Iterative algorithms for variational inclusions,mixed equilibrium and fixed point problems with application to optimization 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 a nonexpansive mapping, and the the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator.     

关于变分包含问题和不动点问题的一些结果及应用

基于一个广义迭代算法,考虑了逼近一类拟变分包含问题解集与一族无限多个非扩张映象公共不动点集的某一公共元问题.在实Hilbert空间的框架下,证明了由次广义迭代算法产生的迭代序列强收敛到某一公共元.     

A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings

In this paper, by using Bregman distance, we introduce a new iterative process involving products of resolvents of maximal monotone operators for approximating a common element of the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings and the solution set of the multiple-sets split feasibility problem and common zeros of maximal monotone operators. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Finally, we mention several corollaries and two applications of our algorithm.     

Regularized and inertial algorithms for common fixed points of nonlinear operators

This paper deals with a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real Hilbert space. The proposed method combines two strategies: viscosity approximations (regularization) and inertial type extrapolation. The first strategy is known to ensure the strong convergence of some successive approximation methods, while the second one is intended to speed up the convergence process. Under classical conditions on the operators and the parameters, we prove that the sequence of iterates generated by our scheme converges strongly to the element of minimal norm in the solution set. This algorithm works, for instance, for approximating common fixed points of infinite families of demicontractive mappings, including the classes of quasi-nonexpansive operators and strictly pseudocontractive ones.     

Extragradient methods for split feasibility problems and generalized equilibrium problems in Banach spaces

The purpose of this paper is the presentation of a new extragradient algorithm in 2‐uniformly convex real Banach spaces. We prove that the sequences generated by this algorithm converge strongly to a point in the solution set of split feasibility problem, which is also a common element of the solution set of a generalized equilibrium problem and fixed points of of two relatively nonexpansive mappings. We give a numerical example to investigate the behavior of the sequences generated by our algorithm.     

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.     

Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings

In this paper, we introduce a new mapping and a Hybrid iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common fixed point of a finite family of nonexpansive mappings which is a solution of the generalized equilibrium problem. The results obtained in this paper extend the recent ones of 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].     

Viscosity approximation methods for generalized 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 generalized equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in the setting of Hilbert spaces. By using well-known Fan-KKM lemma, we derive the existence and uniqueness of a solution of the auxiliary problem for GEP. On account of this result and Nadler’s theorem, we propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping. Furthermore, it is proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping.     

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.     

Existence theorems and iterative approximation methods for generalized mixed equilibrium problems for a countable family of nonexpansive mappings

In this paper, we introduce the new generalized mixed equilibrium problem basing on hemicontinuous and relaxed monotonic mapping. Using the KKM technique, we obtain the existence of solutions for the generalized mixed equilibrium problem in a Banach space. Furthermore, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of a countable family of nonexpansive mappings. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.     

Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems

In this paper, we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of an infinite family of nonexpansive mappings and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters.     

Convergence theorems based on the shrinking projection method for variational inequality and?equilibrium problems

The purpose of this paper is to introduce a hybrid projection algorithm based on the shrinking projection method for two relatively weak nonexpansive mappings. We prove strong convergence theorem which approximate the common element in the fixed point set of two such mappings, the solution set of the variational inequality and the solution set of the equilibrium problem in the framework of Banach spaces. Our results improve and extend previous results.     

A new iterative method for the split common fixed point problem in Hilbert spaces

The split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed-point set. In this paper, we propose a new algorithm for this problem that is completely different from the existing algorithms. Moreover, our algorithm does not need any prior information of the operator norm. Under standard assumptions, we establish a weak convergence theorem of the proposed algorithm and a strong convergence theorem of its variant.     

On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings

In this article we consider the problem of finding a common element in the solution set of generalized equilibrium problems, in the solution set of the classical variational inequality and in the fixed point set of strictly pseudocontractive mappings. Weak convergence theorems of common elements are established in real Hilbert spaces.     

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.     

Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities

We introduce a new iteration method and prove strong convergence theorems for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of monotone and Lipschitz-type continuous Ky Fan inequality. Under certain conditions on parameters, we show that the iteration sequences generated by this method converge strongly to the common element in a real Hilbert space. Some preliminary computational experiences are reported.     

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.     

Hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense

In this paper we introduce an iterative algorithm for finding a common element of the fixed point set of an asymptotically strict pseudocontractive mapping S in the intermediate sense and the solution set of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in Hilbert space. The iterative algorithm is based on several well-known methods including the extragradient method, CQ method, Mann-type iterative method and hybrid gradient projection algorithm with regularization. We obtain a strong convergence theorem for three sequences generated by our iterative algorithm. In addition, we also prove a new weak convergence theorem by a modified extragradient method with regularization for the MP and the mapping S.     

Some results on generalized equilibrium problems involving strictly pseudocontractive mappings

In this paper,we consider an iterative sequence for generalized equilibrium problems and strictly pseudocontractive mappings.We show that the iterative sequence converges strongly to a common element of the solution set of generalized equilibrium problems and of the fixed point set of strictly pseudocontractive mappings.     

An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings

In this paper, an extragradient-type method is introduced for finding a common element in the solution set of generalized equilibrium problems, in the solution set of classical variational inequalities and in the fixed point set of strictly pseudocontractive mappings. It is proved that the iterative sequence generated in the purposed extragradient-type iterative process converges weakly to some common element in real Hilbert spaces.