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

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 $t\in {\mathbb R}$.