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Section
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Mathematics
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Title
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Dynamical system of translations in the space of multi-valued functions with closed images
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Author(-s)
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Panasenko E.A.a
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Affiliations
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Tambov State Universitya
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Abstract
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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 $\big({\rm clos}({\mathbb R}^n), \rho_{{\rm cl}}\big)$ 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 $\rho_{{\rm cl}}$ metric. As consequence, it is also proved that the trajectory's closure of a recurrent or an almost periodic motion is compact in ${\mathfrak R}.$
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Keywords
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space of multivalued functions with closed images, dynamical system of translations, closure of trajectory
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UDC
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517.938.5, 517.911.5
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MSC
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37С99, 34A60
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DOI
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10.20537/vm120203
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Received
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27 December 2011
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Language
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Russian
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Citation
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Panasenko E.A. Dynamical system of translations in the space of multi-valued functions with closed images, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 2, pp. 28-33.
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References
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- Nemytskii V.V., Stepanov V.V. Kachestvennaya teoriya differentsial'nykh uravnenii (Qualitative theory of differential equations), Moscow: Gos. Izd. Tekh. Teor. Lit., 1949, 550 p.
- Bebutov M.V. Dynamical systems in the space of continuous function, Bull. Mat. Inst. Moscow State University, 1940, vol. 2, no. 5, pp. 1-52.
- Zhukovskii E.S., Panasenko E.A. On one metric in the space of nonempty closed subsets of ${\mathbb R}$$n$ , Vestn. Udmurt. Univ. Mat. Mekh. Komp'ut. Nauki, 2012, no. 1, pp. 15-26.
- Panasenko E.A., Tonkov E.L. Invariant and stably invariant sets for differential inclusions, Tr. Mat. Inst. Steklov, 2008, vol. 262, pp. 202-221.
- Panasenko E.A., Tonkov E.L. Extension of E.A. Barbashin's and N.N. Krasovskii's stability theorems to controlled dynamical systems, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 268, suppl. 1, pp. 204-221.
- Panasenko E.A. On existence of recurrent and almost periodic solutions to differential inclusion, Vestn. Udmurt. Univ. Mat. Mekh. Komp'ut. Nauki, 2010, no. 3, pp. 42-57.
- Panasenko E.A., Rodina L.I., Tonkov E.L. The space ${\rm clcv}($ $\mathbb{R}$$n$ $)$ with the Hausdorff-Bebutov metric and differential inclusions, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 275, suppl. 1, pp. 121-136.
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Full text
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