Abstract:
We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained.

Abstract:
We consider and analyze some new proximal extragradient type methods for solving variational inequalities. The modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. These new iterative methods include the projection, extragradient and proximal methods as special cases.

Abstract:
Let be a nonempty closed convex subset of a real Hilbert space . Let be a -Lipschitzian and -strongly monotone operator with constants , be nonexpansive mappings with where denotes the fixed-point set of , and be a -contraction with coefficient . Let and , where . For each , let be a unique solution of the fixed-point equation . We derive the following conclusions on the behavior of the net along the curve : (i) if , as , then strongly, which is the unique solution of the variational inequality of finding such that and (ii) if , as , then strongly, which is the unique solution of some hierarchical variational inequality problem.

Abstract:
We suggest a modified extragradient method for solving the generalized variational inequalities in a Banach space. We prove some strong convergence results under some mild conditions on parameters. Some special cases are also discussed.

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:
In this paper, we generalize the classical extragradient algorithm for solving variational inequality problems by utilizing non-null normal vectors of the feasible set. In particular, two conceptual algorithms are proposed and each of them has three different variants which are related to modified extragradient algorithms. Our analysis contains two main parts: The first part contains two different linesearches, one on the boundary of the feasible set and the other one along the feasible direction. The linesearches allow us to find suitable halfspaces containing the solution set of the problem. By using non-null normal vectors of the feasible set, these linesearches can potentially accelerate the convergence. If all normal vectors are chosen as zero, then some of these variants reduce to several well-known projection methods proposed in the literature for solving the variational inequality problem. The second part consists of three special projection steps, generating three sequences with different interesting features. Convergence analysis of both conceptual algorithms is established, assuming existence of solutions, continuity and a weaker condition than pseudomonotonicity on the operator. Examples, on each variant, show that the modifications proposed here perform better than previous classical variants. These results suggest that our scheme may significantly improve the extragradient variants.

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:
In this paper, we introduce a new approximation scheme based on the extragradient method and viscosity method for finding a common element of the set of solutions of the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces as follows: Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz continuous mapping of C into H. Let S be a nonexpansive mapping of C into H such that , where and , respectively, denote the set of fixed point of S and the solution set of a variational inequality. Let f be a contraction of H into itself and and be sequences generated by for every n=1,2,…, where and are sequences of numbers satisfying and and . Then, and converge strongly to The results in this paper improve some well-known results in the literature.

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.