Skip to main content
Springer Nature Link
Log in

On two numerical-analytic methods for the investigation of periodic solutions

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We discuss two techniques useful in the investigation of periodic solutions of broad classes of non-linear non-autonomous ordinary differential equations, namely the trigonometric collocation and the method based upon periodic successive approximations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Farkas, Autonóm rendszerek periodikus perturbációiról, Alkalmazott Matematikai Lapok, 1 (1975), 197–254.

    MathSciNet  Google Scholar 

  2. N. N. Bogolyubov and Yu. A. Mitropolsky, Asmptotic Methods in the Theory of nonlinear Oscillations, Fizmatgiz, Moscow, 1963 (in Russian).

    Google Scholar 

  3. N. N. Bogolyubov, Yu. A. Mitropolsky and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics, Naukova Dumka, Kiev, 1969 (in Russian).

    Google Scholar 

  4. E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameter and Relaxation Oscillations, Nauka, Moscow, 1975 (in Russian).

    MATH  Google Scholar 

  5. J. Moser, Rapidly convergent method of iterations for nonlinear differential equations, Usp. Mat. Nauk, 23 (1968), 179–238.

    Google Scholar 

  6. V. A. Pliss, Integral Sets of Periodic Systems of Differential Equations, Nauka, Moscow, 1977 (in Russian).

    MATH  Google Scholar 

  7. I. T. Kiguradze, Some singular boundary value problems for ordinary differential equations, Izdat. Tbilis. Univ., Tbilisi, 1975 (in Russian).

    Google Scholar 

  8. I. Kiguradze and S. Mukhigulasvili, On periodic solutions of two-dimensional nonautonomous differential systems, Nonlinear Analysis, 60 (2005), 241–256.

    MATH  MathSciNet  Google Scholar 

  9. M. Ronto and A. M. Samoilenko, Numerical-Analytic Methods in the Theory of Boundary-Value Problems, World Scientific, 2000.

  10. M. A. Krasnoselsky, Functional-analytic methods in the thery of nonlinear oscillations, Proceedings of the Fifth International Conference on Nonlinear Oscillations, 1 (1970), 323–331.

    Google Scholar 

  11. L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, Vol. 1, Wiley, New York, 1963, 149–187.

    Google Scholar 

  12. J. Mawhin, Topological Degree Methods in Nonlinear Boundary-Value Problems, CBMS Regional Conference Series in Mathematics, Vol. 40, American Mathematical Society, Providence, RI, 1979.

    MATH  Google Scholar 

  13. J. K. Hale, Oscillations in Nonlinear systems, McGraw Hill, New York, 1963.

    MATH  Google Scholar 

  14. M. Farkas, Periodic Motions, Applied Mathematical Sciences 104, Springer-Verlag, New York-London, 1994.

    MATH  Google Scholar 

  15. R. E. Gaines and J. L. Mawhin, Coincidence degree, and nonlinear differential equations, Lecture Notes in Mathematics 568, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

    MATH  Google Scholar 

  16. A. Capietto, J. Mawhin and F. Zanolin, A continuation approach to superlinear periodic boundary-value problems, J. Differential Equation, 88 (1990), 347–385.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41–72.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. A. Krasnoselskii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii and V. Ya. Stetsenko, Approximate Solution of Operator Equations, Wolters-Noordoff Scientific Publications Ltd., Groningen, 1972.

    Google Scholar 

  19. M. Feckan, Minimal periods of periodic solutions, Miskolc Mathematical Notes, 7 (2006), 123–141.

    MathSciNet  Google Scholar 

  20. U. M. Ascher, R. M. M. Mattheij and R. D. Russell, Numerical solution of boundary value problems for ordinary differential equations, Classics in Applied Mathematics 13, SIAM, Philadelphia, 1995.

    MATH  Google Scholar 

  21. H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems, Dover Publications Inc., N.Y., 1992.

    Google Scholar 

  22. S.L. Sobolev, Equations of Mathematical Physics, Gostekizdat, Moscow, 1947 (in Russian).

    Google Scholar 

  23. G. M. Vainikko, On the convergence and stability of the collocation method, Differents. Uravn., 1 (1965), 244–254.

    MathSciNet  Google Scholar 

  24. G. M. Vainikko, On the convergence of the collocation method for nonlinear differential equations, Zh. Vychisl. Mat. i Mat. Fiz., 6 (1966), 35–42.

