Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization

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 Fréchet 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, [3–5] 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, [7–16] 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