{"title":"On a class of Besicovitch almost periodic type selections of multivalued maps","authors":"L. I. Danilov","doi":"10.35634/2226-3594-2023-61-04","DOIUrl":null,"url":null,"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 $t\\in {\\mathbb R}$.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":"7 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.35634/2226-3594-2023-61-04","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 $t\in {\mathbb R}$.