Iterative Schemes for Convex Minimization Problems with Constraints

We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Frechet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.