Advertisement

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

  • Sergey Kornev
  • Valeri ObukhovskiiEmail author
  • Pietro Zecca
Article

Abstract

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.

Keywords

Random differential equation Random topological degree Random multivalent guiding function 

Mathematics Subject Classification

34F05 34A34 34C25 

Notes

Acknowledgements

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.

References

  1. 1.
    Andres, J., Górniewicz, L.: Random topological degree and random differential inclusions. Topol. Methods Nonlinear Anal. 40, 337–358 (2012)MathSciNetzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fonda, A.: Guiding functions and periodic solutions to functional differential equations. Proc. Am. Math. Soc. 99(1), 79–85 (1987)MathSciNetzbMATHGoogle 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)Google Scholar
  7. 7.
    Górniewicz, L., Plaskacz, S.: Periodic solutions of differential inclusions in ${\mathbb{R}}^{n}$. Boll. UMI. 7–A, 409–420 (1993)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)Google Scholar
  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)MathSciNetGoogle Scholar
  12. 12.
    Krasnosel’skii, M.A., Zabreiko, P.P.: Geometrical methods of nonlinear analysis. Grundlehren der Mathematischen Wissenschaften, 263. Springer-Verlag, Berlin (1984)Google Scholar
  13. 13.
    Mawhin, J.: Periodic solutions of nonlinear functional differential equations. J. Differ. Equ. 10, 240–261 (1971)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  15. 15.
    Rachinskii, D.I.: Multivalent guiding functions in forced oscillation problems. Nonlinear Anal. Theory Methods Appl. 26, 631–639 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Sergey Kornev
    • 1
  • Valeri Obukhovskii
    • 1
    Email author
  • Pietro Zecca
    • 2
  1. 1.Department of Physics and MathematicsVoronezh State Pedagogical UniversityVoronezhRussia
  2. 2.Dipartimento di Matematica e Informatica “U. Dini”Universita di FirenzeFlorenceItaly

Personalised recommendations