A theorem of variational inclusion problems and various nonlinear mappings

In this paper, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of nonexpansive mappings and the set of solutions of a finite family of variational inclusion problems in Hilbert spaces. Moreover, we utilize our main result to fixed point problems of various nonlinear mappings and the set of solutions of variational inclusion problems.     

Some results on generalized equilibrium problems involving a family of nonexpansive mappings

In this paper, we introduce an iterative method for finding a common element in the solution set of generalized equilibrium problems, in the solution set of variational inequalities and in the common fixed point set of a family of nonexpansive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.     

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 theorem for nonexpansive semigroups and systems of equilibrium problems

Our purpose in this paper is to prove strong convergence theorem for finding a common element of the set of common fixed points of a one-parameter nonexpansive semigroup and the set of solutions to a system of equilibrium problems in a real Hilbert space using a new iterative method. Finally, we give an application of our result in Hilbert spaces.     

A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings

In this paper, we introduce a new viscosity approximation scheme based on the extragradient method for finding a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions to the variational inequality for a monotone, Lipschitz continuous mapping. Several convergence results for the sequences generated by these processes in Hilbert spaces were derived.     

A viscosity approximation method for finding a common solution of fixed points and equilibrium problems in Hilbert spaces

In this paper, we introduce an iterative method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of common fixed points of a countable family of nonexpansive mappings in Hilbert spaces. Using the result we consider a strong convergence theorem in variational inequalities and equilibrium problems. The result present in this paper extend and improve the corresponding result of Qin et al. (Nonlinear Anal 69:3897–3909, 2008), Plubtieng and Punpaeng (J Math Anal Appl 336:455–469, 2007) and many others.     

Hilbert空间上新的变分不等式问题和不动点问题的粘性迭代算法

本文在Hilbert空间上引入了一个新的粘性迭代算法,找到了关于两个逆强单调算子的变分不等式问题的解集与非扩张映射的不动点集的公共元.通过修改的超梯度算法,得到了强收敛定理,也给出了一个数值例子.所得结果改进了许多最新结果.     

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

The purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of variational inequality for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Our results extend the recent results of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515], Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52], Combettes and Hirstoaga [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 486–491], Iiduka and Takahashi, [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and many others.     

Hilbert空间中关于变分不等式问题和不动点问题的粘性隐式中点算法

该文在Hilbert空间中研究了关于两个逆强单调算子的一般变分不等式问题和非扩张映射的不动点问题的粘性隐式中点算法,用修改的超梯度方法,在对参数作适当的限制下,得到了强收敛定理,所得结果推广和提高了许多最新文献中的相应结果.     

A general composite explicit iterative scheme of fixed point solutions of variational inequalities for nonexpansive semigroups

In this paper, we introduce a composite explicit viscosity iteration method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. We prove strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions. Our result extends and improves those announced by Li et al [General iterative methods for a one-parameter nonexpansive semigroup in Hilbert spaces, Nonlinear Anal. 70 (2009) 3065–3071], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Modelling 48 (2008) 279–286], Plubtieng and Wangkeeree [S. Plubtieng, R. Wangkeeree, A general viscosity approximation method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Bull. Korean Math. Soc. 45 (4) (2008) 717–728] and many others.     

General Iterative Methods for Semigroups of Nonexpansive Mappings Related to Optimization Problems

Abstract

In this article, we introduce two general iterative methods for a certain optimization problem of which the constrained set is the set of the solution set of the variational inequality problem for the fixed point set of nonexpansive semigroups in Hilbert spaces. Under some control conditions, we establish the strong convergence of the proposed methods to the fixed point set, which is the unique solution of a certain optimization problem. Applications to solutions of equilibrium problems are also presented.     

A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings

In this paper, we investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for k-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of Kumam [P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turk. J. Math. 33 (2009) 85–98], 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, Article ID 134148, 17 pages, doi:10.1155/2008/134148], Yao et al. [Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an finite family of nonexpansive mappings, Nonlinear Analysis 69 (5–6) (2008) 1644–1654], Qin et al. [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis (69) (2008) 3897–3909], and many others.     

A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings

The purpose of this paper is to consider a new hybrid relaxed extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions. Our results extend and improve the recent results of Cai and Hu [G. Cai, C.S. Hu, A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping, Nonlinear Anal. Hybrid Syst., 3(2009) 395–407], Kangtunyakarn and Suantai [A. Kangtunyakarn, S. Suantai, A new mapping for finding common solution of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Anal., 71(2009) 4448–4460] and Thianwan [S. Thianwan, Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. Hybrid Syst., 3(2009) 605–614] and many others.     

Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems

In this paper, a new iterative scheme based on the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Applications to optimization problems are given.     

Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems

The purpose of this paper is to introduce a general iterative method for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of split variational inclusion problem in the framework Hilbert spaces. Strong convergence theorem of the sequences generated by the purpose iterative scheme is obtained. In the last section, we present some computational examples to illustrate the assumptions of the proposed algorithms.     

Split Common Fixed Point Problem of Nonexpansive Semigroup

In this paper, we first introduce a new algorithm with a viscosity iteration method for solving the split common fixed point problem (SCFP) for a finite family of nonexpansive semigroups. We also present a new algorithm for solving the SCFP for an infinite family of quasi-nonexpansive mappings. We establish strong convergence of these algorithms in an infinite-dimensional Hilbert spaces. As application, we obtain strong convergence theorems for split variational inequality problems and split common null point problems. Our results improve and extend the related results in the literature.     

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

Recently, Ceng, Guu and Yao introduced an iterative scheme by viscosity-like approximation method to approximate the fixed point of nonexpansive mappings and solve some variational inequalities in Hilbert space (see Ceng et al. (2009) [9]). Takahashi and Takahashi proposed an iteration scheme to solve an equilibrium problem and approximate the fixed point of nonexpansive mapping by viscosity approximation method in Hilbert space (see Takahashi and Takahashi (2007) [12]). In this paper, we introduce an iterative scheme by viscosity approximation method for finding a common element of the set of a countable family of nonexpansive mappings and the set of an equilibrium problem in a Hilbert space. We prove the strong convergence of the proposed iteration to the unique solution of a variational inequality.     

An iterative method of solution for equilibrium and optimization problems

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets. The results of this paper extended and improved the results of H. Iiduka and W. Takahashi [Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. Therefore, by using the above result, an iterative algorithm for the solution of a optimization problem was obtained.     

A Simpler Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces

In the present paper, we propose a simpler explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive mappings on Hilbert spaces. A strong convergence theorem is proved under fewer restrictions imposed on the mappings and parameters. An extension and numerical result are also given to illustrate the effectiveness and superiority of the proposed algorithm.     

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.