A hybrid method for a family of relatively quasi-nonexpansive mappings and an equilibrium problem in Banach spaces

We introduce a hybrid method for finding a common element in the solutions set of an equilibrium problem and the common fixed points set of a countable family of relatively quasi-nonexpansive mappings in a Banach space. A strong convergence theorem of the proposed method is established by using the concept of the Mosco convergence when the family {Tn} satisfies the (*)-condition. The examples of three generated mappings which satisfy the (*)-condition are also given. Using the obtained result, we give some applications concerning the variational inequality problem and the convex minimization problem.