Viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems

In this paper, we first introduce a general iterative scheme by Halpern approximation method for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of the variational inequality for an infinite family of αiαi-inverse-strongly monotone mappings and the set of common fixed points for a family of infinitely nonexpansive mappings in a Hilbert space. Then, we extend the Halpern iterative scheme to a general viscosity approximation scheme by a simple and different method. As applications, we utilize our results to investigate some convergence problems for strictly pseudocontractive mappings and maximal monotone operators. The results obtained in this paper extend and improve the recent ones announced by many others.