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Strong convergence and certain control conditions for modified Mann iteration. (English) Zbl 1189.47071
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.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces