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We use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point with the property such that , where , are two pseudocontractive self-mappings of a closed convex subset of a Hilbert space with the set of fixed points . Assume the solution set of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element . Our results improve and extend a recent result of (Lu et al. 2009).

Abstract In this paper, we suggest a hybrid method for finding a common element of the set of solution of a monotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method. Mathematics Subject Classification (2000): 47H05; 47H09; 47H10; 47J05; 47J25.

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 .

We consider the following hierarchical equilibrium problem and variational inequality problem (abbreviated as HEVP): find a point such that , for all , where , are two monotone operators and is the solution of the equilibrium problem of finding such that , for all . We note that the problem (HEVP) includes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solving (HEVP), we propose a double-net algorithm which generates a net . We prove that the net hierarchically converges to the solution of (HEVP); that is, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem, and as , the net converges in norm to the unique solution of (HEVP).

We propose a modified hybrid projection algorithm to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

We consider an iterative scheme for approximating the common fixed points of two asymptotically quasi-nonexpansive mappings in the intermediate sense in Banach spaces. The present results improve and extend some recent corresponding results of Lan (2006) and many others.

The purpose of this paper is to investigate the problem of approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. First, we propose an extragradient method for solving the mixed equilibrium problems and the fixed point problems. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions.

We introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of a nonexpansive mapping .