Convergence theorems of a new iteration for two nonexpansive mappings

Abstract

The purpose of this paper is to introduce the following new general implicit iteration scheme for approximating the common fixed points of a pair of nonexpansive mappings in a uniformly convex Banach space: for any x 0 ∈ C, the iterative process {x n } defined by x n = a n x n-1 + b n Ty n + c n Sx n , y n = a n x n-1 + b n x n + c n Sx n-1 + d n Tx n , where {a n }, {b n }, {c n }, {a n }, {b n }, {c n }, {d n } are seven sequences of real numbers satisfying a n + b n + c n = 1, a n + b n + c n + d n = 1, and T, S : C → C are two nonexpansive mappings. We approximate the common fixed points of these two mappings by weak and strong convergence of the scheme.