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Convergence of three-step iterations for asymptotically nonexpansive mappings. (English) Zbl 1136.65060
A general three-step fixed-point iteration for asymptotically nonexpansive mappings in a Banach space is studied. The method includes the (modified) Noor and Ishikawa iteration schemes [cf. B. Xu and M. Aslam Noor, J. Math. Anal. Appl. 267, No. 2, 444–453 (2002; Zbl 1011.47039)]. The first main result shows strong convergence if the mapping is a nonexpansive, completely continuous mapping of a nonempty, closed, convex subset of a uniformly convex Banach space into itself. The second main result asserts weak convergence in case the mapping is not necessarily completely continuous but the Banach space possesses Opial’s property.

65J15Equations with nonlinear operators (numerical methods)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces