Advertisement

Ukrainian Mathematical Journal

, Volume 51, Issue 9, pp 1399–1418 | Cite as

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

  • M. I. Rontó
  • A. M. Samoilenko
  • S. I. Trofimchuk
Article

Abstract

For the numerical-analytic method suggested by A. M. Samoilenko in 1965, we analyze the application to abstract differential equations, implicit equations, and control problems.

Keywords

Banach Space Control Problem Periodic Solution Ukrainian Academy Linear Continuous Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “Theory of the numerical-analytic method: achievements and new directions of development. I”,Ukr. Mat. Zh.,50, No. 1, 102–117 (1998).Google Scholar
  2. 2.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “Theory of the numerical-analytic method: achievements and new directions of development. II”,Ukr. Mat. Zh.,50, No. 2, 225–243 (1998).Google Scholar
  3. 3.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “Theory of the numerical-analytic method: achievements and new directions of development. III”,Ukr. Mat. Zh.,50, No. 7, 960–979 (1998).Google Scholar
  4. 4.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “Theory of the numerical-analytic method: achievements and new directions of development. IV”,Ukr. Mat. Zh.,50, No. 12, 1656–1672 (1998).zbMATHGoogle Scholar
  5. 5.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “Theory of the numerical-analytic method: achievements and new directions of development. V”,Ukr. Mat. Zh.,51, No. 5, 663–673 (1999).zbMATHGoogle Scholar
  6. 6.
    M. I. Rontó, A. M. Samoilenko, and S. I. Trofimchuk, “Theory of the numerical-analytic method: achievements and new directions of development. VI”,Ukr. Mat. Zh.,51, No. 7, 960–971 (1999).Google Scholar
  7. 7.
    A. M. Samoilenko, “Numerical-analytic method for the investigation of countable systems of periodic differential equations”,Mat. Fiz. Issue, 2, 115–132 (1966).Google Scholar
  8. 8.
    Yu. A. Mitropol'skii, A.M. Samoilenko, and D. I. Martynyuk,Systems of Evolutionary Equations with Periodic and Conditionally Periodic Coefficients [in Russian], Naukova Dumka, Kiev (1985).Google Scholar
  9. 9.
    A. M. Samoilenko and Yu. V. Teplinskii,Countable Systems of Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1993).Google Scholar
  10. 10.
    N. A. Evkhuta and P. P. Zabreiko, “On the Samoilenko method for the determination of periodic solutions of quasilinear differential equations in a Banach space”,Ukr. Mat. Zh.,37, No. 2, 162–168 (1985).zbMATHMathSciNetGoogle Scholar
  11. 11.
    N. A. Perestyuk, “Periodic solutions of specific 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
  12. 12.
    M. Rontó, A. Ronto, and S. I. Trofimchuk,Numerical-Analytic Method for Differential and Difference Equations in Partially Ordered Banach Spaces, and Some Applications, Preprint No. 96-02, University of Miskolc, Institute of Mathematics, Miskolc (1996).Google Scholar
  13. 13.
    M. G. Krein and M. A. Rutman, “Linear operators keeping the cone invariant in a Banach space”,Usp. Mat. Nauk,3, Issue 1 (23), 3–95 (1948).MathSciNetzbMATHGoogle Scholar
  14. 14.
    M. A. Krasnosel'skii,Positive Solutions of Operator Equations [in Russian], Fizmatgiz, Moscow (1962).Google Scholar
  15. 15.
    L. V. Kantorovich and G. P. Akilov,Functional Analysis [in Russian], Nauka, Moscow (1977).Google Scholar
  16. 16.
    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).Google Scholar
  17. 17.
    F. Riesz and B. Sz.-Nagy,Lectures on Functional Analysis [Russian translation], Mir, Moscow (1979).Google Scholar
  18. 18.
    M. Rontó and S. I. Trofimchuk,Numerical-Analytic Method for Non-Linear Differential Equations, Preprint No. 96-01, University of Miskolc, Institute of Mathematics, Miskolc (1996).Google Scholar
  19. 19.
    W. W. Petrishyn and Z. S. Yu, “Periodic solutions of nonlinear second-order differential equations which are not solvable for the highest derivative”,J. Math. Anal. Appl.,89, 462–488 (1982).CrossRefMathSciNetGoogle Scholar
  20. 20.
    Yu. D. Shlapak, “Periodic solutions of second-order nonlinear differential equations unresolved with respect to the highest derivative”,Ukr. Mat. Zh.,26, No. 6, 850–854 (1974).Google Scholar
  21. 21.
    Yu. D. Shlapak, “Periodic solutions of first-order ordinary differential equations unresolved with respect to the derivative”,Ukr. Mat. Zh.,32, No. 5, 638–644 (1980).zbMATHMathSciNetGoogle Scholar
  22. 22.
    B. E. Turbaev, “Periodic solutions of systems of differential equations unresolved with respect to the derivative”,Visn. Kyiv. Univ., Ser. Mat. Mekh., Issue 27, 98–104 (1985).zbMATHMathSciNetGoogle Scholar
  23. 23.
    Kh. Ovezdurdyev,Numerical-Analytic Methods for the Investigation of Solutions of Two-Point Boundary-Value Problems [in Russian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1985).Google Scholar
