Strong convergence theorems for finitely many nonexpansive mappings

Under the assumption that EE is a reflexive Banach space whose norm is uniformly Gêteaux differentiable and which has a weakly continuous duality mapping JφJφ with gauge function φφ, Ceng–Cubiotti–Yao [Strong convergence theorems for finitely many nonexpansive mappings and applications, Nonlinear Analysis 67 (2007) 1464–1473] introduced a new iterative scheme for a finite commuting family of nonexpansive mappings, and proved strong convergence theorems about this iteration. In this paper, only under the hypothesis that EE is a reflexive Banach space which has a weakly continuous duality mapping JφJφ with gauge function φφ, and several control conditions about the iterative coefficient are removed, we present a short and simple proof of the above theorem.