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On the Edelstein Contractive Mapping Theorem

Published online by Cambridge University Press:  20 November 2018

Ludvik Janos
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Abstract

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Let X be a metrizable topological space and f:X→X a continuous selfmapping such that for every x ∈ X the sequence of iterates {fn(x)} converges. It is proved that under these conditions the following two statements are equivalent:

1. There is a metrization of X relative to which f is contractive in the sense of Edelstein.

2. For any nonempty f-invariant compact subset Y of X the intersection of all iterates fn(Y) is a one-point set. The relation between this type of contractivity and the Banach contraction principle is also discussed.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 5 , 01 December 1975 , pp. 675 - 678
Copyright
Copyright © Canadian Mathematical Society 1975

References

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