Even continuity and the Banach contraction principle
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- by J. L. Solomon and Ludvik Janos
- Proc. Amer. Math. Soc. 69 (1978), 166-168
- DOI: https://doi.org/10.1090/S0002-9939-1978-0500891-8
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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.References
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
- Philip R. Meyers, A converse to Banach’s contraction theorem, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 73–76. MR 221469
Bibliographic Information
- © Copyright 1978 American Mathematical Society
- 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