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

This article is cited in 3 scientific papers (total in 3 papers)

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.

Full text: PDF file (288 kB)
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UDC: 517.518.6
MSC: 42A75, 54C65
Received: 17.05.2012

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  mathnet
    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  mathnet
    3. L. I. Danilov, “Shift dynamical systems and measurable selectors of multivalued maps”, Sb. Math., 209:11 (2018), 1611–1643  mathnet  crossref  crossref  adsnasa  isi  elib
  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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