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Journal of Dynamics and Differential Equations

, Volume 22, Issue 3, pp 463–471 | Cite as

Periodic Solutions of Pendulum-Like Perturbations of Singular and Bounded \({\phi}\)-Laplacians

  • Cristian Bereanu
  • Petru Jebelean
  • Jean Mawhin
Article

Abstract

In this paper we study the existence and multiplicity of periodic solutions of pendulum-like perturbations of bounded or singular \({\phi}\)-Laplacians. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.

Keywords

Curvature operators Periodic problem Pendulum-like non linearities Leray-Schauder degree Upper and lower solutions 

Mathematics Subject Classification (2000)

34B15 

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References

  1. 1.
    Benevieri P., do Ó J.M., de Medeiros E.S.: Periodic solutions for nonlinear systems with mean curvature-like operators. Nonlinear Anal. 65, 1462–1475 (2006)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bereanu C., Mawhin J.: Existence and multiplicity results for some nonlinear problems with singular \({\phi}\)-Laplacian. J. Differ. Equ. 243, 536–557 (2007)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bereanu C., Mawhin J.: Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and \({\phi}\)-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 15, 159–168 (2008)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Bereanu C., Mawhin J.: Periodic solutions of nonlinear perturbations of \({\phi}\) -Laplacians with possibly bounded \({\phi}\). Nonlinear Anal. 68, 1668–1681 (2008)MathSciNetMATHGoogle Scholar
  5. 5.
    Bereanu C., Mawhin J.: Boundary value problems for some nonlinear systems with singular \({\phi}\) -Laplacian. J. Fixed Point Theory Appl. 4, 57–75 (2008)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Bereanu C., Mawhin J.: Nonhomogeneous boundary value problems for some nonlinear equations with singular \({\phi}\) -Laplacian. J. Math. Anal. Appl. 352, 218–233 (2009)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cid J.A., Torres P.J.: Solvability for some boundary value problems with \({\phi}\) -Laplacian operators. Discret. Contin. Dynam. Syst. A 23, 727–732 (2009)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Chu J., Lei J., Zhang M.: The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator. J. Differ. Equ. 247, 530–542 (2009)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Ferracuti L., Papalini F.: Boundary-value problems for strongly nonlinear multivalued equations involving different \({\phi}\) -Laplacians. Adv. Differ. Equ. 14, 541–566 (2009)MathSciNetMATHGoogle Scholar
  10. 10.
    Fournier G., Mawhin J.: On periodic solutions of forced pendulum-like equations. J. Differ. Equ. 60, 381–395 (1985)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Kannan R., Ortega R.: Periodic solutions of pendulum-type equations. J. Differ. Equ. 59, 123–144 (1985)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Mawhin, J.: Boundary value problems for nonlinear perturbations of singular \({\phi}\) -Laplacians. In: Staicu, V.: (ed.) Differential Equations, Chaos and Variational Problems. Progr. Nonlin. Diff. Eq. Appl., Birkhauser 75, 247–256 (2007)Google Scholar
  13. 13.
    Mawhin J., Willem M.: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differ. Equ. 52, 264–287 (1984)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Torres P.J.: Periodic oscillations of the relativistic pendulum with friction. Phys. Lett. A 372, 6386–6387 (2008)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Torres, P.J.: Nondegeneracy of the periodically forced Liénard differential equation with \({\phi}\) -Laplacian, Commun. Contemporary Math., to appearGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomânia
  2. 2.Department of MathematicsWest University of TimişoaraTimişoaraRomânia
  3. 3.Institut de Mathématique et PhysiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgique

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