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 $l^2$.     

Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings

In this paper, we introduce and study an iterative scheme by a hybrid method for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a real Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. Using this result, we consider the problem of finding a common fixed point of a finite family of nonexpansive mappings and a strictly pseudocontractive mapping and the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the several recent results in this area.     

A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings

In this paper, we introduce an general iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the two sets. Using this results, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and 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. The results of this paper extended and improved the results of Iiduka and Takahashi (Nonlinear Anal. 61:341–350, 2005).     

An implicit iterative scheme for monotone variational inequalities and fixed point problems

In this paper we introduce an implicit iterative scheme for finding a common element of the set of common fixed points of N$N$ nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The implicit iterative scheme is based on two well-known methods: extragradient and approximate proximal. We obtain a weak convergence theorem for three sequences generated by this implicit iterative scheme. On the basis of this theorem, we also construct an implicit iterative process for finding a common fixed point of N+1$N+1$ mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other N$N$ mappings are nonexpansive.     

Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone 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 the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and 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.     

A hybrid approximation method for equilibrium,variational inequality and fixed point problems

The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the fixed points of $?$-asymptotically nonexpansive mapping, the set of solutions of the equilibrium problem and the set of solutions of the variational inequality for an inverse strongly monotone operator in the framework of Banach spaces. We show that the iterative scheme converges strongly to a common element of the above three sets under appropriate conditions.     

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.     

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.     

A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping

The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in the framework of a Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under appropriate conditions. Additionally, the idea of our results are applied to find a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space.     

A hybrid iteration scheme for equilibrium problems,variational inequality problems and common fixed point problems in Banach spaces

In this paper, we introduce an iterative process which converges strongly to a common element of a set of common fixed points of finite family of closed relatively quasi-nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for an $\alpha$-inverse strongly monotone mapping in Banach 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.     

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 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.     

A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a family of infinitely 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 the framework of a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. Additionally, we utilize our results to study the optimization problem and find a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. Our results improve and extend the results announced by many others.     

Strong convergence of an iterative method for pseudo-contractive and monotone mappings

In this paper, we introduce an iterative process which converges strongly to a common element of fixed points of pseudo-contractive mapping and solutions of variational inequality problem for monotone mapping. As a consequence, we provide an iteration scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudo-contractive mappings and solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.     

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.     

An extragradient method for relaxed cocoercive variational inequality and equilibrium problems

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansivemappings, the set of an equilibrium problem and the set of solutions of the variational inequality problem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao, Takahashi and many others.     

Weak convergence theorems for a countable family of Lipschitzian mappings

This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37–43], Tada–Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 (2007) 359–370], and Plubtieng–Kumam [S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings. J. Comput. Appl. Math. (2008) doi:10.1016/j.cam.2008.05.045]. Some of our results are established with weaker assumptions.