Abstract:
The approximation-solvability of the following nonlinear variational inequality (NVI) problem is presented: Determine an element such that where is a mapping from a nonempty closed convex subset of a real Hilbert space into . The iterative procedure is characterized as a nonlinear variational inequality (for any arbitrarily chosen initial point ) which is equivalent to a double projection formula where denotes the projection of onto .

Abstract:
The purpose of this paper is to introduce a new modified relaxed extragradient method andstudy for finding some common solutions for a general system of variational inequalitieswith inversestrongly monotone mappings and nonexpansive mappings in the framework ofreal Banach spaces. By using the demiclosedness principle, it is proved that the iterativesequence defined by the relaxed extragradient method converges strongly to a commonsolution for the system of variational inequalities and nonexpansive mappings under quitemild conditions.

Abstract:
In this paper, we suggest and analyze a new extragradient methodfor finding a common element of the set of solutions of an equilibrium problem,the set of fixed points of a nonexpansive mapping and the set of solutions ofsome variational inequality. Furthermore, we prove that the proposed iterativealgorithm converges strongly to a common element of the above three sets. Ourresult includes the main result of Bnouhachem, Noor and Hao [A. Bnouhachem,M.A. Noor and Z. Hao, Some new extragradient methods for variational inequalities,Nonlinear Analysis (2008), doi:10.1016/j.na.2008.02.014] as a specialcase.

Abstract:
As a well-known numerical method, the extragradient method solves numerically the variational inequality of finding such that , for all . In this paper, we devote to solve the following hierarchical variational inequality Find such that , for all . We first suggest and analyze an implicit extragradient method for solving the hierarchical variational inequality . It is shown that the net defined by the suggested implicit extragradient method converges strongly to the unique solution of in Hilbert spaces. As a special case, we obtain the minimum norm solution of the variational inequality .

Abstract:
Motivated and inspired by Korpelevich's and Noor's extragradient methods, we suggest an extragradient method by using the sunny nonexpansiveretraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.

Abstract:
We suggest and analyze a modified extragradient method for solving variational inequalities, which is convergent strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space.

Abstract:
Based on the relaxed extragradient method and viscosity method, we introduce a new iterative method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the studied iterative method. The results of this paper extend and improve the results of Ceng et al., (2008), W. Kumam and P. Kumam, (2009), Yao et al., (2010) and many others.

Abstract:
The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradient-like iterative algorithm. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings. 1991 MSC: 47H09; 47J20.

Abstract:
In this paper, an extragradient method for solving variational inequalities was proposed, which extends the method in 1] and converges under the condition that the underlying mapping is pesudomotone. The numerical analysis was also given in this paper.

Abstract:
The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan (2010) and many others.