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The theory of the numerical-analytic method: Achievements and new trends of development. III

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Abstract

We analyze results concerning the application of the numerical-analytic method suggested by Samoilenko to delay differential equations, differential equations with “maxima,” functional-differential, operator-differential, and integro-differential equations.

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References

  1. 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).

    Article  Google Scholar 

  2. 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).

    Article  Google Scholar 

  3. Yu. A. Mitropol’skii, A. M. Samoilenko, and D. I. Martynyuk, Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients [in Russian], Naukova Dumka, Kiev (1985). English translation: Kluwer AP, Dordrecht (1993).

    Google Scholar 

  4. D. I. Martynyuk and A. M. Samoilenko, “On periodic solutions of nonlinear systems with delay,” Mat. Fiz., Issue 3, 128–145 (1967).

    Google Scholar 

  5. D. I. Martynyuk, “On periodic solutions of countable systems of periodic differential equations with delay,” Mat. Fiz., Issue 4, 84–89 (1968).

    Google Scholar 

  6. N. S. Kurpel’, “On two-sided approximations of periodic solutions of differential equations,” in: Proceedings of the Fifth International Conference on Nonlinear Oscillations (Kiev, 1969) [in Russian], Vol. 1, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1970), pp. 119–121.

    Google Scholar 

  7. N. S. Kurpel’ and K. V. Tsidylo, “On two-sided approximations of periodic solutions of systems of differential equations with delay,” Dokl. Akad. Nauk Ukr. SSR, No. 6, 515–520 (1972).

    Google Scholar 

  8. S. V. Yanchuk, “Numerical-analytic method for investigation of periodic solutions of systems of differential equations with delay,” Dokl. Akad. Nauk Ukr., No. 6, 49–52 (1997).

  9. A. Augustinowicz and M. Kwapisz, “On a numerical-analytic method of solving boundary-value problems for functional differential equations of neutral type,” Math. Nachr., 145, 255–269 (1990).

    Article  MathSciNet  Google Scholar 

  10. M. Kwapisz, “On modification of the integral equation of Samoilenko’s numerical-analytic method,” Math. Nachr., 157, 125–135 (1992).

    MATH  MathSciNet  Google Scholar 

  11. M. Kwapisz, “On integral equations arising in a numerical-analytic method of solving boundary-value problems for differential functional equations,” in: Proceedings of the International Conference on Differential Equations (Spain, Barcelona, August 26–31, 1991), Vol. 2, World Scientific, London (1993), pp. 671–677.

    Google Scholar 

  12. M. Kwapisz, “Some remarks on an integral equation arising in applications of a numerical-analytic method of solving boundary-value problems,” Ukr. Mat. Zh., 44, No. 1, 128–132 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  13. E. P. Trofimchuk and A. V. Kovalenko, “A. M. Samoilenko’s numerical-analytic method without determining equation,” Ukr. Mat. Zh., 47, No. 1, 138–140 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  14. A. N. Rontó, “On one iteration scheme of approximate solution of nonlinear boundary-value problems for ordinary differential equations,” in: Collection of Works of Undergraduate and Postgraduate Students of the Kiev University [in Russian], Kiev University, Kiev (1995), pp. 17–22.

    Google Scholar 

  15. A. D. Myshkis, “On some problems in the theory of differential equations with deviating argument,” in: Differential Equations with Deviating Argument [in Russian], Naukova Dumka, Kiev (1977), pp. 221–247.

    Google Scholar 

  16. D. D. Bainov and G. H. Sarafova, “Application of the numerical-analytic method to the investigation of periodic systems of ordinary differential equations with maxima,” Rev. Roum. Sci. Techn. Sér. Méc. Appl., 26, No. 3, 371–382 (1981).

    MathSciNet  Google Scholar 

  17. G. H. Sarafova and D. D. Bainov, “Application of A.M. Samoilenko’s numerical-analytic method to the investigation of periodic linear differential equations with maxima,” Rev. Roum. Sci. Techn. Sér. Méc. Appl., 26, No. 4, 595–603 (1981).

    MATH  MathSciNet  Google Scholar 

  18. G. H. Sarafova and D. D. Bainov, “Application of A. M. Samoilenko’s numerical-analytic method to the investigation of periodic linear differential equations with maxima,” Stud. Sci. Math. Hungar., 17, No. 1-4, 221–228 (1982).

    MATH  MathSciNet  Google Scholar 

  19. 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), pp. 136–146.

    Google Scholar 

  20. V. P. Shpakovich and V. I. Muntyan, “Periodic solutions of integro-differential equations with ‘maxima,’” in: Some Problems in the Theory of Asymptotic Processes of Nonlinear Mechanics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986), pp. 186–190.

    Google Scholar 

  21. V. I. Muntyan, Justification of Asymptotic Methods for Equations with “Maxima” [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev University, Kiev (1987).

    Google Scholar 

  22. T. K. Yuldashev, “Periodic solutions of nonlinear systems of integro-differential equations with ‘maxima,’” Vopr. Vych. Prikl. Mat. Akad. Nauk. Resp. Uzb., No. 33, 119–129 (1992).

  23. G. D. Zavalykut, “Numerical-analytic method for the investigation of periodic solutions of one class of differential-operator equations,” Differents. Uravn., 19, No. 4, 569–575 (1983).

