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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 4, Pages 3–21 (Mi vuu345)

MATHEMATICS

Recurrent and almost recurrent multivalued maps and their selections. II

L. I. Danilov

Physical–Technical Institute, Ural Branch of the Russian Academy of Sciences, Izhevsk, Russia

Abstract: In the paper, we consider the problem of existence of recurrent and almost recurrent selections of multivalued mappings $\mathbb R\ni t\mapsto F(t)\in\operatorname{comp}U$ with nonempty compact sets $F(t)$ in a complete metric space $U$. The set $\operatorname{comp}U$ 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 $(\operatorname{comp}U,\mathrm{dist})$. 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.

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UDC: 517.518.6
MSC: 42A75, 54C65

Citation: L. I. Danilov, “Recurrent and almost recurrent multivalued maps and their selections. II”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 4, 3–21

Citation in format AMSBIB
\Bibitem{Dan12} \by L.~I.~Danilov \paper Recurrent and almost recurrent multivalued maps and their selections.~II \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2012 \issue 4 \pages 3--21 \mathnet{http://mi.mathnet.ru/vuu345} 

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This publication is cited in the following articles:
1. L. I. Danilov, “Ravnomernaya approksimatsiya rekurrentnykh i pochti rekurrentnykh funktsii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 4, 36–54
2. L. I. Danilov, “Rekurrentnye i pochti rekurrentnye mnogoznachnye otobrazheniya i ikh secheniya. III”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 4, 25–52
3. L. I. Danilov, “Shift dynamical systems and measurable selectors of multivalued maps”, Sb. Math., 209:11 (2018), 1611–1643
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