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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, Issue 4, Pages 25–52 (Mi vuu449)

MATHEMATICS

Recurrent and almost recurrent multivalued maps and their selections. III

L. I. Danilov

Physical Technical Institute, Ural Branch of the Russian Academy of Sciences, ul. Kirova, 132, Izhevsk, 426000, Russia

Abstract: Let $(U,\rho)$ be a complete metric space and let $\mathcal R^p(\mathbb R,U),$ $p\geqslant 1$, and $\mathcal R(\mathbb R,U)$ be the spaces of (strongly) measurable functions $f\colon\mathbb R\to U$ for which the Bochner transforms $\mathbb R\ni t\mapsto f^B_l(t;\cdot)=f(t+\cdot)$ are recurrent functions with ranges in the metric spaces $L^p([-l,l],U)$ and $L^1([-l,l],(U,\rho'))$ where $l>0$, and $(U,\rho')$ is the complete metric space with the metric $\rho'(x,y)=\min\{1,\rho(x,y)\}$, $x,y\in U$. Analogously, we define the spaces $\mathcal R^p(\mathbb R,\mathrm{cl}_bU)$ and $\mathcal R(\mathbb R,\mathrm{cl}_bU)$ of functions (multivalued mappings) $F\colon\mathbb R\to\mathrm{cl}_bU$ with ranges in the complete metric space $(\mathrm{cl}_bU,\mathrm{dist})$ of nonempty closed bounded subsets of the metric space $(U,\rho)$ with the Hausdorff metric $\mathrm{dist}$ (while defining the multivalued mappings $F\in\mathcal R(\mathbb R,\mathrm{cl}_bU)$ the metric $\mathrm{dist}'(X,Y)=\min\{1,\mathrm{dist}(X,Y)\}$, $X,Y\in\mathrm{cl}_bU$, is also considered). We prove the existence of selectors $f\in\mathcal R(\mathbb R,U)$ (accordingly $f\in\mathcal R^p(\mathbb R,U)$) of multivalued maps $F\in\mathcal R(\mathbb R,\mathrm{cl}_bU)$ (accordingly $F\in\mathcal R^p(\mathbb R,\mathrm{cl}_bU)$) for which the sets of almost periods are subordinated to the sets of almost periods of multivalued maps $F$. For functions $g\in\mathcal R(\mathbb R,U),$ the conditions for the existence of selectors $f\in\mathcal R(\mathbb R,U)$ and $f\in\mathcal R^p(\mathbb R,U)$ such that $\rho(f(t),g(t))=\rho(g(t),F(t))$ for a.e. $t\in\mathbb R$ are obtained. On the assumption that the function $g$ and the multivalued map $F$ have relatively dense sets of common $\varepsilon$-almost periods for all $\varepsilon>0$, we also prove the existence of selectors $f\in\mathcal R(\mathbb R,U)$ such that $\rho(f(t),g(t))\leqslant\rho(g(t),F(t))+\eta(\rho(g(t),F(t)))$ for a.e. $t\in\mathbb R$, where $\eta\colon[0,+\infty)\to[0,+\infty)$ is an arbitrary nondecreasing function for which $\eta(0)=0$ and $\eta(\xi)>0$ for all $\xi>0$, and, moreover, $f\in\mathcal R^p(\mathbb R,U)$ if $F\in\mathcal R^p(\mathbb R,\mathrm{cl}_bU)$. To prove the results we use the uniform approximation of functions $f\in\mathcal R(\mathbb R,U)$ by elementary functions belonging to the space $\mathcal R(\mathbb R,U)$ which have the sets of almost periods subordinated to the sets of almost periods of the functions $f$.

Keywords: recurrent function, selector, multivalued map.

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UDC: 517.518.6
MSC: 42A75, 54C65

Citation: L. I. Danilov, “Recurrent and almost recurrent multivalued maps and their selections. III”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 4, 25–52

Citation in format AMSBIB
\Bibitem{Dan14} \by L.~I.~Danilov \paper Recurrent and almost recurrent multivalued maps and their selections.~III \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2014 \issue 4 \pages 25--52 \mathnet{http://mi.mathnet.ru/vuu449} 

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This publication is cited in the following articles:
1. L. I. Danilov, “Rekurrentnye i pochti avtomorfnye secheniya mnogoznachnykh otobrazhenii”, Izv. IMI UdGU, 2015, no. 2(46), 45–52
2. L. I. Danilov, “Shift dynamical systems and measurable selectors of multivalued maps”, Sb. Math., 209:11 (2018), 1611–1643
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