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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.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems
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