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Ukrainian Mathematical Journal

, Volume 25, Issue 5, pp 481–489 | Cite as

On the averaging principle for functional — Differential equations of neutral type

  • R. R. Akhmerov
Article

Keywords

Differential Equation Average Principle Neutral Type 
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Literature cited

  1. 1.
    N. M. Krylov and N. N. Bogolyubov, Introduction to Nonlinear Mechanics [in Russian], Izd. AN UkrSSR, Kiev (1937).Google Scholar
  2. 2.
    I. I. Gikhman, “On a theorem of N. N. Bogolyubov,” Ukrainsk. Matem. Zh.,4, No.2 (1952).Google Scholar
  3. 3.
    M. A. Krasnosel'skii and S. G. Krein, “On the averaging principle in nonlinear mechanics,” Usp. Matem. Nauk,10, No. 3 (1955).Google Scholar
  4. 4.
    P. P. Zabreiko and I. B. Ledovskaya, Existence Theorem for Differential Equations in a Banach Space and the Averaging Principle, in: Problems in the Mathematical Analysis of Complex Systems [in Russian], No.3, Voronezh State University, Voronezh (1968).Google Scholar
  5. 5.
    Yu. L. Daletskii and M. G. Krein, The Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).Google Scholar
  6. 6.
    A. Halanay, “The averaging method for systems of differential equations with a retarded argument.” Rev. Math. Pures Appl.,IV, No. 3 (1959).Google Scholar
  7. 7.
    V. I. Fodchuk, “On the continuous dependence on the parameter of the solutions of differential equations with a retarded argument,” Ukrainsk. Matem. Zh.,16, No.2 (1964).Google Scholar
  8. 8.
    J. K. Hale, Averaging Methods for Differential Equations with Retarded Arguments and a Small Parameter, Brown University Technical Report (1964).Google Scholar
  9. 9.
    Yu. A. Mitropol'skii and D. I. Martynyuk, Lectures on the Theory of Vibrations of Systems with a Delay [in Russian], Naukova Dumka, Kiev (1968).Google Scholar
  10. 10.
    V. I. Fodchuk, “The averaging method for differential-difference equations of neutral type,” Ukrainsk. Matem. Zh.,20, No.2 (1968).Google Scholar
  11. 11.
    A. N. Filatov, Averaging Methods in Differential and Integrodifferential Equations [in Russian], Fan, Tashkent (1971).Google Scholar
  12. 12.
    V. I. Fodchuk, On the Problem of Justifying the Averaging Principle for Differential Equations with a Retarded Argument, III Konferenz über Nichtlineare Schwingungen, Vol.I, Akademie-Verlag, Berlin (1965).Google Scholar
  13. 13.
    J. K. Hale, “Averaging methods for differential equations with retarded arguments and a small parameter,” J. Differential Equations,2, No.1 (1966).Google Scholar
  14. 14.
    Yu. A. Mitropol'skii and V. I. Fodchuk, “The second theorem of N. N. Bogolyubov on the averaging method for differential equations with a retarded argument,” Ukrainsk. Matem. Zh.,24, No.1 (1972).Google Scholar
  15. 15.
    Yu. A. Mitropol'skii, The Averaging Method in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1971).Google Scholar
  16. 16.
    R. R. Akhmerov and M. I. Kamenskii, On the Second Theorem of N. N. Bogolyubov for Functional-Differential Equations of Neutral Type. Subject Material of the Third All-Union Inter-Higher-Educational-Institution-Conference on the Theory and Applications of Differential Equations with a Perturbed Argument [in Russian], Chernovits (1972).Google Scholar
  17. 17.
    L. M. Averina and B. N. Sadovskii, “On the local solvability of a functional - differential equation of neutral type,” Trudy Matem. Fakul'teta, No. 3, Voronezh State University, Voronezh (1971).Google Scholar
  18. 18.
    L. É. Él'sgol'ts and S. B. Norkin, Introduction to the Theory of Differential Equations with a Perturbed Argument [in Russian], Nauka, Moscow (1971).Google Scholar
  19. 19.
    B. N. Sadovskii, On Measures of Noncompactness and Densifying Operators, in: Problems in the Mathematical Analysis of Complex Systems [in Russian], No.2, Voronezh State University, Voronezh (1968).Google Scholar
  20. 20.
    B. N. Sadovskii, “Limitly compact and densifying operators,” Usp. Matem. Nauk,27, No. l (1972).Google Scholar
  21. 21.
    A. E. Rodkina, On the Extendability of the Solutions of Systems of Functional - Differential Equations of Neutral Type. Subject Matter of the Third Ail-Union Inter-Higher-Educational-Institution Conference on the Theory and Applications of Differential Equations with a Perturbed Argument [in Russian], Chernovits (1972).Google Scholar
  22. 22.
    B. P. Demidovich, Lectures on the Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).Google Scholar
  23. 23.
    I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1970).Google Scholar
  24. 24.
    C. Banfi, “Sull' approssimazione di processi in mecanica non lineare,” Boll. Un. Mat. Ital.,22 (1967).Google Scholar
  25. 25.
    A. L. Badoev and B. N. Sadovskii, “An example of a densifying operator in the theory of differential equations with a perturbed argument of neutral type,” Dokl. Akad. Nauk SSSR,186, No. 6 (1969).Google Scholar

Copyright information

© Consultants Bureau 1974

Authors and Affiliations

  • R. R. Akhmerov
    • 1
  1. 1.Voronezh State UniversityUSSR

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