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

 

A fixed point theorem for mappings with a nonexpansive iterate


Author: W. A. Kirk
Journal: Proc. Amer. Math. Soc. 29 (1971), 294-298
MSC: Primary 47.85
MathSciNet review: 0284887
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Abstract: Let X be a reflexive Banach space which has strictly convex norm and suppose K is a nonempty, bounded, closed and convex subset of X. Suppose $ T:K \to K$ has the property that, for some positive integer $ N,{T^N}$ is nonexpansive ( $ \left\Vert {{T^N}x - {T^N}y} \right\Vert \leqq \left\Vert {x - y} \right\Vert$ for all $ x,y \in K$). A function $ \gamma (N)$ is determined, $ \gamma (N) > 1$, such that if $ \left\Vert {{T^j}x - {T^j}y} \right\Vert \leqq k\left\Vert {x - y} \right\Vert$ for all $ x,y \in K,1 \leqq j \leqq N - 1$, where $ k < \gamma (N)$, then T has a fixed point in K.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0284887-3
PII: S 0002-9939(1971)0284887-3
Keywords: Fixed point theory, nonexpansive mappings, uniformly convex Banach spaces, normal structure, nonexpansive iterate
Article copyright: © Copyright 1971 American Mathematical Society