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## Archive of Issues

Russia Izhevsk
Year
2012
Issue
4
Pages
3-21
 Section Mathematics Title Recurrent and almost recurrent multivalued maps and their selections. II Author(-s) Danilov L.I.a Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa Abstract In the paper, we consider the problem of existence of recurrent and almost recurrent selections of multivalued mappings ${\mathbb R}\ni t\mapsto F(t)\in {\mathrm {comp}}\, U$ with nonempty compact sets $F(t)$ in a complete metric space $U.$ The set ${\mathrm {comp}}\, U$ is equipped with the Hausdorff metric ${\mathrm {dist}}$. Recurrent and almost recurrent multivalued maps are defined as the functions with values in the metric space $({\mathrm {comp}}\, U,{\mathrm {dist}}).$ It is proved that there are recurrent (almost recurrent) selections of multivalued recurrent (almost recurrent) uniformly absolutely continuous maps. We also consider mappings ${\mathbb R}\ni t\mapsto F(t)$ with the sets $F(t)$ consisting of a finite number of points (the number depends on the $t\in {\mathbb R}$). We prove that if such a map is almost recurrent, then it has an almost recurrent selection. A multivalued recurrent mapping $t\mapsto F(t)$ with sets $F(t)$ consisting of at most $n$ points (where $n\in {\mathbb N}$) has a recurrent selection. If the sets $F(t)$ of a multivalued recurrent (almost recurrent) mapping $t\mapsto F(t)$ consist of $n$ points for all $t\in {\mathbb R},$ then all $n$ continuous selections of the map $F$ are recurrent (almost recurrent). Keywords recurrent function, selection, multivalued mapping UDC 517.518.6 MSC 42A75, 54C65 DOI 10.20537/vm120401 Received 17 May 2012 Language Russian Citation Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections. II, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 4, pp. 3-21. References Michael E. Continuous selections. I, Ann. Math., 1956, vol. 63, no. 2, pp. 361–381. Kikuchi N., Tomita Y. On the absolute continuity of multi-functions and orientor fields, Funkcialaj Ekvacioj, 1971, vol. 14, pp. 161–170. Hermes H. On continuous and measurable selections and the existence of solutions of generalized differential equations, Proc. Amer. Math. Soc., 1971, vol. 29, no. 3, pp. 535–542. 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. Nemytskii V.V., Stepanov V.V. Kachestvennaya teoriya differentsial’nykh uravnenii (Qualitative theory of differential equations), Moscow – Izhevsk: RCD, 2004, 456 p. 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. Borisovich Y.G., Gel’man B.D., Myshkis A.D., Obukhovskii V.V. Vvedeniye v teoriyu mnogoznachnykh otobrazhenii i differentsial’nykh vklyuchenii (Introduction to the theory of set-valued mappings and differential inclusions), Moscow: KomKniga, 2005, 216 p. Birkhoff G.D. Dynamical Systems, New York, 1927. Translated under the title Dinamicheskie sistemy, Izhevsk: Udmurt. Univ., 1999, 408 p. Irisov A.E., Tonkov E.L. Sufficient conditions for optimality of Birkhoff recurrent motions for differential inclusion, Vestn. Udmurt. Univ. Mat., 2005, no. 1, pp. 59–74. Panasenko E.A. On existence of recurrent and almost periodic solutions to differential inclusions, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 3, pp. 42–57. Danilov L.I. Almost periodic selections of multivalued maps, Izv. Otd. Mat. Inform. Udmurt. Gos. Univ., Izhevsk, 1993, no. 1, pp. 16–78. Lyusternik L.A., Sobolev V.I. Kratkii kurs funktsional’nogo analiza (A short course of functional analysis), Moscow: Vysshaya shkola, 1982, 271 p. Full text