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Method of Guiding Functions for Existence Problems for Periodic Solutions of Differential Equations

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Abstract

We provide a review and systematic explanation of various generalizations of the guiding function method. The current state of the said method and its applications to various kinds of problems for nonlinear periodic systems described by differential and functional differential equations are considered.

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References

  1. Yu. G. Borisovich, B. D. Gel’man, A. D. Myshkis, and V. V. Obukhovskiĭ, Introduction to the Theory of Multivalued Mappings and Differential Inclusions [in Russian], Librokom, Moscow (2011).

    MATH  Google Scholar 

  2. A. Fonda, “Guiding functions and periodic solutions to functional differential equations,” Proc. Amer. Math. Soc., 99, No. 1, 79–85 (1987).

    MathSciNet  MATH  Google Scholar 

  3. S. V. Kornev, “On the method of multivalent guiding functions in the problem of periodic solutions of differential inclusions,” Autom. Remote Control, 64, No. 3, 409–419 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. V. Kornev and V. V. Obukhovskiĭ, “On integral guiding functions for functional differential inclusions,” In: Topol. Methods Nonlin. Anal., Voronezh. Gos. Univ., Voronezh, 87–107 (2000).

  5. S. V. Kornev and V. V. Obukhovskiĭ, “Localization of the method of guiding functions in the problem of periodic solutions of differential inclusions,” Russian Math. (Iz. VUZ), 53, No. 5, 19–27 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. M. Krasnoselskii, M. A. Krasnoselskii, J. Mawhin, and A. Pokrovskii, “Generalized guiding functions in a problem on high-frequency forced oscillations,” Nonlinear Anal., 22, No. 11, 1357–1371 (1994).

    Article  MathSciNet  Google Scholar 

  7. M. A. Krasnosel’skiĭ, Displacement Operators along Trajectories of Differential Equations [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  8. M. A. Krasnosel’skiĭ and A. I. Perov, “On a certain principle of existence of bounded, periodic and almost periodic solutions of systems of ordinary differential equations,” Dokl. Akad. Nauk SSSR, 123, No. 2, 235–238 (1958).

    MathSciNet  Google Scholar 

  9. M. A. Krasnosel’skiĭ and P. P. Zabreĭko, Geometric Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  10. J. L. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, Am. Math. Soc., Providence (1979).

    Book  MATH  Google Scholar 

  11. J. L. Mawhin and H. B. Thompson, “Periodic or bounded solutions of Carathéodory systems of ordinary differential equations,” J. Dynam. Differ. Equ., 15, No. 2-3, 327–334 (2003).

    Article  MATH  Google Scholar 

  12. J. L. Mawhin and J. R. Ward Jr., “Guiding-like functions for periodic or bounded solutions of ordinary differential equations,” Discrete Contin. Dyn. Syst., 8, No. 1, 39–54 (2002).

    MathSciNet  MATH  Google Scholar 

  13. V. V. Obukhovskiĭ, P. Zecca, N. V. Loi, and S. Kornev, Method of Guiding Functions in Problems of Nonlinear Analysis, Springer, Berlin (2013).

    Book  MATH  Google Scholar 

  14. A. I. Perov and V. K. Evchenko, Methods of Guiding Functions [in Russian], Voronezh State Univ., Voronezh (2012).

    Google Scholar 

  15. D. I. Rachinskiĭ, “Forced oscillations in control systems under near-resonance conditions,” Autom. Remote Control, 56, No. 11, Part 1, 1575–1584 (1996).

  16. D. I Rachinskii, “Multivalent guiding functions in forced oscillation problems,” Nonlinear Anal., 26, No. 3, 631–639 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  17. V. G. Zvyagin, Introduction to Topological Methods of Nonlinear Analysis [in Russian], Voronezh State Univ., Voronezh (2014).

    Google Scholar 

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Zvyagin, V.G., Kornev, S.V. Method of Guiding Functions for Existence Problems for Periodic Solutions of Differential Equations. J Math Sci 233, 578–601 (2018). https://doi.org/10.1007/s10958-018-3944-4

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  • DOI: https://doi.org/10.1007/s10958-018-3944-4

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