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Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2023, Volume 61, Pages 57–75
DOI: https://doi.org/10.35634/2226-3594-2023-61-04
(Mi iimi442)
 

MATHEMATICS

On a class of Besicovitch almost periodic type selections of multivalued maps

L. I. Danilov

Udmurt Federal Research Center, Ural Branch of the Russian Academy of Sciences, ul. T. Baramzinoi, 34, Izhevsk, 426067, Russia
References:
Abstract: Let ${\mathcal B}$ be a Banach space and let ${\mathcal M}^p({\mathbb R};{\mathcal B})$, $p\geqslant 1$, be the Marcinkiewicz space with a seminorm $\| \cdot \| _{{\mathcal M}^p}$. By $\widetilde {\mathfrak B}^p_c({\mathbb R};{\mathcal B})$ we denote the set of functions ${\mathcal F}\in {\mathcal M}^p({\mathbb R};{\mathcal B})$ that satisfy the following three conditions: (1) $\| {\mathcal F}(\cdot )-{\mathcal F}(\cdot +\tau )\| _{{\mathcal M}^p}\to 0$ as $\tau \to 0$, (2) for every $\varepsilon >0$ the set of ($\varepsilon ,\| \cdot \| _{{\mathcal M}^p}$)-almost periods of the function ${\mathcal F}$ is relatively dense, (3) for every $\varepsilon >0$ there exists a set $X(\varepsilon )\subseteq {\mathbb R}$ such that $\| \chi _{X(\varepsilon )}\| _{{\mathcal M}^1({\mathbb R};{\mathbb R})}<\varepsilon $ and the set $\{ {\mathcal F}(t):t\in {\mathbb R}\, \backslash \, X(\varepsilon )\} $ has a finite $\varepsilon $-net. Let $\widetilde {\mathcal M}^{p,\circ }({\mathbb R};{\mathcal B})$ be the set of functions ${\mathcal F}\in {\mathcal M}^p({\mathbb R};{\mathcal B})$ that satisfy the condition (3) and the following condition: for any $\varepsilon >0$ there is a number $\delta >0$ such that the estimate $\| \chi _X{\mathcal F}\| _{{\mathcal M}^p}<\varepsilon $ is fulfilled for all sets $X\subseteq {\mathbb R}$ with $\| \chi _X\| _{{\mathcal M}^1({\mathbb R};{\mathbb R})}<\delta $. The sets $\widetilde {\mathfrak B}^p_c({\mathbb R};U)$ and $\widetilde {\mathcal M}^{p,\circ }({\mathbb R};U)$ for a complete metric space $(U,\rho )$ are defined analogously. By ${\mathrm {cl}}\, U$ denote the metric space of nonempty, closed, and bounded subsets of the space $(U,\rho )$ with Hausdorff metrics. In the paper, in particular, for any $F\in \widetilde {\mathfrak B}^p_c({\mathbb R};{\mathrm {cl}}\, U)$, $p\geqslant 1$, and $u\in U$, $\varepsilon >0$, we prove under the condition $\rho (u,F(\cdot ))\in \widetilde {\mathcal M}^{p,\circ }({\mathbb R};{\mathbb R})$ the existence of a function ${\mathcal F}\in \widetilde {\mathfrak B}^p_c({\mathbb R};U)\cap \widetilde {\mathcal M}^{p,\circ }({\mathbb R};U)$ such that ${\mathcal F}(t)\in F(t)$ and $\rho (u,{\mathcal F}(t))<\varepsilon +\rho (u,F(t))$ for almost every $t\in {\mathbb R}$.
Keywords: Besicovitch almost periodic type functions, selections, multivalued maps.
Received: 28.01.2023
Accepted: 20.03.2023
Bibliographic databases:
Document Type: Article
UDC: 517.518.6
MSC: 28B20, 42A75
Language: Russian
Citation: L. I. Danilov, “On a class of Besicovitch almost periodic type selections of multivalued maps”, Izv. IMI UdGU, 61 (2023), 57–75
Citation in format AMSBIB
\Bibitem{Dan23}
\by L.~I.~Danilov
\paper On a class of Besicovitch almost periodic type selections of multivalued maps
\jour Izv. IMI UdGU
\yr 2023
\vol 61
\pages 57--75
\mathnet{http://mi.mathnet.ru/iimi442}
\crossref{https://doi.org/10.35634/2226-3594-2023-61-04}
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