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Proceedings of the American Mathematical Society

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Fixed points and iteration of a nonexpansive mapping in a Banach space


Author: Shiro Ishikawa
Journal: Proc. Amer. Math. Soc. 59 (1976), 65-71
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9939-1976-0412909-X
MathSciNet review: 0412909
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Abstract: The following result is shown. If $ T$ is a nonexpansive mapping from a closed convex subset $ D$ of a Banach space into a compact subset of $ D$ and $ {x_1}$ is any point in $ D$, then the sequence $ \{ {x_n}\} $ defined by $ {x_{n + 1}} = {2^{ - 1}}({x_n} + T{x_n})$ converges to a fixed point of $ T$. As a matter of fact, a theorem which includes this result is proved. Furthermore, a similar result is obtained under certain restrictions which do not imply the assumption on the compactness of $ T$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0412909-X
Keywords: Iteration method, nonexpansive mapping
Article copyright: © Copyright 1976 American Mathematical Society

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