AN ITERATIVE METHOD FOR GENERALIZED MIXED VECTOR EQUILIBRIUM PROBLEMS AND FIXED POINT OF NONEXPANSIVE MAPPINGS AND VARIATIONAL INEQUALITIES

Abstract

In this paper, we study the problem of finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the generalized mixed vector equilibrium problem and the solution set of a variational inequality problem with a monotone Lipschitz continuous mapping in Hilbert spaces. We first consider an auxiliary problem for the generalized mixed vector equilibrium problem and prove the existence and uniqueness of the solution for the auxiliary problem. We then introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the generalized mixed vector equilibrium problem and the solution set of a variational inequality problem with a monotone Lipschitz continuous mapping. The results presented in this paper can be considered as a generalization of some known results due to Peng and Yao [16, 17].