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Russia Tambov
Section Mathematics
Title On fixed points of multi-valued maps in metric spaces and differential inclusions
Author(-s) Zhukovskiy E.S.a, Panasenko E.A.a
Affiliations Tambov State Universitya
Abstract A generalization of the Nadler fixed point theorem for multi-valued maps acting in metric spaces is proposed. The obtained result allows to study the existence of fixed points for multi-valued maps that have as images any arbitrary sets of the corresponding metric space and are not necessarily contracting, or even continuous, with respect to the Hausdorff metric. The mentioned result can be used for investigating differential and functional-differential equations with discontinuities and inclusions generated by multi-valued maps with arbitrary images. In the second part of the paper, as an application, conditions of existence and continuation of solutions to the Cauchy problem for a differential inclusion with noncompact in $\mathbb{R}^n$ right-hand side are derived.
Keywords multi-valued map, fixed point, differential inclusion
UDC 515.126.83, 515.126.4, 517.911.5
MSC 47H04, 47H10, 34A60
DOI 10.20537/vm130202
Received 1 February 2013
Language English
Citation Zhukovskiy E.S., Panasenko E.A. On fixed points of multi-valued maps in metric spaces and differential inclusions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 2, pp. 12-26.
  1. Panasenko E.A., Tonkov E.L. Invariant and stably invariant sets for differential inclusions, Tr. Mat. Inst. Steklov, 2008, vol. 262, pp. 202-221.
  2. Panasenko E.A., Tonkov E.L. Extension of E.A. Barbashin’s and N.N. Krasovskii’s stability theorems to controlled dynamical systems, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 268, suppl. 1, pp. 204–221.
  3. Panasenko E.A., Rodina L.I., Tonkov E.L. The space ${\rm clcv}($ $\mathbb{R}$$n$ $)$ with the Hausdorff–Bebutov metric and differential inclusions, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 275, suppl. 1, pp. 121–136.
  4. Zhukovskii E.S., Panasenko E.A. On one metric in the space of nonempty closed subsets of $\mathbb{R}$$n$ , Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 1, pp. 15–26.
  5. Zhukovskiy E.S., Panasenko E.A. On multi-valued maps with images in the space of closed subsets of a metric space, Fixed Point Theory and Applications, 2013, 2013:10, doi:10.1186/1687-1812-2013-10.
  6. Filippov A.F. Classical solutions of differential inclusions with multi-valued right-hand side, SIAM J. Control, 1967, vol. 5, no. 4, pp. 609–621.
  7. Himmelberg C.J., Van Vleck F.S. Lipschitzian generalized differential equations, Rend. Sem. Mat. Padova, 1972, vol. 48, pp. 159–169.
  8. Ioffe A.D., Tikhomirov V.M. Teoriya ekstremal’nykh zadach (Theory of extremal problems), Moscow: Nauka, 1974, 479 p.
  9. Gel’man B.D. Multivalued contraction maps and their applications, Vestn. Voronezh. Univ. Ser. Fiz. Mat., 2009, no. 1, pp. 74–86.
  10. Dunford N., Schwartz J. Lineinye operatory. Obshchaya teoriya (Linear operators. General theory), Moscow: Editorial URSS, 2004, 896 p.
  11. Filippov A.F. Differentsial’nye uravneniya s razryvnoi pravoi chast’yu (Differential equations with discontinuous right-hand sides), Мoscow: Nauka, 1985, 224 p.
  12. Borisovich Yu.G., Gel’man B.D., Myshkis A.D., Obukhovskii V.V. Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differentsial’nykh vklyuchenii (Introduction to the theory of multi-valued maps and differential inclusions), Moscow: KomKniga, 2005, 216 p.
  13. Kolmogorov А.N., Fomin S.V. Elementy teorii funktsii i funktsional’nogo analiza (Elements of functions theory and functional analysis), Мoscow: Nauka, 1976, 544 p.
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