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Russia Izhevsk
Section Mathematics
Title Recurrent and almost recurrent multivalued maps and their selections. II
Author(-s) Danilov L.I.a
Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa
Abstract 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).
Keywords recurrent function, selection, multivalued mapping
UDC 517.518.6
MSC 42A75, 54C65
DOI 10.20537/vm120401
Received 17 May 2012
Language Russian
Citation 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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