Weakly nonlinear boundary-value problem in a special critical case

  • S. M. Chuiko

We investigate the problem of the determination of conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for finding the generating solution of a weakly nonlinear Noetherian boundary-value problem turns into an identity. We improve the classification of critical cases and construct an iterative algorithm for finding solutions of weakly nonlinear Noetherian boundary-value problems in the special critical case.


Generate Solution Iterative Procedure Critical Case Solvability Condition Periodic Problem 
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.


  1. 1.
    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht (2004).zbMATHGoogle Scholar
  2. 2.
    A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Generalized Inverse Operators and Noetherian Boundary-Value Problems [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1995).Google Scholar
  3. 3.
    A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods for the Investigation of Solutions of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1986).Google Scholar
  4. 4.
    E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods for Analysis of Nonlinear Systems [in Russian], Nauka, Moscow (1979).Google Scholar
  5. 5.
    I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1956).zbMATHGoogle Scholar
  6. 6.
    A. A. Boichuk, S. M. Chuiko, and A. S. Chuiko, “Nonautonomous periodic boundary-value problems in a special critical case,” Nelin. Kolyvannya, 7, No. 1, 53–66 (2004).MathSciNetGoogle Scholar
  7. 7.
    P. L. Kapitsa, “Dynamical stability of a pendulum with oscillating pivot point,” Zh. Éksp. Teor. Fiz., 21, No. 5, 499–597 (1951).Google Scholar
  8. 8.
    M. M. Postnikov, Introduction to Morse Theory [in Russian], Nauka, Moscow (1971).Google Scholar
  9. 9.
    V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1989).Google Scholar
  10. 10.
    S. M. Chuiko, “Noetherian boundary-value problem in a special critical case,” Dopov. Nats. Akad. Nauk Ukr., No. 2, 26–30 (2007).Google Scholar
  11. 11.
    J. E. Dennis, Jr., and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ (1983).zbMATHGoogle Scholar
  12. 12.
    A. S. Chuiko, “Domain of convergence of an iterative procedure for a weakly nonlinear boundary-value problem,” Nelin. Kolyvannya, 8, No. 2, 278–288 (2005).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • S. M. Chuiko
    • 1
  1. 1.Slavyansk Pedagogic UniversitySlavyanskUkraine

Personalised recommendations