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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A fixed point theorem for asymptotically nonexpansive mappings
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by K. Goebel and W. A. Kirk
Proc. Amer. Math. Soc. 35 (1972), 171-174
DOI: https://doi.org/10.1090/S0002-9939-1972-0298500-3

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 $\left \| {{F^i}x - {F^i}y} \right \| \leqq {k_i}\left \| {x - y} \right \|,x,y \in K$. 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.
References
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Bibliographic Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 35 (1972), 171-174
  • MSC: Primary 47H10
  • DOI: https://doi.org/10.1090/S0002-9939-1972-0298500-3
  • MathSciNet review: 0298500