Skip to main content
SpringerLink
Log in

Periodic solutions of a second order differential equation with one-sided growth restrictions on the restoring term

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. J. W. Bebernes, A simple alternative problem for finding periodic solutions of second order ordinary differential systems. Proc. Amer. Math. Soc.42, 121–127 (1974).

    Google Scholar 

  2. A. Capozzi, On subquadratic Hamiltonian systems. J. Nonlinear Anal. T.M.A.8, 553–562 (1984).

    Google Scholar 

  3. S. N. Chow andJ. A. Sanders, On the number of critical points of the period. J. Differential Equations64, 51–66 (1986).

    Google Scholar 

  4. V. Coti Zelati, Periodic solutions of dynamical systems with bounded potential. J. Differential Equations67, 400–413 (1987).

    Google Scholar 

  5. E. N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations. Bull. Austral. Math. Soc.15, 321–328 (1976).

    Google Scholar 

  6. T. Ding, Boundedness of solutions of Duffing's equations. J. Differential Equations61, 178–207 (1986).

    Google Scholar 

  7. M. L. C.Fernandes, On an elementary phase-plane analysis for solving second order BVPs. To appear.

  8. A. Fonda andF. Zanolin, Periodic solutions to second order differential equations of Liénard type with jumping nonlinearities. Comment. Math. Univ. Carolin.28, 33–41 (1987).

    Google Scholar 

  9. S. Fucik, Boundary value problems with jumping nonlinearities. Časopi Pěst. Mat.101, 69–87 (1976).

    Google Scholar 

  10. S.Fučik, Solvability of nonlinear equations and boundary value problems. Dortrecht 1980.

  11. S. Fucik andV. Lovicar, Periodic solutions of the equationx″+g(x)=p. Časopis Pěst. Mat.100, 160–175 (1975).

    Google Scholar 

  12. R.E.Gaines and J.Mawhin, Coincidence degree and nonlinear differential equations. LNM568, Berlin-Heidelberg-New York 1977.

  13. J. K.Hale, Ordinary differential equations. New York 1969.

  14. R.Iannacci, M. N.Nkashama, P.Omari and F.Zanolin, Periodic solutions of forced Liénard equations with jumping nonlinearities under non uniform conditions. To appear.

  15. H.-W. Knobloch, Eine neue Methode zur Approximation periodischer Lösungen nichtlinearer Differentialgleichungen zweiter Ordnung. Math. Z.82, 177–197 (1963).

    Google Scholar 

  16. J. P.La Salle and S.Lefschetz, Stability by Liapunov's direct method with appiications. New York-London 1963.

  17. J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces. J. Differential Equations12, 610–636 (1972).

    Google Scholar 

  18. J. Mawhin, J. R. Ward Jr., Periodic solutions of some forced Liénard differential equations at resonance. Arch. Math.41, 337–351 (1983).

    Google Scholar 

  19. J. Mawhin andM. Willem, Critical points of convex perturbations of some indefinite quadratic forms and semi-linear boundary value problems at resonance. Ann. Inst. Henri Poincaré, Analyse non linéaire3, 431–453 (1986).

    Google Scholar 

  20. P. Omari, G. Villari andF. Zanolin, Periodic solutions of the Liénard equation with one sided growth restrictions. J. Differential Equations67, 278–293 (1987).

    Google Scholar 

  21. P.Omari and F.Zanolin, Some remarks about the paper “Periodic solutions of the Liénard equation with one-sided growth restrictions”. Unpublished internal report, Trieste 1986.

  22. P. Omari andF. Zanolin, On the existence of periodic solutions of forced Liénard differential equations. J. Nonlinear Anal. T.M.A.11, 275–284 (1987).

    Google Scholar 

  23. Z. Opial, Sur les solutions périodiques de l'équation différentiellex″+g(x)=p(t). Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.8, 151–156 (1960).

    Google Scholar 

  24. Z. Opial, Sur l'existence des solutions périodiques de l'équation différentiellex″+f(x, x′)x′+g(x)=p(t). Ann. Polon. Math.11 149–159 (1961).

    Google Scholar 

  25. Z. Opial, Sur les périodes des solutions de l'équation différentiellex″+g(x)=0. Ann. Polon. Math.10, 49–72 (1961).

    Google Scholar 

  26. R. Reissig, Schwingungssätze für die verallgemeinerte Liénardsche Differentialgleichung. Abh. Math., Sem. Univ. Hamburg44, 45–51 (1975).

    Google Scholar 

  27. R. Reissig, Periodic solutions of a second order differential equation including a one-sided restoring term. Arch. Math.33, 85–90 (1979).

    Google Scholar 

  28. R. Reissig, G. Sansone undR. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen. Cremonese, Roma 1963.

    Google Scholar 

  29. H. L.Royden, Real Analysis. New York 1963.

  30. K. Schmitt, Periodic solutions of nonlinear second order differential equations. Math. Z.98, 200–207 (1967).

    Google Scholar 

  31. K. Schmitt, Periodic solutions of a forced nonlinear oscillator involving a one-sided restoring force. Arch. Math.31, 70–73 (1978).

    Google Scholar 

  32. G. Seifert, A note on periodic solutions of second order differential equations without damping. Proc. Amer. Math. Soc.10, 396–398 (1959).

    Google Scholar 

  33. M.Urabe, Nonlinear autonomous oscillations. Analytical theory. New York-London 1967.

  34. J. R. Ward Jr., Periodic solutions for systems of second order ordinary differential equations. J. Math. Anal. Appl.81, 92–98 (1981).

    Google Scholar 

  35. M. Willem, Subharmonic oscillations of convex hamiltonian systems. J. Nonlinear Anal. T.M.A.9, 1303–1311 (1985).

    Google Scholar 

Download references

Author information

Author notes

  1. Lourdes Fernandes

    Present address: International School for Advanced Studies, Strada Costiera 11, 34014, Trieste, Italy

  2. Fabio Zanolin

    Present address: Dipartimento di Matematica e Informatica, Università, Via Zanon 6, 33100, Udine, Italy

Authors and Affiliations

    Authors
    1. Lourdes Fernandes
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. Fabio Zanolin
      View author publications

      You can also search for this author in PubMed Google Scholar

    Additional information

    On leave of absence from Faculdade de Ciências da Universidade de Lisboa, with a scholarship of Fundação Calouste Gulbenkian.

    Work performed under the auspicies of G.N.A.F.A.-C.N.R. and supported by the fund M.P.I. 60%, ZAN 6Q4.

    Rights and permissions

    Reprints and permissions

    About this article

    Cite this article

    Fernandes, L., Zanolin, F. Periodic solutions of a second order differential equation with one-sided growth restrictions on the restoring term. Arch. Math 51, 151–163 (1988). https://doi.org/10.1007/BF01206473

    Download citation

    • Received:

    • Issue Date:

    • DOI: https://doi.org/10.1007/BF01206473

    Keywords

    Search

    Navigation