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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 1, Pages 15–25 (Mi vuu306)

This article is cited in 5 scientific papers (total in 5 papers)

MATHEMATICS

On one metric in the space of nonempty closed subsets of $\mathbb R^n$

E. S. Zhukovskii, E. A. Panasenko

Department of Algebra and Geometry, Tambov State University named after G. R. Derzhavin, Tambov, Russia

Abstract: In the work, there is presented a new metric in the space $\operatorname{clos}(\mathbb R^n)$ of all nonempty closed (not necessarily bounded) subsets of $\mathbb R^n$. The convergence of sets in this metric is equivalent to convergence in the Hausdorff metric of the intersections of the given sets with the balls of any positive radius centered at zero united then with the corresponding spheres. It is proved that, with respect to the metric considered, the space $\operatorname{clos}(\mathbb R^n)$ is complete, and its subspace of nonempty closed convex subsets of $\mathbb R^n$ is closed. There are also derived the conditions that guarantee the equivalence of convergence in this metric to convergence in the Hausdorff metric, and to convergence in the Hausdorff–Bebutov metric. The results obtained can be applied to studying control problems and differential inclusions.

Keywords: complete metric space of nonempty closed subsets of ${\mathbb R}^n,$ subspaces, convergence.

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UDC: 515.124+517.911.5
MSC: 54E50, 34A60
Received: 12.10.2011

Citation: E. S. Zhukovskii, E. A. Panasenko, “On one metric in the space of nonempty closed subsets of $\mathbb R^n$”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 1, 15–25

Citation in format AMSBIB
\Bibitem{ZhuPan12} \by E.~S.~Zhukovskii, E.~A.~Panasenko \paper On one metric in the space of nonempty closed subsets of~$\mathbb R^n$ \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2012 \issue 1 \pages 15--25 \mathnet{http://mi.mathnet.ru/vuu306} 

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This publication is cited in the following articles:
1. E. A. Panasenko, “Dinamicheskaya sistema sdvigov v prostranstve mnogoznachnykh funktsii s zamknutymi obrazami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 2, 28–33
2. 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
3. E. S. Zhukovskiy, E. A. Panasenko, “On fixed points of multi-valued maps in metric spaces and differential inclusions”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 2, 12–26
4. 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
5. 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
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