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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.

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