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Ukrainian Mathematical Journal

, Volume 58, Issue 4, pp 529–550 | Cite as

Once again on the Samoilenko numerical-analytic method of successive periodic approximations

  • I. I. Korol’
  • M. O. Perestyuk
Article

Abstract

A new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear periodic systems of differential equations dx/dt = A(t) x+ ƒ(t, x) in the critical case is developed. The problem of the existence of solutions and their approximate construction is studied. Estimates for the convergence of successive periodic approximations are obtained.

Keywords

Periodic Solution Critical Case Fundamental Matrix Odic Solution Linear Homogeneous System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).zbMATHGoogle Scholar
  2. 2.
    A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Method for Investigation of Periodic Solutions [in Russian], Vyshcha Shkola, Kiev (1976).Google Scholar
  3. 3.
    A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods for Investigation of Solutions of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1985).Google Scholar
  4. 4.
    A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods in the Theory of Boundary-Value Problems for Ordinary Differential Equations [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  5. 5.
    A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Generalized Inverse Operators and Noetherian Boundary-Value Problems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1995).Google Scholar
  6. 6.
    A. Yu. Luchka, Projection-Iterative Methods [in Russian], Naukova Dumka, Kiev (1993).Google Scholar
  7. 7.
    I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1956).zbMATHGoogle Scholar
  8. 8.
    E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods for Analysis of Nonlinear Systems [in Russian], Nauka, Moscow (1979).Google Scholar
  9. 9.
    V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients [in Russian], Nauka, Moscow (1972).Google Scholar
  10. 10.
    A. M. Samoilenko, “A numerical-analytic method for investigation of periodic systems of ordinary differential equations. I,” Ukr. Mat. Zh., 17, No. 4, 82–93 (1965).Google Scholar
  11. 11.
    A. M. Samoilenko, “A numerical-analytic method for investigation of periodic systems of ordinary differential equations. II,” Ukr. Mat. Zh., 18, No. 2, 50–59 (1966).Google Scholar
  12. 12.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “The theory of the numerical-analytic method: achievements and new trends of development. I,” Ukr. Mat. Zh., 50, No. 1, 102–117 (1998).CrossRefGoogle Scholar
  13. 13.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “The theory of the numerical-analytic method: achievements and new trends of development. II,” Ukr. Mat. Zh., 50, No. 2, 225–243 (1998).CrossRefGoogle Scholar
  14. 14.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “The theory of the numerical-analytic method: achievements and new trends of development. III,” Ukr. Mat. Zh., 50, No. 7, 960–979 (1998).Google Scholar
  15. 15.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “The theory of the numerical-analytic method: achievements and new trends of development. IV,” Ukr. Mat. Zh., 50, No. 12, 1656–1672 (1998).CrossRefzbMATHGoogle Scholar
  16. 16.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “The theory of the numerical-analytic method: achievements and new trends of development. V,” Ukr. Mat. Zh., 51, No. 5, 663–673 (1999).CrossRefzbMATHGoogle Scholar
  17. 17.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “The theory of the numerical-analytic method: achievements and new trends of development. VI,” Ukr. Mat. Zh., 51, No. 7, 960–971 (1999).Google Scholar
  18. 18.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “The theory of the numerical-analytic method: achievements and new trends of development. VII,” Ukr. Mat. Zh., 51, No. 9, 1244–1261 (1999).CrossRefzbMATHGoogle Scholar
  19. 19.
    A. M. Samoilenko, “On one sequence of polynomials and the radius of convergence of its Poisson-Abel sum,” Ukr. Mat. Zh., 55, No. 7, 926–934 (2003).zbMATHCrossRefGoogle Scholar
  20. 20.
    B. P. Demidovich, Lectures on Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).Google Scholar
  21. 21.
    A. A. Boichuk, Constructive Methods for Analysis of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1990).Google Scholar
  22. 22.
    F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).zbMATHGoogle Scholar
  23. 23.
    M. G. Krein and M. A. Rutman, “Linear operators leaving invariant a cone in a Banach space,” Usp. Mat. Nauk, 3, Issue 1, 3–95 (1948).MathSciNetGoogle Scholar
  24. 24.
    A. M. Ronto, Numerical-Analytic Methods for Investigation of Boundary-Value Problems [in Ukrainian], Author’s Abstract of Candidate-Degree Thesis (Physics and Mathematics), Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (1997).Google Scholar
  25. 25.
    N. A. Evtukha and P. P. Zabreiko, “On the Samoilenko method for finding periodic solutions of quasilinear differential equations in a Banach space,” Ukr. Mat. Zh., 37, No. 2, 162–168 (1985).Google Scholar
  26. 26.
    N. A. Perestyuk, “On periodic solutions of some systems of differential equations,” in: Asymptotic and Qualitative Methods in the Theory of Nonlinear Oscillations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1971).Google Scholar
  27. 27.
    S. M. Kopystra, “On 2π-periodic solutions of nonlinear systems of first-order differential equations,” Visn. Kyiv. Univ., Ser. Fiz.-Mat. Nauk, No. 1, 69–80 (1997).Google Scholar
  28. 28.
    I. I. Korol’, “On periodic solutions of one class of systems of differential equations,” Ukr. Mat. Zh., 57, No. 4, 483–495 (2005).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • I. I. Korol’
    • 1
  • M. O. Perestyuk
    • 1
  1. 1.Shevchenko Kyiv National UniversityKyiv

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