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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Contractification of a semigroup of maps
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by Hwei Mei Ko and Kok Keong Tan PDF
Proc. Amer. Math. Soc. 77 (1979), 267-275 Request permission

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 $\lambda$.
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Additional 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