Viscosity Approximation Methods for Strong Vector Equilibrium Problems and Fixed Point Problems

In this paper,we study the problem of finding a common element of the set of solutions of a strong vector equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space.We first give the auxiliary problem for the strong vector equilibrium problem and obtain some results of existence and uniqueness of solutions of the auxiliary problem under suitable conditions.Then,by virtue of these results,we propose a viscosity iterative scheme for finding a common element of the set of solutions of the strong vector equilibrium problem and the set of fixed points of the nonexpansive mapping.Furthermore,we prove that,under suitable conditions,the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of the strong vector equilibrium problem and the set of fixed points of the nonexpansive mapping.