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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 2, Pages 28–33 (Mi vuu319)

MATHEMATICS

Dynamical system of translations in the space of multi-valued functions with closed images

E. A. Panasenko

Department of Algebra and Geometry, Tambov State University, Tambov, Russia

Abstract: In the work there is considered the dynamical system of translations in the space $\mathfrak R$ of continuous multi-valued functions with images in complete metric space $(\mathrm{clos}(\mathbb R^n),\rho_\mathrm{cl})$ of nonempty closed subsets of $\mathbb R^n$. The distance between such functions is measured by means of the metric analogous to the Bebutov metric constructed for the space of continuous real-valued functions defined on the whole real line. It is shown that for compactness of the trajectory's closure in $\mathfrak R$ it is sufficient to have initial function bounded and uniformly continuous in the $\rho_\mathrm{cl}$ metric. As consequence, it is also proved that the trajectory's closure of a recurrent or an almost periodic motion is compact in $\mathfrak R$.

Keywords: space of multivalued functions with closed images, dynamical system of translations, closure of trajectory.

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UDC: 517.938.5+517.911.5
MSC: 37Ñ99, 34A60

Citation: E. A. Panasenko, “Dynamical system of translations in the space of multi-valued functions with closed images”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 2, 28–33

Citation in format AMSBIB
\Bibitem{Pan12} \by E.~A.~Panasenko \paper Dynamical system of translations in the space of multi-valued functions with closed images \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2012 \issue 2 \pages 28--33 \mathnet{http://mi.mathnet.ru/vuu319} 

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This publication is cited in the following articles:
1. L. I. Rodina, E. L. Tonkov, “O mnozhestve dostizhimosti upravlyaemoi sistemy bez predpolozheniya kompaktnosti geometricheskikh ogranichenii na dopustimye upravleniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 4, 68–79
2. E. S. Zhukovskii, E. A. Panasenko, “Definition of the metric on the space $\mathrm{clos}_{\varnothing}(X)$ of closed subsets of a metric space $X$ and properties of mappings with values in $\mathrm{clos}_{\varnothing}(\mathbb{R}}^n)$”, Sb. Math., 205:9 (2014), 1279–1309
3. E. A. Panasenko, “On the Metric Space of Closed Subsets of a Metric Space and Set-Valued Maps with Closed Images”, Math. Notes, 104:1 (2018), 96–110
4. L. I. Danilov, “Shift dynamical systems and measurable selectors of multivalued maps”, Sb. Math., 209:11 (2018), 1611–1643
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