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Proceedings of the American Mathematical Society

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The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces


Authors: Kok-Keong Tan and Hong Kun Xu
Journal: Proc. Amer. Math. Soc. 114 (1992), 399-404
MSC: Primary 47H09; Secondary 47A35, 47H10
MathSciNet review: 1068133
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Abstract: Let $ X$ be a uniformly convex Banach space with a Frechet differentiable norm, $ C$ a bounded closed convex subset of $ X$, and $ T:C \to C$ an asymptotically nonexpansive mapping. It is shown that for each $ x$ in $ C$, the sequence $ \{ {T^n}x\} $ is weakly almost-convergent to a fixed point $ y$ of $ T$, i.e., $ (1/n)\sum\nolimits_{i = 0}^{n - 1} {{T^{k + i}}x \to y} $ weakly as $ n$ tends to infinity uniformly in $ k = 0,1,2, \ldots $


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DOI: https://doi.org/10.1090/S0002-9939-1992-1068133-2
Keywords: Asymptotically nonexpansive mapping, nonlinear ergodic theorem, fixed point
Article copyright: © Copyright 1992 American Mathematical Society