In this paper, the authors propose a new modified Mann iteration for computing fixed points of nonexpansive mappings in a Banach space setting. This new iterative scheme combines the modified Mann iteration introduced by T. H. Kim
and H. K. Xu
[“Strong convergence of modified Mann iterations”, Nonlinear Anal., Theory Methods Appl. 61, No. 1–2 (A), 51–60 (2005; Zbl 1091.47055
)] and the viscosity approximation method introduced by A. Moudafi
[“Viscosity approximation methods for fixed-points problems”, J. Math. Anal. Appl. 241, No. 1, 46–55 (2000; Zbl 0957.47039
)]. The main results extend and improve some due to Xu and Kim in the aforementioned paper.