Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.     

Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process     

Some iterative methods for finding fixed points and for solving constrained convex minimization problems

The present paper is divided into two parts. In the first part, we introduce implicit and explicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. We establish results on the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping. Such a fixed point is also a solution of a variational inequality defined on the set of fixed points. In the second part, we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem. Such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping. The results of this paper extend and improve several results presented in the literature in the recent past.     

An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.      

A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications

The purpose of this paper is to prove by using a new hybrid method a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem, the set of solutions for a variational inequality problem and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space. As applications, we utilize our results to obtain some new results for finding a solution of an equilibrium problem, a fixed point problem and a common zero-point problem for maximal monotone mappings in Banach spaces.     

Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings

In this paper, we introduce an iterative process for finding the common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality problem for an α$\alpha$-inverse-strongly-monotone mapping. We obtain a weak convergence theorem for a sequence generated by this process. Moreover, we apply our result to the problem for finding a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem of a monotone mapping.     

A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping

Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point set are provided in the final section.     

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.     

Strong convergence theorems for variational inequality problems and fixed point problems in uniformly smooth and uniformly convex Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings and the set of common fixed points for a nonexpansive mapping in a uniformly smooth and uniformly convex Banach space. We obtain a strong convergence theorem under some suitable conditions. Our results improve and extend the recent ones announced by many others in the literature.     

A relaxed extragradient-like method for a generalized mixed equilibrium problem,a general system of generalized equilibria and a fixed point problem

Very recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025–1033] suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a Hilbert space. In this paper, based on Takahashi–Takahashi’s iterative method and well-known extragradient method we introduce a relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a Hilbert space and then obtain a strong convergence theorem. Utilizing this theorem, we establish some new strong convergence results in fixed point problems, variational inequalities, mixed equilibrium problems and systems of generalized equilibria.     

均衡问题和不动点问题的粘滞近似方法及其应用

用粘滞近似方法产生了一个新的迭代序列,并证明了该迭代序列强收敛于一个非扩张映射的不动点,同时该不动点也是一个变分不等式和一个均衡问题的共同解.作为应用,另外证明了一个关于非扩张映射和严格伪压缩映射的定理.     

A hybrid extragradient method for general variational inequalities

In this paper, we introduce and study a hybrid extragradient method for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. An iterative algorithm is proposed by virtue of the hybrid extragradient method. Under two sets of quite mild conditions, we prove the strong convergence of this iterative algorithm to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively. L. C. Zeng’s research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J. C. Yao’s research was partially supported by a grant from the National Science Council of Taiwan.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43-52], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (1) (2007) 506-515], and many others.     

Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce an iterative scheme based on the extragradient approximation method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of the variational inequality problem for a monotone L-Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Applications to optimization problems are given. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications (2008) 17. doi:10.1155/2008/134148. Article ID 134148], Kumam and Katchang [P. Kumam, P. Katchang, A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings, Nonlinear Anal. Hybrid Syst. (2009) doi:10.1016/j.nahs.2009.03.006] and many others.     

Convergence and certain control conditions for hybrid viscosity approximation methods

Very recently, Yao, Chen and Yao [20] proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition β_n→0 is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space.     

Banach空间中非膨胀映象和α-逆强增生算子的强收敛性

在2-一致光滑的Banach空间中,引入一种新的迭代算法研究非膨胀映象的不动点集与α-逆强增生算子的变分不等式解集的公共元素,并获得了迭代算法的强收敛性定理.而且应用这些结果考虑了非膨胀映象和严格伪压缩映象公共不动点的收敛性问题.     

Hilbert空间中广义平衡问题和不动点问题的粘滞逼近法

在Hilbert空间,我们用粘滞逼近法建立了一迭代序列来逼近两个集合的公共点,这两个集合分别是广义平衡问题的解集和渐进非扩张映射的不动点集.我们表明这一迭代序列强收敛到这两个集合的公共点,而且这一公共点还是一变分不等式的解.用这一结果,还研究了三个强收敛问题和优化问题.     

Fixed point solutions of generalized equilibrium problems for nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Then, strong convergence of the scheme to a common element of the two sets is proved. As an application, problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem is solved. Moreover, solution is given to the problem of finding a common element of fixed points set of nonexpansive mappings and the set of solutions of a variational inequality problem.     

Iterative method for fixed point problem, variational inequality and generalized mixed 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 mixed 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 present other applications.     

Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping, the set of solutions of the variational inequality for the monotone mapping and the set of solutions of an equilibrium problem in a 2-uniformly convex and uniformly smooth Banach space. Then we show that the iterative sequence converges strongly to a common element of the three sets. In this paper, we also give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping in Banach space $l^2$.