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A Topological Banach Fixed Point Theorem for Compact Hausdorff Spaces

  • Juris Steprans (a1), Stephen Watson (a1) and Winfried Just (a2)

Abstract

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.

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References

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1. Edelstein, M., A shorter proof of Janos’ theorem, Proc. Amer. Math. Soc. 20(1969), 509510.
2. Janos, L., A converse to Banachs contraction mapping theorem, Proc. Amer. Math. Soc. 18(1967), 287289.
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A Topological Banach Fixed Point Theorem for Compact Hausdorff Spaces

  • Juris Steprans (a1), Stephen Watson (a1) and Winfried Just (a2)

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