%0 Journal Article
%T Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization
%A A. E. Al-Mazrooei
%A A. Latif
%A J. C. Yao
%J Abstract and Applied Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/587865
%X We propose implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fr¨¦chet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions. 1. Introduction and Problems Formulation Let be a real Hilbert space with inner product and norm , let be a nonempty closed convex subset of , and let be the metric projection of onto . Let be a self-mapping on . We denote by the set of fixed points of and by the set of all real numbers. Recall that a mapping is said to be -Lipschitz continuous if there exists a constant such that In particular, if , then is called a nonexpansive mapping [1], and if , then is called a contraction. Recall that a mapping is called(i)monotone if (ii) -strongly monotone if there exists a constant such that (iii) -inverse strongly monotone if there exists a constant such that It is obvious that if is -inverse strongly monotone, then is monotone and -Lipschitz continuous. Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP): find a point such that The solution set of VIP (5) is denoted by . The VIP (5) was first discussed by Lions [2] and is now well known. The VIP (5) has many potential applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and so on; see, for example, [3¨C5] and the references therein. In 1976, Korpelevich [6] proposed an iterative algorithm for solving the VIP (5) in Euclidean space : with , a given number which is known as the extragradient method. The literature on the VIP is vast and Korpelevich¡¯s extragradient method has received great attention given by many researchers. See, for example, [7¨C16] and the references therein. In particular, motivated by the idea of Korpelevich¡¯s extragradient method [6], Nadezhkina and Takahashi [17] introduced an extragradient iterative scheme: where is a monotone, -Lipschitz continuous mapping, is a nonexpansive mapping, for some , and for some . They proved the weak convergence of to an element of . Let be a real-valued function, let be a nonlinear
%U http://www.hindawi.com/journals/aaa/2014/587865/