    MATH  MathSciNet  Google Scholar 

  25. G. M. Vainikko, Approximate methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Anal., 2 (1978), 647–687.

    Article  MATH  MathSciNet  Google Scholar 

  26. G. M. Vainikko, On the convergence of the difference method in the problem of periodic solutions of ordinary differential equations, Zh. Vychisl. Mat. i Mat. Fiz., 15 (1975), 87–100.

    MATH  MathSciNet  Google Scholar 

  27. L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer, Berlin, 1963.

    MATH  Google Scholar 

  28. A. M. Samoilenko, Numerical-analytic method for the investigation of periodic systems of ordinary differential equations I, Ukr. Mat. Zh., 17 (1965), 82–93.

    Google Scholar 

  29. A. M. Samoilenko, Numerical-analytic method for the investigation of periodic systems of ordinary differential equations II, Ukr. Mat. Zh., 18 (1966), 50–59.

    Article  Google Scholar 

  30. A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods for the Investigation of Periodic Solutions, Mir, Moscow, 1979.

    Google Scholar 

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

    Google Scholar 

  32. A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods in the Theory of Boundary-Value Problems for Ordinary Differential Equations, Naukova Dumka, Kiev, 1992 (in Russian).

    Google Scholar 

  33. M. Ronto and R. M. Tégen, Successive approximation method for three point boundary-value problem with singular matrixes, Mathematica Pannonica, 5 (1994), 15–29.

    MATH  MathSciNet  Google Scholar 

  34. M. Rontó and A. Tuzson, Construction of periodic solutions of differential equations with impulse effect, Publ. Math. Debrecen, 44 (1994), 335–357.

    MATH  MathSciNet  Google Scholar 

  35. M. Rontó and J. Mészáros, Some remarks on the convergence analysis of the numerical-analytic method based upon successive approximations, Ukr. Mat. Zh., 48 (1996), 90–95.

    Article  Google Scholar 

  36. M. Rontó, Numerical-analytic successive approximation method for non-linear boundary value problems, Nonlinear Anal., 30 (1997), 3179–3188.

    Article  MATH  MathSciNet  Google Scholar 

  37. M. Rontó, On some existence results for parametrised boundary value problems, Publ. Univ. Miskolc Ser. D Nat. Sci. Math., 37 (1997), 95–103.

    MATH  MathSciNet  Google Scholar 

  38. M. Rontó and A. Galántai, A computational modification of the numerical-analytic method for periodic BVPs, Nonlinear Oscillations, 2 (1999), 109–114.

    MATH  Google Scholar 

  39. M. Rontó and R. M. Tégen, Numerical-analytic method for investigating three point boundary value problems with parameters, Publ. Univ. Miskolc Ser. D Nat. Sci. Math., 40 (1999), 67–77.

    MATH  MathSciNet  Google Scholar 

  40. M. Rontó and A. Tuzson, Modification of trigonometric collocation method for impulsive periodic BVP effect, Computers & Mathematics with Applications, 38 (1999), 117–123.

    Article  MATH  Google Scholar 

  41. A. Ronto and M. Rontó, On the investigation some boundary value problems with non-linear conditions, Math. Notes (Miskolc), 1 (2000), 43–55.

    MATH  MathSciNet  Google Scholar 

  42. M. Rontó, On the investigation of parametrized non-linear boundary value problems, Proc. 3rd World Congress of Nonlinear Analysts (July 19–26, 2000, Catania) Nonlinear Analysis, 47 (2001), 4409–4420.

    Article  MATH  MathSciNet  Google Scholar 

  43. M. Rontó, On nonlinear boundary value problems containing parameter, CDDE 2000 Proc., Archivum Mathematicum (Brno), 36 (2000), 583–593.

    Google Scholar 

  44. A. Ronto and M. Rontó, A note on the numerical-analytic method for non-linear two-point boundary value problems, Nonlinear Oscillations, 4 (2001), 112–128.

