Strong convergence of projection methods for a countable family of nonexpansive mappings and applications to constrained convex minimization problems

In this paper, we introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a real Hilbert space, which solves a corresponding variational inequality. Furthermore, we propose explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem. Our results improve and generalize some known results in the current literature.