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Zbl 1136.65060
Yao, Yonghong; Noor, Muhammad Aslam
Convergence of three-step iterations for asymptotically nonexpansive mappings. (English)
[J] Appl. Math. Comput. 187, No. 2, 883-892 (2007). ISSN 0096-3003

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. {\it B. Xu} and {\it 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.
[Etienne Emmrich (Berlin)]

MSC 2000:: *65J15 Equations with nonlinear operators (numerical methods)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: asymptotically nonexpansive mapping; Noor iteration method; convergence; three-step fixed-point iteration; Ishikawa iteration schemes; Opial's property; uniformly convex Banach space

Citations: Zbl 1011.47039

Cited in: Zbl 1141.65365

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