Iterative algorithm by using the hybrid method in mathematical programming for solving variational inequality problems and equilibrium problems

Abstract

In this paper, we introduce an iterative scheme by a new hybrid method for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α -inverse-strongly monotone mappings in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets, which solves some fixed point problems, variational inequality problems and equilibrium problems by using the hybrid method in mathematical programming which connected with optimization problems. The results are connected with Kumam’s results [11, 12], Shinzato and Takahashi’s result [19], Tada and Takahashi’s result [23] and Takahashi’s et al. result [26] and many others.