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Strong convergence theorems for relatively nonexpansive mappings in a Banach space. (English) Zbl 1124.47046
In [J. Math. Anal. Appl. 279, No. 2, 372–379 (2003; Zbl 1035.47048)], K. Nakajo and W. Takahashi defined a modified Mann iteration for a single nonexpansive map T to obtain strong convergence results in Hilbert space. In [Nonlinear Anal., Theory Methods Appl. 64, No. 11 (A), 2400–2411 (2006; Zbl 1105.47060)], C. Martinez–Yanes and H. K. Xu defined modified Ishikawa and Halpern iterations to prove interesting convergence results. In the paper under review, the authors, guided by the first mentioned article, prove strong convergence theorems for relatively nonexpansive mappings in uniformly convex and uniformly smooth Banach space.

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