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Proceedings of the American Mathematical Society

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

 
 

 

Even continuity and the Banach contraction principle


Authors: J. L. Solomon and Ludvik Janos
Journal: Proc. Amer. Math. Soc. 69 (1978), 166-168
MSC: Primary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1978-0500891-8
MathSciNet review: 0500891
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Abstract: In 1967, Philip R. Meyers established a nice converse to the Banach Contraction Mapping Theorem. We provide a counterexample to one of his corollaries and show that if X is a metrizable topological space, f a continuous self-map on X such that: (a)f has a fixed point p which has a compact neighborhood; (b) $ {f^n}(x) \to p$ as $ n \to \infty $ for each x in X, then the following are equivalent: (1) f is a contraction relative to a suitable metric on X; (2) the sequence of iterates $ \{ {f^n}\} _{n = 1}^\infty $ is evenly continuous.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0500891-8
Keywords: Banach contraction principle, evenly continuous family, fixed point
Article copyright: © Copyright 1978 American Mathematical Society

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