    MathSciNet  Google Scholar 

  24. I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon, Oxford (1991).

    MATH  Google Scholar 

  25. M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate Solution of Operator Equations [in Russian], Nauka, Moscow (1969). English translation: Akademie-Verlag, Berlin (1973).

    Google Scholar 

  26. M. Rontó, A. N. Rontó, and S. I. Trofimchuk, Numerical-Analytic Method for Differential and Difference Equations in Partially Ordered Banach Spaces and Some Applications [in Russian], Preprint No. 96-02, Institute of Mathematics, University of Miskolc, Miskolc (1996).

    Google Scholar 

  27. A. M. Samoilenko and M. I. Rontó, Numerical-Analytic Methods of Investigating Periodic Solutions, Mir, Moscow (1979).

    Google Scholar 

  28. F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  29. G. D. Zavalykut and O. D. Nurzhanov, “On a periodic boundary-value problem for one class of differential-operator equations,” Ukr. Mat. Zh., 39, No. 3, 299–303 (1987).

    MATH  MathSciNet  Google Scholar 

  30. A. M. Samoilenko and M. I. Rontó, “A periodic boundary-value problem for autonomous systems,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 9, 20–23 (1984).

  31. A. M. Samoilenko and M. I. Rontó, Numerical-Analytic Methods for the Investigation of Solutions of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1985).

    Google Scholar 

  32. G. Vakhabov, “Numerical-analytic method for the investigation of a periodic systems of integro-differential equations,” Ukr. Mat. Zh., 21, No. 5, 675–683 (1969).

    Google Scholar 

  33. G. Vakhabov, On Some Methods for the Investigation of Oscillations in Nonlinear Systems of Integro-Differential Equations [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1970).

    Google Scholar 

  34. M. G. Gospodinov, “Numerical-analytic method for the investigation of one class of periodic systems of first-order integro-differential equations,” Godishn. Vyssh. Uchebn. Zaved. Prilozh. Mat., 16, No. 1, 191–202 (1980).

    MathSciNet  Google Scholar 

  35. O. D. Nurzhanov, “On periodic solutions of nonlinear integro-differential equations,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 595–599 (1977).

  36. R. N. Butris, Periodic Solutions of Nonlinear Integral and Differential Equations [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev University, Kiev (1992).

    Google Scholar 

  37. G. H. Sarafova and D. D. Bainov, “Numerical-analytic method for the investigation of countable systems of periodic integro-differential equations,” Godishn. Vyssh. Uchebn. Zaved. Prilozh. Mat., 16, No. 1, 135–146 (1980).

    MathSciNet  Google Scholar 

  38. G. H. Sarafova and D. D. Bainov, “Periodic solutions of nonlinear integro-differential equations with an impulse effect,” Period. Math. Hungar., 18, No. 2, 99–113 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  39. D. D. Bainov and G. H. Sarafova, “On application of the numerical-analytic method of A. M. Samoilenko for investigation of periodic systems of integro-differential equations,” Arch. Math., 15, No. 2, 67–80 (1979).

    MathSciNet  Google Scholar 

  40. A. T. Alymbaev, “Numerical-analytic method for the investigation of autonomous systems of integro-differential equations,” in: Asymptotic Methods in the Theory of Differential and Integro-Differential Equations and Their Applications [in Russian], Kirghiz State University, Frunze (1981), pp. 27–36.

    Google Scholar 

  41. O. D. Nurzhanov and A. T. Alymbaev, “Numerical-analytic method for the investigation of autonomous systems of integro-differential equations,” Ukr. Mat. Zh., 33, No. 4, 540–547 (1981).

    MathSciNet  Google Scholar 

  42. A. T. Alymbaev, “Periodic solutions of systems of nonlinear integro-differential equations,” in: Investigations of Integro-Differential Equations [in Russian], Issue 21, Ilim, Frunze (1988), pp. 116–123.

    Google Scholar 

  43. B. Wujtowicz, “Numerical-analytic method for the investigation of integro-differential equations,” Vestn. Kiev. Univ., Ser. Mat. Mekh., No. 24, 14–21 (1982).

  44. A. T. Alymbaev, “Periodic solutions of a system of autonomous integro-differential equations with unbounded delay,” in: Investigations of Integro-Differential Equations [in Russian], Issue 20, Ilim, Frunze (1988), pp. 15–23.

    Google Scholar 

  45. B. E. Turbaev, Asymptotic Methods for Integration of Nonlinear Integro-Differential Equations [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev University, Kiev (1985).

    Google Scholar 

  46. Yu. A. Mitropol’skii, G. P. Khoma, and M. I. Gromyak, Asymptotic Methods for the Investigation of Quasiwave Equations of Hyperbolic Type [in Russian], Naukova Dumka, Kiev (1991).

    Google Scholar 

  47. V. Z. Chornyi, Investigation of Periodic Solutions of Some Classes of Second-Order Differential Equations [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992).

    Google Scholar 

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 960–979, July, 1998.

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Rontó, M.I., Samoilenko, A.M. & Trofimchuk, S.I. The theory of the numerical-analytic method: Achievements and new trends of development. III. Ukr Math J 50, 1091–1114 (1998). https://doi.org/10.1007/BF02528821

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