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A Topological Banach Fixed Point Theorem for Compact Hausdorff Spaces

Published online by Cambridge University Press:  20 November 2018

Juris Steprans ,
Stephen Watson  and
Winfried Just
Juris Steprans
Affiliation:
Department of Mathematics, Ohio University Athens, Ohio 45701-2979
Stephen Watson
Affiliation:
Department of Mathematics, Ohio University Athens, Ohio 45701-2979
Winfried Just
Affiliation:
Department of Mathematics, York University North York, Ontario M3J 1P3
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Abstract

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We propose an analogue of the Banach contraction principle for connected compact Hausdorff spaces. We define a J-contraction of a connected compact Hausdorff space. We show that every contraction of a compact metric space is a J-contraction and that any J-contraction of a compact metrizable space is a contraction for some admissible metric. We show that every J-contraction has a unique fixed point and that the orbit of each point converges to this fixed point.

Keywords

Primary: 54H2554E40secondary: 54E4554D0554D30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 4 , 01 December 1994 , pp. 552 - 555
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Edelstein, M., A shorter proof of Janos’ theorem, Proc. Amer. Math. Soc. 20(1969), 509–510.Google Scholar
2. Janos, L., A converse to Banachs contraction mapping theorem, Proc. Amer. Math. Soc. 18(1967), 287–289.Google Scholar