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Uniform Contractification

Published online by Cambridge University Press:  20 November 2018

Kok-Keong Tan
Kok-Keong Tan*
Affiliation:
Mathematics Department, Dalhousie University Halifax, Nova Scotia Canada B3H 4H8
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Abstract

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Let (X, τ) be a metrizable space and {fn: n = 1,2,…} be a commuting family of continuous mappings on X with a common fixed point ξ ∈ X such that (I) for each k = l, 2, … and (II) Then for each c ∈ (0,1), there exists a metric d on X inducing the topology τ such that , for all x, y∈X and n = l, 2, …. The above result is also generalized to Tychonoff spaces.

Keywords

54.3554.654.85d-contractionLipschitz constantuniform contractificationtopologically uniformly contractifiable
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 1 , 01 March 1978 , pp. 93 - 97
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Chew, K. P. and Tan, K. K., Remetrization and family of commuting contractive type mappings, to appear in the Proceedings of the Seminar on fixed point theory and applications, Dalhousie University, 1975.Google Scholar
2. Goldman, A. J. and Meyers, P. R., Simultaneous contractification, J. Res. NBS, 73B (Math, and Math. Phys.) No. 4, 301-305.Google Scholar
3. Janos, L., Contraction property of the operator of integration, Can. Math. Bull. 18 (1975), 367-369.Google Scholar
4. Jones, L., A converse of Banach's contraction theorem, Proc. Amer. Math. Soc. 18 (1967), 287-289.Google Scholar