Doklady Mathematics

, Volume 98, Issue 2, pp 425–429 | Cite as

Frequency Tests for the Existence and Stability of Bounded Solutions to Differential Equations of Higher Order

  • A. I. Perov
  • I. D. Kostrub


To study a vector-matrix differential equation of order n, the method of integral equations is used. When the Lipschitz condition holds, an existence and uniqueness theorem for a bounded solution and its estimates are obtained. This solution is almost periodic if the nonlinearity is almost periodic, and it is asymptotically Lyapunov stable if the matrix characteristic polynomial is a Hurwitz polynomial. Under a Lipschitztype condition, a theorem on the existence of at least one bounded solution is proved; among the bounded solutions, there is at least one recurrent solution if the nonlinearity is almost periodic. The equation is S-dissipative if the matrix characteristic polynomial is a Hurwitz polynomial.


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  1. 1.
    M. A. Krasnosel’skii, V. Sh. Burd, and Yu. S. Kolesov, Nonlinear Almost Periodic Oscillations (Nauka, Moscow, 1970) [in Russian].Google Scholar
  2. 2.
    B. P. Demidovich, Mat. Sb. 40 (1), 73–94 (1956).MathSciNetGoogle Scholar
  3. 3.
    V. G. Zvyagin and S. V. Kornev, Method of Guiding Functions and Its Modifications (Lenand, Moscow, 2018) [in Russian].zbMATHGoogle Scholar
  4. 4.
    A. I. Perov and I. D. Kostrub, Sib. Math. J. 57 (4), 650–665 (2016).MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. M. Levitan, Almost Periodic Functions (Gostekhizdat, Moscow, 1953) [in Russian].zbMATHGoogle Scholar
  6. 6.
    V. G. Kurbatov and I. V. Kurbatova, Comput. Methods Appl. Math. (2017). doi 10.1515/cmam-2017-0042Google Scholar
  7. 7.
    S. Bochner, Lectures on Fourier Integrals (Princeton Univ. Press, Princeton, NJ, 1959).zbMATHGoogle Scholar
  8. 8.
    A. V. Pokrovskii, Autom. Remote Control 56, 1397–1404 (1985).Google Scholar
  9. 9.
    A. G. Baskakov, Dokl. Akad. Nauk SSSR 333 (3), 282–284 (1993).Google Scholar
  10. 10.
    A. G. Baskakov, Differ. Equations 39 (3), 447–450 (2003).MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. F. Bylov and D. M. Grobman, Usp. Mat. Nauk 17 (3), 159–161 (1962).Google Scholar
  12. 12.
    O. I. Nikitin and A. I. Perov, Differ. Uravn. 19 (11), 2001–2004 (1983).Google Scholar
  13. 13.
    B. P. Demidovich, Lectures on Mathematical Stability Theory (Nauka, Moscow, 1967) [in Russian].zbMATHGoogle Scholar
  14. 14.
    M. A. Krasnosel’skii and A. V. Pokrovskii, Dokl. Akad. Nauk SSSR 233 (3), 293–296 (1977).MathSciNetGoogle Scholar
  15. 15.
    A. N. Tikhonov, Math. Ann. 111, 767–776 (1935).MathSciNetCrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Voronezh State UniversityVoronezhRussia

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