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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Fixed point iteration processes for asymptotically nonexpansive mappings


Authors: Kok-Keong Tan and Hong Kun Xu
Journal: Proc. Amer. Math. Soc. 122 (1994), 733-739
MSC: Primary 47H17; Secondary 47H09, 47H10
MathSciNet review: 1203993
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Abstract: Let X be a uniformly convex Banach space which satisfies Opial's condition or has a Fréchet differentiable norm, C a bounded closed convex subset of X, and $ T:C \to C$ an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by $ {x_{n + 1}} = {t_n}{T^n}{x_n} + (1 - {t_n}){x_n}$ and $ {x_{n + 1}} = {t_n}{T^n}({s_n}{T^n}{x_n} + (1 - {s_n}){x_n}) + (1 - {t_n}){x_n}$, respectively, converge weakly to a fixed point of T.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1203993-5
PII: S 0002-9939(1994)1203993-5
Keywords: Fixed point, asymptotically nonexpansive mapping, fixed point iteration process, uniformly convex Banach space, Fréchet differentiable norm, Opial's condition
Article copyright: © Copyright 1994 American Mathematical Society