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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A fixed point theorem for asymptotically nonexpansive mappings

Authors: K. Goebel and W. A. Kirk
Journal: Proc. Amer. Math. Soc. 35 (1972), 171-174
MSC: Primary 47H10
MathSciNet review: 0298500
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Abstract: Let K be a subset of a Banach space X. A mapping $ F:K \to K$ is said to be asymptotically nonexpansive if there exists a sequence $ \{ {k_i}\} $ of real numbers with $ {k_i} \to 1$ as $ i \to \infty $ such that $ \left\Vert {{F^i}x - {F^i}y} \right\Vert \leqq {k_i}\left\Vert {x - y} \right\Vert,x,y \in K$. It is proved that if K is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if $ F:K \to K$ is asymptotically nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.

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Keywords: Fixed point theorem, nonexpansive mapping, asymptotically nonexpansive mapping, uniformly convex Banach space, lipschitzian mapping
Article copyright: © Copyright 1972 American Mathematical Society