Skip to main content
SpringerLink
Log in

Periodic solutions for a nonlinear (q,p)-Laplacian dynamical system with impulsive effects

  • Computational mathematics
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, some existence theorems are obtained for periodic solutions of a nonlinear (q,p)-Laplacian dynamical system with impulsive effects by using the least action principle and mountain pass theorem.

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

Similar content being viewed by others

References

  1. Berger, M.S., Schechter, M.: On the solvability of semilinear gradient operator equations. Adv. Math. 25, 97–132 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Mawhin, J.: Semi-coercive monotone variational problems. Bull. Cl. Sci., Acad. R. Belg. 73, 118–130 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)

    MATH  Google Scholar 

  4. Mawhin, J., Willem, M.: Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 431–453 (1986)

    MathSciNet  MATH  Google Scholar 

  5. Tang, C.L.: Periodic solutions of nonautonomous second order systems with γ-quasisubadditive potential. J. Math. Anal. Appl. 189, 671–675 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Tang, C.L.: Periodic solutions of nonautonomous second order systems. J. Math. Anal. Appl. 202, 465–469 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tang, C.L.: Periodic solutions of nonautonomous second order systems with sublinear nonlinearity. Proc. Am. Math. Soc. 126, 3263–3270 (1998)

    Article  MATH  Google Scholar 

  8. Tang, C.L., Wu, X.P.: Periodic solutions for second order systems with not uniformly coercive potential. J. Math. Anal. Appl. 259, 386–397 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Willem, M.: Oscillations forcées de systèmes hamiltoniens. In: Public. Sémin. Analyse Non Linéaire, Univ. Besancon (1981)

  10. Wu, X.: Saddle point characterization and multiplicity of periodic solutions of non-autonomous second order systems. Nonlinear Anal. TMA 58, 899–907 (2004)

    Article  MATH  Google Scholar 

  11. Wu, X.P., Tang, C.L.: Periodic solutions of a class of nonautonomous second order systems. J. Math. Anal. Appl. 236, 227–235 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhao, F., Wu, X.: Periodic solutions for a class of non-autonomous second order systems. J. Math. Anal. Appl. 296, 422–434 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhao, F., Wu, X.: Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity. Nonlinear Anal. 60, 325–335 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Tang, C.L., Wu, X.P.: A note on periodic solutions of nonautonomous second-order systems. Proc. Am. Math. Soc. 132(5), 1295–1303 (2003)

    Article  MathSciNet  Google Scholar 

  15. Tang, X.H., Meng, Q.: Solutions of a second-order Hamiltonian system with periodic boundary conditions. Nonlinear Anal. 11(5), 3722–3733 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, Z., Zhang, J.: Periodic solutions of a class of second order non-autonomous Hamiltonian systems. Nonlinear Anal. 72(12), 4480–4487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Schechter, M.: Periodic non-autonomous second-order dynamical systems. J. Differ. Equ. 223(2), 290–302 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schechter, M.: Periodic solutions of second-order non-autonomous dynamical systems. Bound. Value Probl. 2006, Art. ID 25104 (2006)

    Article  MathSciNet  Google Scholar 

  19. Ma, J., Tang, C.L.: Periodic solutions for some nonautonomous second-order systems. J. Math. Anal. Appl. 275, 482–494 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhao, F., Wu, X.: Saddle point reduction method for some non-autonomous second order systems. J. Math. Anal. Appl. 291, 653–665 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nieto, J.J., O’Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal. Real World Appl. 10, 680–690 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tian, Y., Ge, W.G.: Applications of variational methods to boundary value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51, 509–527 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhou, J.W., Li, Y.K.: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. 71, 2856–2865 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhou, J.W., Li, Y.K.: Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 72, 1594–1603 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ding, W., Qian, D.: Periodic solutions for sublinear systems via variational approach. Nonlinear Anal. Real World Appl. 11(4), 2603–2609 (2011)

    Article  MathSciNet  Google Scholar 

  26. Sun, J.T., Chen, H.B., Nieto, J.J.: Infinitely many solutions for second-order Hamiltonian system with impulsive effects. Math. Comput. Model. 54, 544–555 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sun, J.T., Chen, H.B., Zhou, T.J.: Multiplicity of solutions for a fourth-order impulsive differential equation via variational methods. Bull. Aust. Math. Soc. 82, 446–458 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pasca, D., Tang, C.L.: Some existence results on periodic solutions of nonautonomous second-order differential systems with (q,p)-Laplacian. Appl. Math. Lett. 23, 246–251 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pasca, D.: Periodic solutions of a class of nonautonomous second-order differential systems with (q,p)-Laplacian. Bull. Belg. Math. Soc. Simon Stevin 17, 841–850 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Pasca, D., Tang, C.L.: Some existence results on periodic solutions of ordinary (q,p)-laplacian systems. J. Appl. Math. Inform. 29(1–2), 39–48 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Tian, Y., Ge, W.G.: Periodic solutions of non-autonomous second order systems with p-Laplacian. Nonlinear Anal. 66(1), 192–203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7, 241–273 (1983)

    Article  MathSciNet  Google Scholar 

  33. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conf. Ser. in Math., vol. 65. Am. Math. Soc., Providence (1986)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410083, P.R. China

    Xiaoxia Yang & Haibo Chen

Authors
  1. Xiaoxia Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haibo Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoxia Yang.

Additional information

This work is supported by NFSC (10871206).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, X., Chen, H. Periodic solutions for a nonlinear (q,p)-Laplacian dynamical system with impulsive effects. J. Appl. Math. Comput. 40, 607–625 (2012). https://doi.org/10.1007/s12190-012-0556-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-012-0556-x

Keywords

Mathematics Subject Classification

Search

Navigation