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Proceedings of the American Mathematical Society

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

 

 

An abstract fixed point theorem for nonexpansive mappings


Author: W. A. Kirk
Journal: Proc. Amer. Math. Soc. 82 (1981), 640-642
MSC: Primary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1981-0614894-6
MathSciNet review: 614894
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Abstract: A class $ \mathcal{S}$ of subsets of a bounded metric space is said to be normal if each member of $ \mathcal{S}$ contains a nondiametral point. An induction proof is given for the following. Suppose $ M$ is a nonempty bounded metric space which contains a class $ \mathcal{S}$ of subsets which is countably compact, normal, stable under arbitrary intersections, and which contains the closed balls in $ M$. Then every nonexpansive self-mapping of $ M$ has a fixed point.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0614894-6
Keywords: Fixed point theorem, nonexpansive mappings
Article copyright: © Copyright 1981 American Mathematical Society