  24. 24.
    M. Kwapisz, “On modifications of the integral equation of Samoilenko's numerical-analytic method of solving boundary-value problems”,Math. Nachr.,157, 125–135 (1992).zbMATHMathSciNetGoogle Scholar
  25. 25.
    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).MathSciNetGoogle Scholar
  26. 26.
    S. V. Martynyuk,Investigation of Solutions of Boundary-Value Problems for Systems of Nonlinear Differential Equations [in Russian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1992).Google Scholar
  27. 27.
    A. M. Samoilenko and M. I. Rontó,Numerical-Analytic Method for the Investigation of Periodic Solutions [in Russian], Vyshcha Shkola, Kiev (1976).Google Scholar
  28. 28.
    E. P. Trofimchuk “Integral operators of the method of successive periodic approximations”,Mat. Fiz. Nelin. Mekh., Issue 13 (47), 31–36 (1990).MathSciNetGoogle Scholar
  29. 29.
    R. I. Sobkovich, “Periodic control problem for systems of differential equations of the second order”, in:Analytic Methods in Nonlinear Mechanics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1981), pp. 125–133.Google Scholar
  30. 30.
    R. I. Sobkovich,Numerical-Analytic Methods for the Investigation of Boundary-Value Control Problems [in Russian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1983).Google Scholar
  31. 31.
    Yu. A. Mitropol'skii, D. I. Martynyuk, and A. I. Yurchik, “Control problems for systems of second-order delay differential equations”,Ukr. Mat. Zh.,37, No. 5, 594–599 (1985).MathSciNetGoogle Scholar
  32. 32.
    Yu. A. Mitropol'skii, D. I. Martynyuk, and A. I. Yurchik, “Solution of one control problem for delay systems by the method of two-sided approximations”,Ukr. Mat. Zh.,37, No. 4, 462–467 (1985).MathSciNetGoogle Scholar
  33. 33.
    D. I. Martynyuk and A. I. Yurchik, “Control problem for delay differential equations”, in:Nonlinear Differential Equations in Applied Problems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984), pp. 129–132.Google Scholar
  34. 34.
    D. I. Martynyuk and A. I. Yurchik, “Periodic control problem for systems of difference equations”, in:Specific Problems of the Theory of Asymptotic Methods in Nonlinear Mechanics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986), pp. 138–141.Google Scholar
  35. 35.
    D. I. Martynyuk and A. I. Yurchik, “Solution of one periodic control problem in systems of difference equations by the method of two-sided approximations”,Mat. Fiz. Nelin. Mekh., Issue 7 (41), 145–151 (1987).MathSciNetGoogle Scholar
  36. 36.
    A. I. Yurchik, “One periodic problem of control of nonlinear delay systems”, in:Approximate Methods of Analysis of Nonlinear Oscillations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984), pp. 165–169.Google Scholar
  37. 37.
    E. P. Trofimchuk,Iterative Methods for the Investigation of Differential Systems with Singularities [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Kiev (1992).Google Scholar
  38. 38.
    A. M. Samoilenko, M. I. Rontó, and V. A. Ronto, “Two-point boundary-value problem with parameters in the boundary conditions”,Dokl. Akad. Nauk Ukr. SSR. Ser. A, No. 7, 23–26 (1985).Google Scholar
  39. 39.
    M. I. Rontó and V. A. Ronto, “One method for the investigation of boundary-value problems with parameters”, in:Boundary-Value Problems of Mathematical Physics [in Russian], Naukova Dumka, Kiev (1990), pp. 3–10.Google Scholar
  40. 40.
    A. M. Samoilenko and M. I. Rontó,Numerical-Analytic Methods in the Theory of Boundary-Value Problems for Ordinary Differential Equations [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  41. 41.
    M. I. Rontó and I. I. Korol', “Investigation and solution of boundary-value problems with parameters by the numerical-analytic method”,Ukr. Mat. Zh.,46, No. 8, 1031–1043 (1994).CrossRefGoogle Scholar
  42. 42.
    I. I. Korol', “Investigation of two-point boundary-value problems for systems of second-order ordinary differential equations with parameters”.Dopov. Akad. Nauk Ukrainy, No. 9, 6–12 (1995).Google Scholar
  43. 43.
    I. I. Korol',Numerical-Analytic Methods for the Investigation of Solutions of Two-Point Boundary-Value Problems with Parameters [in Ukrainian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1996).Google Scholar
  44. 44.
    M. Rontó, “On numerical-analytic method for BVPs with parameters”,Publ. Univ. Miskolc. Ser. D, Natur. Sci. Math.,36, No. 2, 125–132 (1996).zbMATHGoogle Scholar
  45. 45.
    M. Rontó, “On some existence results for parametrized boundary value problems”,Publ. Univ. Miskolc. Ser. D, Natur. Sci. Math.,37, No. 2, 95–104 (1997).zbMATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • M. I. Rontó
  • A. M. Samoilenko
    • 1
  • S. I. Trofimchuk
  1. 1.AcademicianUkrainian Academy of SciencesUSSR

Personalised recommendations