The approximation of common element for maximal monotone operator, generalized mixed equilibrium problem and fixed point problem

The purpose of this paper is to get strong convergence theorems for a countable family of relatively quasi-nonexpansive mappings {Sn}n=0∞, a maximal monotone operator T, and a generalized mixed equilibrium problem in a uniformly smooth and uniformly convex Banach space lacking condition UARC. Two examples are given to support our results. One is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another is uniformly closed but not satisfies condition UARC. Many recent results in this field have been unified and improved.