for the sequences produced by the methods. Finally, we apply these results for approximating zeroes of accretive operators. Our results improve and generalize many known results in the current literature.     

Iterative convergence theorems for maximal monotone operators and relatively nonexpansive mappings

In this paper, some iterative schemes for approximating the common element of the set of zero points of maximal monotone operators and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Some strong convergence theorems are obtained, to extend the previous work.     

On the zero point problem of monotone operators in Hadamard spaces

Numerical Algorithms - In this paper, by using products of finitely many resolvents of monotone operators, we propose an iterative algorithm for finding a common zero of a finite family of monotone...     

Banach空间中均衡问题和弱相对非扩展映射的强收敛定理及其应用(英文)

本文在Banach空间中设计了一些新的杂交迭代算法用以逼近一类均衡问题解集和弱相对非扩展映射不动点集或极大单调算子零点集的公共元.得到了一些强收敛的结论,并将它们推广到逼近一类均衡问题解集和有限个弱相对非扩展映射公共不动点集或有限个极大单调算子公共零点集的公共元的情形.最后,展示了本文的迭代算法在最优化问题上的应用.     

Implicit iterative algorithms of the split common fixed point problem for Bregman quasi-nonexpansive mapping in Banach spaces

In this paper, we study a modified implicit rule for finding a solution of split common fixed point problem of a Bregman quasi-nonexpansive mapping in Banach spaces. We propose a new iterative algorithm and prove the strong convergence theorem under appropriate conditions. As an application, the results are applied to solving the zero problem and the equilibrium problem.