Iterative algorithms for families of variational inequalities fixed points and equilibrium problems

We introduce an iterative algorithm for finding a common element‎ ‎of the set of fixed points for an infinite family of nonexpansive‎ ‎mappings‎, ‎the set of solutions‎ ‎of the variational inequalities for a family of‎ ‎$\alpha$-inverse-strongly monotone mappings and the set of‎ ‎solutions of a system of equilibrium problems in a Hilbert space‎. ‎We prove the strong convergence of the proposed iterative‎ ‎algorithm to the unique solution of a variational inequality‎, ‎which is the optimality condition for a minimization problem‎. ‎Moreover‎, ‎we apply our result to the problem of finding a common‎ ‎fixed point of a family of strictly pseudocontractive mappings‎.