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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
4
Pages
25-52
 Section Mathematics Title Recurrent and almost recurrent multivalued maps and their selections. III Author(-s) Danilov L.I.a Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa 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 $f:{\mathbb R}\to U$ 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 $L^1([-l,l],(U,\rho ^{ \prime }))$ where $l>0$, and $(U,\rho ^{ \prime })$ is the complete metric space with the metric $\rho ^{ \prime }(x,y)=\min \{ 1,\rho (x,y)\} ,$ $x, y\in U.$ Analogously, we define the spaces ${\mathcal R}^p({\mathbb R}, {\mathrm {cl}}\,_{ b}\, U)$ and ${\mathcal R} ({\mathbb R},{\mathrm {cl}}\,_{ b}\, U)$ of functions (multivalued mappings) $F:{\mathbb R}\to {\mathrm {cl}}\,_{ b}\, U$ with ranges in the complete metric space $({\mathrm {cl}}\,_{ b}\, U,{\mathrm {dist}})$ of nonempty closed bounded subsets of the metric space $(U,\rho )$ with the Hausdorff metric ${\mathrm {dist}}$ (while defining the multivalued mappings $F\in {\mathcal R} ({\mathbb R},{\mathrm {cl}}\,_{ b}\, U)$ the metric ${\mathrm {dist}} ^{ \prime }(X,Y)=\min \{ 1,{\mathrm {dist}}(X,Y)\} ,$ $X, Y\in {\mathrm {cl}}\,_{ b}\, U$, 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 $F\in {\mathcal R} ({\mathbb R},{\mathrm {cl}}\,_{ b}\, U)$ (accordingly $F\in {\mathcal R}^p ({\mathbb R},{\mathrm {cl}}\,_{ b}\, U)$) 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 $\eta :[0,+\infty ) \to [0,+\infty )$ 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 $F\in {\mathcal R}^p({\mathbb R},{\mathrm {cl}}\,_{ b}\, U).$ 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 UDC 517.518.6 MSC 42A75, 54C65 DOI 10.20537/vm140403 Received 18 October 2014 Language Russian Citation Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections. III, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 4, pp. 25-52. References Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 2, pp. 19-51 (in Russian). Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections. II, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 4, pp. 3-21 (in Russian). Danilov L.I. The uniform approximation of recurrent functions and almost recurrent functions, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 4, pp. 36-54 (in Russian). Danilov L.I. Almost periodic selections of multivalued mappings, Izv. Otd. Mat. Inform. Udmurt. Gos. 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Markova i ikh dal'neishee razvitie (The Markov moment problem and extremal problems. Ideas and problems of P.L. Chebyshev and A.A. Markov and their further development), Moscow: Nauka, 1973, 561 p. Full text