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.     

Minimization of equilibrium problems,variational inequality problems and fixed point problems

In this paper, we devote to find the solution of the following quadratic minimization problem

$\min_{x\in \Omega}\|x\|^2,$

where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping and the solution set of some variational inequality. In order to solve the above minimization problem, we first construct an implicit algorithm by using the projection method. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. Finally, we prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem.

     

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.     

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.     

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.     

Strong convergence theorems for variational inequality, equilibrium and fixed point problems with applications

In this paper, we introduce a new iterative scheme for finding a common element of the set of common solutions of a finite family of equilibrium problems with relaxed monotone mappings, of the set of common solutions of a finite family of variational inequalities and of the set of common fixed points of an infinite family of nonexpansive mappings in a Hilbert space. Strong convergence for the proposed iterative scheme is proved. As an application, we solve a multi-objective optimization problem using the result of this paper. Our results improve and extend the corresponding ones announced by others.     

A viscosity approximation method for equilibrium problems,fixed point problems of nonexpansive mappings and a general system of variational inequalities

In this paper, we introduce and study a new iterative scheme for finding the common element of the set of common fixed points of a sequence of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the general system of variational inequality for α and μ-inverse-strongly monotone mappings. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main theorem extends a recent result of Ceng et al. (Math Meth Oper Res 67:375–390, 2008) and many others.     

A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce and analyze a new general iterative scheme by the viscosity approximation method for finding the common element of the set of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings and the set solutions of the variational inequality problems for an ξ-inverse-strongly monotone mapping in Hilbert spaces. We show that the sequence converge strongly to a common element of the above three sets under some parameters controlling conditions. The result extends and improves a recent result of Chang et al. (Nonlinear Anal. 70:3307–3319, 2009) and many others.     

Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points F(S) of a nonexpansive mapping S and the set of solutions Ω _A of the variational inequality for a monotone, Lipschitz continuous mapping A. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of \({F(S)\cap\Omega_{A}}\). As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.     

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 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.     

解变分不等式问题的混合方法

By using Fukushima‘s differentiable merit function,Taji,Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993. In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix F(x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty, polyhedral,closed and convex is proposed. Armijo-type line search and trust region strategies as well as Fukushima‘s differentiable merit function are incorporated into the method. It is then shown that the method is well defined and globally convergent and that,under the same assumptions as those of Taji et al. ,the method reduces to the basic Newton method and hence the rate of convergence is quadratic. Computational experiences show the efficiency of the proposed method.     

On the hybrid projection method for fixed point and equilibrium problems

In this paper, fixed point and equilibrium problems are considered based on a hybrid projection method. Strong convergence theorems of common elements are established in the framework of real Hilbert spaces.     

Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems

In this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of the solutions of the variational inequality problem by using a new hybrid method. We obtain a new result for finding a solution of an equilibrium problem and the solutions of the variational inequality problem.     

Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems

Let C be a nonempty closed convex subset of a real Hilbert space H. Let Q:C→C be a fixed contraction and S,T:C→C be two nonexpansive mappings such that Fix(T)≠?. Consider the following two-step iterative algorithm: $$\begin{array}{@{}rll}x_{n+1}&=&\alpha_{n}Qx_{n}+(1-\alpha_{n})Ty_{n},\\[1.5mm]y_{n}&=&\beta_{n}Sx_{n}+(1-\beta_{n})x_{n},\quad n\geq0.\end{array}$$ It is proven that under appropriate conditions, the above iterative sequence {x _n } converges strongly to $\tilde{x}\in \mathrm{Fix}(T)$ which solves some variational inequality depending on a given criterion S, namely: find $\tilde{x}\in H$ ; $0\in (I-S)\tilde{x}+N_{\mathrm{Fix}(T)}\tilde{x}$ , where N _Fix(T) denotes the normal cone to the set of fixed points of T.     

Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems

Numerical Algorithms - The purpose of this paper is to study and analyze two different kinds of extragradient-viscosity-type iterative methods for finding a common element of the set of solutions...     

Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems

The subgradient extragradient method can be considered as an improvement of the extragradient method for variational inequality problems for the class of monotone and Lipschitz continuous mappings. In this paper, we propose two new algorithms as combination between the subgradient extragradient method and Mann-like method for finding a common element of the solution set of a variational inequality and the fixed point set of a demicontractive mapping.     

New extragradient methods for solving variational inequality problems and fixed point problems

In this paper, we introduce two new iterative algorithms for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the variational inequality problem with a monotone and Lipschitz continuous mapping in real Hilbert spaces, by combining a modified Tseng’s extragradient scheme with the Mann approximation method. We prove weak and strong convergence theorems for the sequences generated by these iterative algorithms. The main advantages of our algorithms are that the construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the Lipschitz constant of cost operators. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.