Contractification of a semigroup of maps
Hwei Mei Ko and Kok Keong Tan
Proc. Amer. Math. Soc. 77 (1979), 267-275
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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 .
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L. Janos, H.-M. Ko and K.-K. Tan, Edelstein's contractivity and attractors, Proc. Amer. Math. Soc. (to appear).
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(58 #18398), http://dx.doi.org/10.1090/S0002-9939-1978-0500891-8
- L. Janos, On the Edelstein contractive mapping theorem, Canad. Math. Bull. 18 (1975), 675-678. MR 0420589 (54:8603)
- L. Janos, H.-M. Ko and K.-K. Tan, Edelstein's contractivity and attractors, Proc. Amer. Math. Soc. (to appear).
- J. L. Kelley, General topology, Van Nostrand Company Inc., Princeton, N. J., 1955. MR 0070144 (16:1136c)
- S. Leader, A topological characterization of Banach contractions, Pacific J. Math. 69 (1977), 461-466. MR 0436093 (55:9044)
- 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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