Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 799–803 | Cite as

Autonomous Noether Boundary-Value Problems not Solved with Respect to the Derivative

Article
  • 3 Downloads

Abstract

In monographs of N. V. Azbelev, A. M. Samoilenko, and A. A. Boichuk, constructive methods of study of Noether boundary-value problems have been developed. These methods continue the investigation of periodic problems stated by H. Poincaré, A. M. Lyapunov, N. M. Krylov, N. N. Bogolyubov, I. G. Malkin, and O. Veivoda by the methods of small parameter. We propose an improved scheme of study of autonomous Noether boundary-value problems for nonlinear systems in critical cases. In the case of multiple roots of the equation for generating constants, we obtain sufficient conditions of existence of solutions to an autonomous boundary-value problem not solved with respect to the derivative. The effectiveness of the scheme proposed is illustrated by an example of the periodic problem for the Liénard equation.

Keywords and phrases

autonomous boundary-value problem ordinary differential equation Liénard equation 

AMS Subject Classification

34B15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Linear Functional Differential Equations, World Federation Publ., Atlanta (1995).Google Scholar
  2. 2.
    A. A. Boichuk and S. M. Chuiko, “Autonomous weakly nonlinear boundary-value problems,” Differ. Uravn., 28, No. 10, 1668–1674 (1992).MathSciNetMATHGoogle Scholar
  3. 3.
    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht, Boston (2004).Google Scholar
  4. 4.
    S. M. Chuiko, “On an approximate solution of boundary-value problems by the least square method,” Nonlin. Oscill., 11, No. 4, 585–604 (2008).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    S. M. Chuiko and A. S. Chuiko, “On approximate solution of autonomous Noether boundary-value problems by the least square method,” Dinam. Sist., 29, 103–111 (2011).MATHGoogle Scholar
  6. 6.
    S. M. Chuiko and O. V. Starkova, “Autonomous boundary-value problems in a particular critical case,” Dinam. Sist., 27, 127–142 (2009).MATHGoogle Scholar
  7. 7.
    S. M. Chuiko and O. V. Starkova, “On the approximate solution of autonomous boundary value problems by the least square method,” Nonlin. Oscill., 12, No. 4, 556–573 (2009).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    S. M. Chuiko, O. V. Starkova, and O. E. Pirus, “Nonlinear Noether boundary-value problems not solved with respect to the derivative,” Dinam. Sist., 30, Nos. 1-2, 169–186 (2012).Google Scholar
  9. 9.
    I. G. Malkin, Problems of the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1956).MATHGoogle Scholar
  10. 10.
    O. Vejvoda, “On perturbed nonlinear boundary-value problems,” Czech. Math. J., No. 11, 323-364 (1961).Google Scholar
  11. 11.
    V. Volterra, Le¸cons sur la Th´eorie Mathematique de la Lutte pour la Vie, Gauthiers-Villars, Paris (1931).Google Scholar
  12. 12.
    V. F. Zaitsev and A. D. Polyanin, Handbook of Ordinary Differential Equations [in Russian], Fizmatlit, Moscow (2001).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Donbass State Pedagogical UniversitySlavyanskUkraine
  2. 2.Institute of Applied Mathematics and MechanicsNational Academy of Sciences of UkraineSlavyanskUkraine

Personalised recommendations