On Multivalent Guiding Functions Method in the Periodic Problem for Random Differential Equations


By applying the random topological degree theory we develop the methods of random multivalent guiding functions and use them for the study of periodic solutions for random differential equations in finite dimensional spaces.

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  1. 1.

    Andres, J., Górniewicz, L.: Random topological degree and random differential inclusions. Topol. Methods Nonlinear Anal. 40, 337–358 (2012)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Borisovich, Yu.G., Gelman, B.D., Myshkis, A.D., Obukhovskii, V.V.: Introduction to the Theory of Multivalued Maps and Differential Inclusions, 2nd edn. Librokom, Moscow (2011). (in Russian)

    Google Scholar 

  3. 3.

    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977)

    Google Scholar 

  4. 4.

    De Blasi, F.S., Górniewicz, L., Pianigiani, G.: Topological degree and periodic solutions of differential inclusions. Nonlinear Anal. 37, 217–245 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Fonda, A.: Guiding functions and periodic solutions to functional differential equations. Proc. Am. Math. Soc. 99(1), 79–85 (1987)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings, 2nd edition. Topological Fixed Point Theory and Its Applications, 4. Springer, Dordrecht (2006)

  7. 7.

    Górniewicz, L., Plaskacz, S.: Periodic solutions of differential inclusions in ${\mathbb{R}}^{n}$. Boll. UMI. 7–A, 409–420 (1993)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Kornev, S.V.: On the method of multivalent guiding functions to the periodic problem of differential inclusions. Autom. Remote Control 64, 409–419 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Kornev, S.V.: Multivalent guiding function in a problem on existence of periodic solutions of some classes of differential inclusions. Russian Mathematics (Iz. VUZ) 11, 14–26 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Krasnosel’skii, M.A.: The Operator of Translation Along the Trajectories of Differential Equations, Translations of Mathematical Monographs 19. Amer. Math. Soc, Providence, R.I. (1968)

  11. 11.

    Krasnosel’skii, M.A., Perov, A.I.: On a certain priciple of existence of bounded, periodic and almost periodic solutions of systems of ordinary differential equations. Dokl. Akad. Nauk SSSR 123(2), 235–238 (1958). (in Russian)

    MathSciNet  Google Scholar 

  12. 12.

    Krasnosel’skii, M.A., Zabreiko, P.P.: Geometrical methods of nonlinear analysis. Grundlehren der Mathematischen Wissenschaften, 263. Springer-Verlag, Berlin (1984)

  13. 13.

    Mawhin, J.: Periodic solutions of nonlinear functional differential equations. J. Differ. Equ. 10, 240–261 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Obukhovskii, V., Zecca, P., Loi ans, N.V., Kornev, S.: Method of Guiding Functions in Problems of Nonlinear Analysis, Lecture Notes in Math., 2076, Springer, Berlin (2013)

  15. 15.

    Rachinskii, D.I.: Multivalent guiding functions in forced oscillation problems. Nonlinear Anal. Theory Methods Appl. 26, 631–639 (1996)

    MathSciNet  Article  MATH  Google Scholar 

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The authors are grateful to the anonymous referees for their helpful remarks. The work on the paper was carried out during Prof. V. Obukhovskii’s and Prof. S. Kornev’s visit to Dipartimento di Matematica e Informatica “Ulisse Dini”, Universita di Firenze, Firenze, Italy in 2017. They would like to express their gratitude to the members of the Dipartimento di Matematica e Informatica “Ulisse Dini” for their kind hospitality.

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Correspondence to Valeri Obukhovskii.

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The work is supported by the Ministry of Education and Science of the Russian Federation in the frameworks of the project part of the state work quota (Project No. 1.3464.2017/4.6) and from INDAM and the University of Firenze. The third author is partially supported by GNANPA and the University of Firenze (Italy).

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Kornev, S., Obukhovskii, V. & Zecca, P. On Multivalent Guiding Functions Method in the Periodic Problem for Random Differential Equations. J Dyn Diff Equat 31, 1017–1028 (2019). https://doi.org/10.1007/s10884-019-09734-5

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  • Random differential equation
  • Random topological degree
  • Random multivalent guiding function

Mathematics Subject Classification

  • 34F05
  • 34A34
  • 34C25