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

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

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]**L. Janos,*On the Edelstein contractive mapping theorem*, Canad. Math. Bull.**18**(1975), 675-678. MR**0420589 (54:8603)****[2]**L. Janos, H.-M. Ko and K.-K. Tan,*Edelstein's contractivity and attractors*, Proc. Amer. Math. Soc. (to appear).**[3]**J. L. Kelley,*General topology*, Van Nostrand Company Inc., Princeton, N. J., 1955. MR**0070144 (16:1136c)****[4]**S. Leader,*A topological characterization of Banach contractions*, Pacific J. Math.**69**(1977), 461-466. MR**0436093 (55:9044)****[5]**J. L. Solomon and L. Janos,*Even continuity and Banach contraction principle*, Proc. Amer. Math. Soc.**69**(1978), 166-168. MR**0500891 (58:18398)**

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Additional Information

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