Journal of Dynamical and Control Systems

, Volume 18, Issue 3, pp 339–354 | Cite as

Multiple periodic solutions for a class of non-autonomous hamiltonian systems with even-typed potentials



In this paper, we investigate the periodic solutions for a class of non-autonomous Hamiltonian systems. By using a decomposition technique of space and variational approaches we give new sufficient conditions for the existence of multiple periodic solutions.

Key words and phrases

Periodic solution critical point second-order Hamiltonian systems variational approach 

2000 Mathematics Subject Classiffication

35B10 47J30 58E05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Brezis, L. Nirenberg. Remarks on finding critical points, Commun. Pure Appl. math. 44 (1991), 939–963.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Guo Dajun. Nonlinear Functional Analysis, Shandong science and technology Press, Shandong, China 1985.Google Scholar
  3. 3.
    J. Mawhin, M. Willem. Critical Point Theory and Hamiltonian Systems, Springer-Verlag, Berlin 1989.MATHGoogle Scholar
  4. 4.
    J. J. Nieto, D. O’Regan. Variational approach to impulsive differential equations, Nonlinear Anal.: Real World Applications 10 (2009), 680–690.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    P. H. Rabinowitz. Minimax Methods in Critical Point Theory with Applicatins to Differential Equations, CBMS Regional Conf. Ser. in Math., 65, American Mathematical Society, Providence, RI 1986.Google Scholar
  6. 6.
    B. Ricceri. On a three critical points theorem, Arch. Math. (Basel) 75 (2000), 220–226.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    B. Ricceri. A general multiplicity theorem for certain nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc. 133 (2005), 3255–3261.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    C. Tang. Periodic solutions of non-autonomous second order systems with γ-quasisubadditive potential, J. Math. Anal. Appl. 189 (1995), 671–675.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    C. Tang. Existence and multiplicity of periodic solutions for nonautonomous second order systems, Nonlinear Anal. TMA 32(3) (1998), 299–304.MATHCrossRefGoogle Scholar
  10. 10.
    C. Tang. Periodic solutions of non-autonomous second order systems, J. Math. Anal. Appl. 202 (1996), 465–469.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    C. Tang, X. Wu. Periodic solutions for second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc. 126(11) (1998), 3263–3270.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    C. Tang, X. Wu. Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems, J. Math. Anal. Appl. 275 (2002), 870–882.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Y. Tian, W. G. Ge. Periodic solutions of non-autonomous second-order systems with a p-Laplacian, Nonlinear Anal. 66 (2007), 192–203.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Y. Tian, W. G. Ge. Multiple positive solutions for a second-order Sturm-Liouville boundary value problem with a p-Laplacian via variational methods, Rocky Mountain J. Math. 39(1) 2009, 325–342.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Y. Tian, W. G. Ge. Applications of Variational Methods to Boundary Value Problem for Impulsive Differential Equations, Proceedings of Edinburgh Mathematical Society 51 (2008) 509–527.MathSciNetMATHGoogle Scholar
  16. 16.
    Q. Wang, Z. Wang, J. Shi. Subharmonic oscillations with prescribed minimal period for a class of Hamiltonian systems, Nonlinear Anal. 28 (1996), 1273–1282.CrossRefGoogle Scholar
  17. 17.
    X. Wu, C. Tang. Periodic solutions of nonautonomous second-order Hamiltonian systems with even-typed potentials, Nonlinear Anal. 55 (2003), 759–769.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    J. Yu, Subharmonic solutions with prescribed minimal period of a class of nonautonomous Hamiltonian systems, J. Dyn. Diff. Equat. 20 (2008), 787–796.MATHCrossRefGoogle Scholar
  19. 19.
    E. Zeidler. Nonlinear Functional Analysis and its Applications, III Springer, 1985.Google Scholar
  20. 20.
    J. Zhou, Y. Li. Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects, Nonlinear Anal. 71 (2009), 2856–2865.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    J. Zhou, Y. Li. Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects, Nonlinear Anal. 72 (2010), 1594–1603.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingP. R. China
  2. 2.Department of Applied MathematicsBeijing Institute of TechnologyBeijingP. R. China

Personalised recommendations