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 Izv. IMI UdGU, 2015, Issue 2(46), Pages 45–52 (Mi iimi301)

Recurrent and almost automorphic selections of multivalued mappings

L. I. Danilov

Physical Technical Institute of the Ural Branch of the Russian Academy of Sciences, ul. Kirova, 132, Izhevsk, 426001, Russia

Abstract: Let $(U,\rho )$ be a complete metric space and $({\mathrm {cl}}_{ b} U,{\mathrm {dist}})$ be the metric space of nonempty closed bounded subsets of the space $U$ with the Hausdorff metric ${\mathrm {dist}}$. On the set $M({\mathbb R},U)$ of strongly measurable functions $f\colon{\mathbb R}\to U$ we introduce the metric $d^{(\rho )}$ such that the convergence in this metric is equivalent to the convergence in Lebesgue measure on every closed interval $[-l,l]$, $l>0$. The metric $d^{({\mathrm {dist}})}$ on the set $M({\mathbb R},{\mathrm {cl}}_{ b} U)$ of strongly measurable multivalued mappings $f\colon{\mathbb R}\to {\mathrm {cl}}_{ b} U$ (which are considered as functions with the range in ${\mathrm {cl}}_{ b} U$) is defined by analogy with the metric $d^{(\rho )}.$ The spaces $M({\mathbb R},U)$ and $M({\mathbb R},{\mathrm {cl}}_{ b} U)$ are the phase spaces of the dynamical systems of translations. For a multivalued Stepanov-like recurrent mapping $F\in {\mathcal R}({\mathbb R},{\mathrm {cl}}_{ b} U)\subseteq M({\mathbb R},{\mathrm {cl}}_{ b} U)$ and for any $x_0\in U$ and any nondecreasing function $\eta \colon[0,+\infty )\to [0,+\infty )$ for which $\eta (0)=0$ and $\eta (\xi )>0$ for $\xi >0$, it is proved that there exists a homomorphism of dynamical systems ${\mathcal F}:\overline {{\mathrm {orb}} F}=\overline {\{ F(\cdot +t):t\in {\mathbb R}\} }\to M({\mathbb R},U)$ such that $({\mathcal F}F^{ \prime })(t)\in F^{ \prime }(t)$ and $\rho (({\mathcal F}F^{ \prime })(t),x_0)\leqslant \rho (x_0,F^{ \prime }(t))+\eta ( \rho (x_0,F^{ \prime }(t)))$ for all $F^{ \prime }\in \overline {{\mathrm {orb}} F}$ and a.e. $t\in {\mathbb R}$. Furthermore, the functions ${\mathcal F}F^{ \prime }$ are Stepanov-like recurrent. If the multivalued mapping $F$ is Stepanov-like almost automorphic, then the function ${\mathcal F}F$ is Stepanov-like almost automorphic as well.

Keywords: recurrent function, almost automorphic function, selector, multivalued mapping.

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UDC: 517.518.6
MSC: 42A75, 54C65

Citation: L. I. Danilov, “Recurrent and almost automorphic selections of multivalued mappings”, Izv. IMI UdGU, 2015, no. 2(46), 45–52

Citation in format AMSBIB
\Bibitem{Dan15} \by L.~I.~Danilov \paper Recurrent and almost automorphic selections of multivalued mappings \jour Izv. IMI UdGU \yr 2015 \issue 2(46) \pages 45--52 \mathnet{http://mi.mathnet.ru/iimi301} \elib{http://elibrary.ru/item.asp?id=25030021} 

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This publication is cited in the following articles:
1. L. I. Danilov, “Shift dynamical systems and measurable selectors of multivalued maps”, Sb. Math., 209:11 (2018), 1611–1643
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