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 Mat. Sb., 2014, Volume 205, Number 9, Pages 65–96 (Mi msb8240)

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 $\mathrm{clos}_{\varnothing}(\mathbb{R}}^n)$

E. S. Zhukovskii, E. A. Panasenko

Institute of Mathematics, Physics and Information Science, Tambov State University

Abstract: The paper is concerned with the extension of tests for superpositional measurability, Filippov's implicit function lemma and the Scorza Dragoni property to set-valued (and, as a corollary, to single-valued) mappings that fail to satisfy the Carathéodory conditions (the upper Carathéodory conditions) and are not continuous (upper semicontinuous) in the phase variable. To obtain the corresponding results the space $\mathrm{clos}_{\varnothing}(X)$ of all closed subsets (including the empty set) of an arbitrary metric space $X$ is introduced; a metric on $\mathrm{clos}_{\varnothing}(X)$ is proposed; the space $\mathrm{clos}_{\varnothing}(X)$ is shown to be complete whenever the original space $X$ is; a criterion for convergence of a sequence is put forward; mappings with values in $\mathrm{clos}_\varnothing(X)$ are studied. Some results on set-valued mappings satisfying the Carathéodory conditions and having compact values in $\mathbb R^n$ are shown to hold for mappings with values in $\mathrm{clos}_\varnothing(\mathbb R^n)$, measurable in the first argument, and continuous in the proposed metric in the second argument.
Bibliography: 22 titles.

Keywords: superpositional measurability, Filippov's implicit function lemma, Scorza Dragoni property, the space of closed subsets of a metric space, set-valued mapping.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 2014/285

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8240

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English version:
Sbornik: Mathematics, 2014, 205:9, 1279–1309

Bibliographic databases:

UDC: 515.124+515.126.83
MSC: 54C60, 54C65, 54E35

Citation: 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 $\mathrm{clos}_{\varnothing}(\mathbb{R}}^n)$”, Mat. Sb., 205:9 (2014), 65–96; Sb. Math., 205:9 (2014), 1279–1309

Citation in format AMSBIB
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\issue 9
\pages 65--96
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\transl
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\yr 2014
\vol 205
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\pages 1279--1309
\crossref{https://doi.org/10.1070/SM2014v205n09ABEH004418}
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• http://mi.mathnet.ru/eng/msb8240
• https://doi.org/10.4213/sm8240
• http://mi.mathnet.ru/eng/msb/v205/i9/p65

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Panasenko E.A., “O vpolne ogranichennykh i kompaktnykh mnozhestvakh v prostranstve zamknutykh podmnozhestv metricheskogo prostranstva”, Vestnik Tambovskogo universiteta. Seriya: Estestvennye i tekhnicheskie nauki, 20 (2015), 1340–1344
2. 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
3. E. S. Zhukovskii, E. M. Yakubovskaya, “O suschestvovanii i otsenkakh reshenii funktsionalnykh vklyuchenii”, Tr. IMM UrO RAN, 25, no. 1, 2019, 45–54
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