Section
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Mathematics
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Title
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Recurrent and almost recurrent multivalued maps and their selections. II
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Author(-s)
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Danilov L.I.a
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Affiliations
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Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa
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Abstract
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In the paper, we consider the problem of existence of recurrent and almost recurrent selections of multivalued mappings ${\mathbb R}\ni t\mapsto F(t)\in {\mathrm {comp}}\, U$ with nonempty compact sets $F(t)$ in a complete metric space $U.$ The set ${\mathrm {comp}}\, U$ is equipped with the Hausdorff metric ${\mathrm {dist}}$. Recurrent and almost recurrent multivalued maps are defined as the functions with values in the metric space $({\mathrm {comp}}\, U,{\mathrm {dist}}).$ It is proved that there are recurrent (almost recurrent) selections of multivalued recurrent (almost recurrent) uniformly absolutely continuous maps. We also consider mappings ${\mathbb R}\ni t\mapsto F(t)$ with the sets $F(t)$ consisting of a finite number of points (the number depends on the $t\in {\mathbb R}$). We prove that if such a map is almost recurrent, then it has an almost recurrent selection. A multivalued recurrent mapping $t\mapsto F(t)$ with sets $F(t)$ consisting of at most $n$ points (where $n\in {\mathbb N}$) has a recurrent selection. If the sets $F(t)$ of a multivalued recurrent (almost recurrent) mapping $t\mapsto F(t)$ consist of $n$ points for all $t\in {\mathbb R},$ then all $n$ continuous selections of the map $F$ are recurrent (almost recurrent).
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Keywords
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recurrent function, selection, multivalued mapping
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UDC
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517.518.6
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MSC
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42A75, 54C65
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DOI
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10.20537/vm120401
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Received
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17 May 2012
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Language
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Russian
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Citation
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Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections. II, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 4, pp. 3-21.
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References
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