Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings

Abstract

The purpose of this paper is to introduce and consider a hybrid shrinking projection method for finding a common element of the set E P of solutions of a generalized equilibrium problem, the set ⋂ n = 0 ∞ F ( S n ) of common fixed points of a countable family of relatively nonexpansive mappings { S n } n = 0 ∞ and the set T − 1 0 of zeros of a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space. It is proven that under appropriate conditions, the sequence generated by the hybrid shrinking projection method, converges strongly to some point in E P ∩ T − 1 0 ∩ ( ⋂ n = 0 ∞ F ( S n ) ) . This new result represents the improvement, complement and development of the previously known ones in the literature.