    MATH  MathSciNet  Google Scholar 

  45. A. Ronto, M. Rontó, A. M. Samoilenko and S. I. Trofimchuk, On periodic solutions of autonomous difference equations, Georgian Mathematical Journal, 8 (2001), 135–164.

    MATH  MathSciNet  Google Scholar 

  46. A. Ronto and M. Rontó, On some symmetry properties of periodic solutions, Nonlinear Oscillations, 6 (2003), 83–108.

    MATH  MathSciNet  Google Scholar 

  47. M. Rontó and N. Shchobak, On the numerical-analytic investigations of parametrized problems with nonlinear boundary conditions, Nonlinear Oscillations, 6 (2003), 482–510.

    MATH  MathSciNet  Google Scholar 

  48. A. Ronto and M. Rontó, On the (τ, E) property of periodic solutions, Folia Facultatis Scientiarum Naturalium, Universitas Masarykianae Brunensis, Mathematica, 13 (2003), 247–267.

    Google Scholar 

  49. M. Rontó and N. Shchobak, On parametrized problems with nonlinear boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, (2004), 24 pp.

  50. A. M. Rontó, M. Rontó and N. Shchobak, On parametrization of three point nonlinear boundary value problems, Nonlinear Oscillations, 7 (2004), 395–413.

    MATH  MathSciNet  Google Scholar 

  51. G. Bognár and M. Rontó, Numerical-analytic investigation of the radially symmetric solutions for some nonlinear PDEs, Computers & Mathematics with Applications, 50 (2005), 983–991.

    Article  MATH  MathSciNet  Google Scholar 

  52. G. Bognár, M. Rontó and N. Rajabov, On initial value problems related to p-Laplacian and pseudo-Laplacian, Acta Math. Hungar., 108 (2005), 1–12.

    Article  MATH  MathSciNet  Google Scholar 

  53. N. I. Ronto, A. M. Samoilenko and S. I. Trofimchuk, The theory of the numerical analytic method: achievements and new directions of development I, Ukrainian Math. J., 50 (1998), 116–135.

    MathSciNet  Google Scholar 

  54. N. I. Ronto, A. M. Samoilenko and S. I. Trofimchuk, The theory of the numerical analytic method: achievements and new directions of development II, Ukrainian Math. J., 50 (1998), 255–277.

    MathSciNet  Google Scholar 

  55. N. I. Ronto, A. M. Samoilenko and S. I. Trofimchuk, The theory of the numerical analytic method: achievements and new directions of development III, Ukrainian Math. J., 50 (1998), 1091–1114.

    Article  MathSciNet  Google Scholar 

  56. N. I. Ronto, A. M. Samoilenko and S. I. Trofimchuk, The theory of the numerical analytic method: achievements and new directions of development IV, Ukrainian Math. J., 50 (1998), 1888–1907.

    MathSciNet  Google Scholar 

  57. N. I. Ronto, A. M. Samoilenko and S. I. Trofimchuk, The theory of the numerical analytic method: achievements and new directions of development V, Ukrainian Math. J., 51 (1999), 735–747.

    Article  MathSciNet  Google Scholar 

  58. N. I. Ronto, A. M. Samoilenko and S. I. Trofimchuk, The theory of the numerical analytic method: achievements and new directions of development VI, Ukrainian Math. J., 51 (1999), 1079–1094.

    Article  MathSciNet  Google Scholar 

  59. N. I. Ronto, A. M. Samoilenko and S. I. Trofimchuk, The theory of the numerical analytic method: achievements and new directions of development VII, Ukrainian Math. J., 51 (1999), 1399–1418.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Analysis, University of Miskolc, 3515, Miskolc-Egyetemváros, Miskolc, Hungary

    Miklós Rontó

Authors
  1. Miklós Rontó
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Miklós Rontó.

Additional information

Dedicated to the memory of Professor Miklós Farkas

Supported in part by the Hungarian NFSR OTKA through Grant No. K68311.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rontó, M. On two numerical-analytic methods for the investigation of periodic solutions. Period Math Hung 56, 121–135 (2008). https://doi.org/10.1007/s10998-008-5121-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-008-5121-3

Mathematics subject classification number

Key words and phrases