## Contractification of a semigroup of maps

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- by Hwei Mei Ko and Kok Keong Tan
- Proc. Amer. Math. Soc.
**77**(1979), 267-275 - DOI: https://doi.org/10.1090/S0002-9939-1979-0542096-1
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## Abstract:

Let $(X,\tau )$ be a metrizable topological space, $\mathcal {P}(\tau )$ be the family of all metrics on*X*whose metric topologies are $\tau$. Assume that the semigroup

*F*of maps from

*X*into itself, with composition as its semigroup operation, is equicontinuous under some $d \in \mathcal {P}(\tau )$; then we have the following results: I. There exists $d’ \in \mathcal {P}(\tau )$ such that

*f*is nonexpansive under $d’$ for each $f \in F$. II. If

*F*is countable, commutative, and for each $f \in F$, there is ${x_f} \in X$ such that the sequence $({f^n}(x))_{n = 1}^\infty$ converges to ${x_f},\forall x \in X$, then there exists $d'' \in \mathcal {P}(\tau )$ such that

*f*is contractive under $d''$ for each $f \in F$. III. If there is $p \in X$ such that (1) ${\lim _{n \to \infty }}{f^n}(x) = p,\forall x \in X$ and $\forall f \in F$, (2) there is a neighbourhood

*B*of

*p*such that ${\lim _{m \to \infty }}{f_{{n_1}}}{f_{{n_2}}} \cdots {f_{{n_m}}}(B) = \{ p\}$ for any choice of ${f_{{n_i}}} \in F,i = 1, \ldots ,m$, and the limit depends on

*m*only, then for each $\lambda$ with $0 < \lambda < 1$, there exists $d''’ \in \mathcal {P}(\tau )$ such that each

*f*in

*F*is a Banach contraction under $d''’$ with Lipschitz constant $\lambda$.

## References

- Ludvik Janos,
*On the Edelstein contractive mapping theorem*, Canad. Math. Bull.**18**(1975), no. 5, 675–678. MR**420589**, DOI 10.4153/CMB-1975-118-8
L. Janos, H.-M. Ko and K.-K. Tan, - John L. Kelley,
*General topology*, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR**0070144** - Solomon Leader,
*A topological characterization of Banach contractions*, Pacific J. Math.**69**(1977), no. 2, 461–466. MR**436093** - J. L. Solomon and Ludvik Janos,
*Even continuity and the Banach contraction principle*, Proc. Amer. Math. Soc.**69**(1978), no. 1, 166–168. MR**500891**, DOI 10.1090/S0002-9939-1978-0500891-8

*Edelstein’s contractivity and attractors*, Proc. Amer. Math. Soc. (to appear).

## Bibliographic Information

- © Copyright 1979 American Mathematical Society
- 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