Strong convergence of composite iterative methods for equilibrium problems and fixed point problems

We introduce a new composite iterative scheme by viscosity approximation method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping. Our results substantially improve the corresponding results of Takahashi and Takahashi [A. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506-515]. Essentially a new approach for finding solutions of equilibrium problems and the fixed points of nonexpansive mappings is provided.     

A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings

In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mapping in Hilbert spaces. We prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main result improve and extend Plubtieng and Punpaeng’s corresponding result [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 197 (2008), 548–558]. Using this theorem, we obtain three corollaries.     

无限族demi压缩映射与平衡问题的收敛定理

在Hilbert空间中,为了找到无限个demi压缩映射公共不动点集和广义混合平衡问题解的公共元,本文介绍了一种迭代算法,得到关于公共元的强收敛定理,并给出例子说明结果.     

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 hybrid iterative scheme for mixed equilibrium problems and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a mixed equilibrium problem (MEP) and the set of common fixed points of finitely many nonexpansive mappings in a real Hilbert space. First, by using the well-known KKM technique we derive the existence and uniqueness of solutions of the auxiliary problems for the MEP. Second, by virtue of this result we introduce a hybrid iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings. Furthermore, we prove that the sequences generated by the hybrid iterative scheme converge strongly to a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings.     

强向量均衡问题与不动点问题的粘性逼近算法

讨论了强向量均衡问题与非扩张映射不动点问题的公共解.首先,给出了强向量均衡问题的辅助问题,并在适当的条件下,证明了其解的存在性和唯一性结果.然后,利用这些结果,提出了强向量均衡问题与非扩张映射不动点问题公共解的粘性逼近算法,并进一步证明了,在适当的条件下,由该算法产生的迭代序列强收敛于强向量均衡问题和非扩张映射不动点问题的公共解.     

A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems

We introduce an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems and common solutions of finite family of variational inequality problems for monotone mappings in Banach spaces. Our theorem extends and unifies most of the results that have been proved for this important class of nonlinear operators.     

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.     

Convergence theorems of common elements for equilibrium problems and fixed point problems

In this paper, we introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of two asymptotically nonexpansive mappings in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the two sets.     

Mixed equilibrium problems and optimization problems

In this paper, we introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings. Furthermore, we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions. We also discuss some special cases. Results obtained in this paper improve the previously known results in this area.     

Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications

In this paper, we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems, the set of common fixed points of a family of infinite non-expansive mappings. Strong convergence theorems are established in a real Hilbert space under suitable conditions. As some applications, we consider convex feasibility problems and equilibrium problems. The results presented improve and extend the corresponding results of many others.     

A system of mixed equilibrium problems, fixed point problems of strictly pseudo-contractive mappings and nonexpansive semi-groups

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed points for an infinite family of strictly pseudo-contractive mappings and the set of common fixed points for nonexpansive semi-groups in Hilbert space. Under suitable conditions some strong convergence theorem are proved. The results presented in the paper extend and improve some recent results.     

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.     

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.     

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.     

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.     

带扰动映像的广义平衡问题和不动点问题的混合粘性逼近问题

引入一个用于寻求带扰动映像的广义平衡问题解集以及可数无穷多非扩张映像之族公共不动点集的公共解的新的迭代算法. 证明了由此算法生成的序列的强收敛性. 所得的结果推广改进了先前许多作者的结果.     

An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets. Our results extend and improve the corresponding results of Ceng, Wang, and Yao [L.C. Ceng, C.Y. Wang, J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. 67 (2008) 375–390], Ceng and Yao [L.C. Ceng, J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. doi:10.1016/j.cam.2007.02.022], 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] and many others.     

An iterative method for finding common solutions of equilibrium and fixed point problems

We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.     

Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications

We use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi-Lipschitzian mappings. As a consequence, several convergence theorems for quasi-nonexpansive mappings and asymptotically κ-strict pseudo-contractions are deduced. We also establish strong convergence of iterative sequences for finding a common element of the set of fixed point, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain the corresponding results due to Tada-Takahashi and Nakajo-Shimoji-Takahashi.