phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2014 - 4 issue » pp. 25-52

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
  1. 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).
  2. 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).
  3. 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).
  4. Danilov L.I. Almost periodic selections of multivalued mappings, Izv. Otd. Mat. Inform. Udmurt. Gos. Univ., 1993, no. 1, pp. 16-78. (in Russian).
  5. Danilov L.I. On Weyl almost periodic selections of multivalued maps, J. Math. Anal. Appl., 2006, vol. 316, no. 1, pp. 110-127.
  6. Danilov L.I. On Besicovich almost periodic selections of multivalued maps, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2008, no. 1, pp. 97-120 (in Russian).
  7. Danilov L.I. On a class of Weyl almost periodic selections of multivalued maps, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2009, no. 1, pp. 34-55 (in Russian).
  8. Dolbilov A.M., Shneiberg I.Y. Multivalued almost periodic mappings and selections of them, Siberian Math. J., 1991, vol. 32, no. 2, pp. 326-328.
  9. Fryszkowski A. Continuous selections for a class of non-convex multivalued maps, Studia Math., 1983, vol. 76, no. 2, pp. 163-174.
  10. Danilov L.I. On almost periodic sections of multivalued maps, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2008, no. 2, pp. 34-41 (in Russian).
  11. Michael E. Continuous selections. I, Ann. Math., 1956, vol. 63, no. 2, pp. 361-381.
  12. Nemytskii V.V., Stepanov V.V. Kachestvennaya teoriya differentsial'nykh uravnenii (Qualitative theory of differential equations), Moscow-Izhevsk: RCD, 2004, 456 p.
  13. Anosov D.V., Aranson S.Kh., Arnold V.I., Bronshtein I.U., Grines V.Z., Il'yashenko Yu.S. Ordinary differential equations and smooth dynamical systems, Berlin-Heidelberg-New York: Springer-Verlag, 1997.
  14. Levitan B.M. Pochti periodicheskie funktsii (Almost periodic functions), Moscow: GITTL, 1953, 396 p.
  15. Danilov L.I. Measure-valued almost periodic functions and almost periodic selections of multivalued maps, Sbornik: Mathematics, 1997, vol. 188, no. 10, pp. 1417-1438.
  16. Danilov L.I. On almost periodic multivalued maps, Mathematical Notes, 2000, vol. 68, no. 1, pp. 71-77.
  17. Danilov L.I. Uniform approximation of Stepanov almost periodic functions and almost periodic selections of multivalued maps, Physical Technical Institute of Ural Branch of the Russian Academy of Sciences, Izhevsk, 2003, 70 p. Deposited in VINITI 21.02.2003, no. 354-B2003 (in Russian).
  18. Danilov L.I., Ivanov A.G. On a pointwise maximum theorem in the almost periodic case, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1994, no. 6, pp. 50-59 (in Russian).
  19. Ivanov A.G. Elements of the mathematical apparatus in the almost periodic optimization problems. I, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2002, no. 1 (24), pp. 3-100 (in Russian).
  20. Danilov L.I. On the uniform approximation of a function that is almost periodic in the sense of Stepanov, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1998, no. 5, pp. 10-18 (in Russian).
  21. Krein M.G., Nudel'man A.A. Problema momentov Markova i ekstremal'nye zadachi. Idei i problemy P.L. Chebysheva i A.A. 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
/doc/issues-2014/14-04-03.pdf
<< Previous article
Next article >>