Generalized mixed equilibria, variational inequalities and constrained convex minimization

In this paper, we introduce one multistep relaxed implicit extragradient-like scheme and another multistep relaxed explicit extragradient-like scheme 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 strongly monotone mappings in a real Hilbert space. Under suitable control conditions, we establish the strong convergence of these two multistep relaxed extragradient-like schemes to the same common element of the above three sets, which is also the unique solution of a variational inequality defined over the intersection of the above three sets.