Contractification of a semigroup of maps

Authors:
Hwei Mei Ko and Kok Keong Tan

Journal:
Proc. Amer. Math. Soc. **77** (1979), 267-275

MSC:
Primary 54E10

MathSciNet review:
542096

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a metrizable topological space, be the family of all metrics on *X* whose metric topologies are . Assume that the semigroup *F* of maps from *X* into itself, with composition as its semigroup operation, is equicontinuous under some ; then we have the following results:

I. There exists such that *f* is nonexpansive under for each .

II. If *F* is countable, commutative, and for each , there is such that the sequence converges to , then there exists such that *f* is contractive under for each .

III. If there is such that (1) and , (2) there is a neighbourhood *B* of *p* such that for any choice of , and the limit depends on *m* only, then for each with , there exists such that each *f* in *F* is a Banach contraction under with Lipschitz constant .

**[1]**Ludvik Janos,*On the Edelstein contractive mapping theorem*, Canad. Math. Bull.**18**(1975), no. 5, 675–678. MR**0420589****[2]**L. Janos, H.-M. Ko and K.-K. Tan,*Edelstein's contractivity and attractors*, Proc. Amer. Math. Soc. (to appear).**[3]**John L. Kelley,*General topology*, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. MR**0070144****[4]**Solomon Leader,*A topological characterization of Banach contractions*, Pacific J. Math.**69**(1977), no. 2, 461–466. MR**0436093****[5]**J. L. Solomon and Ludvik Janos,*Even continuity and the Banach contraction principle*, Proc. Amer. Math. Soc.**69**(1978), no. 1, 166–168. MR**0500891**, 10.1090/S0002-9939-1978-0500891-8

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DOI:
https://doi.org/10.1090/S0002-9939-1979-0542096-1

Keywords:
Contractification,
nonexpansive,
contractive,
equicontinuous,
evenly continuous,
one point compactification

Article copyright:
© Copyright 1979
American Mathematical Society