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An application of combinatorial techniques to a topological problem

Published online by Cambridge University Press:  17 April 2009

Ludvik Janos
Ludvik Janos
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle, New South Wales.
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Abstract

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The following statement is proved: Let X be a set having at most continuously many elements and f: X → X a mapping such that each iteration fn (n = 1, 2, …) has a unique fixed point. Then for every number c ∈ (0, 1) there exists a metric p on X such that the metric space (X, p) is separable and the mapping f is a.contraction with the Lipschitz constant c.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 3 , December 1973 , pp. 439 - 443
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Bacon, Philip, “Extending a complete metric”, Amer. Math. Monthly 75 (1968), 642–643.CrossRefGoogle Scholar
[2]Bessaga, C., “On the converse of the Banach ‘fixed-point principle’”, Colloq. Math. 7 (1959), 41–42.CrossRefGoogle Scholar
[3]de Groot, J. and de Vries, H., “Metrization of a set which is mapped into itself”, Quart. J. Math. Oxford (2) 9 (1958), 144–148.CrossRefGoogle Scholar
[4]Meyers, Philip R., “A converse to Banach's contraction theorem”, J. Res. Nat. Bur. Standards Seat. B 71 (1967), 73–76.CrossRefGoogle Scholar