Skip to main content
SpringerLink
Log in

New results on periodic solutions for second order damped vibration systems

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

The purpose of this paper is to study the multiplicity of periodic solutions for a class of non-autonomous second-order damped vibration systems. New results are obtained by using Fountain theorem. These results improve the related ones in the literature.

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. Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonanno, G., Livrea, R.: Periodic solutions for a class of second-order Hamiltonian systems. Electron. J. Differ. Equ. 2005(115), 357–370 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Cerami, G.: An existence criterion for the critical points on unbounded manifolds. Istit. Lombardo Accad. Sci. Lett. Rend. A 112, 332–336 (1978)

    MathSciNet  MATH  Google Scholar 

  4. Chen, X.F., Guo, F., Liu, P.: Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions. Front. Math. China 13, 1313–1323 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, X.F., Guo, F.: Existence and multiplicity of periodic solutions for nonautonomous second order Hamiltonian systems. Bound. Value Probl 138, 1–10 (2016)

    MathSciNet  Google Scholar 

  6. Fei, G., Kim, S.K., Wang, T.: Periodic solutions of classical Hamiltonian systems without Palais–Smale condition. J. Math. Anal. Appl. 267, 665–678 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gu, H., An, T.: Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems. Bound. Value Probl. 2013(1), 1–8 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jiang, Q., Tang, C.L.: Periodic and subharmonic solutions of a class subquadratic second order Hamiltonian systems. J. Math. Anal. Appl. 328, 380–389 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Khachnaoui, K.: Existence and multiplicity of periodic solutions for a class of dynamical systems. Nonlinear Stud. 23(1), 103–110 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Long, Y.M.: Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials. Nonlinear Anal. 24, 1665–1671 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, L., Schechter, M.: Existence solutions for second order Hamiltonian systems. Nonlinear Anal. Real World Appl. 27, 283–296 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  13. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., Rhode Island (1986)

    Book  MATH  Google Scholar 

  14. Tang, C.: Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proc. Am. Math. Soc. 126(11), 3263–3270 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tang, C., Wu, X.: Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems. J. Math. Anal. Appl. 275(2), 870–882 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Tang, X., Jiang, J.: Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems. Comput. Math. Appl. 59(12), 3646–3655 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Tang, X.H., Jiang, J.C.: Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems. Comput. Math. Appl. 59, 3646–3655 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Tang, C.L., Wu, X.P.: Periodic solutions of a class of new superquadratic second order Hamiltonian systems. Appl. Math. Lett. 34, 65–71 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, Z.Y., Xiao, J.Z.: On periodic solutions of subquadratic second order nonautonomous Hamiltonian systems. Appl. Math. Lett. 40, 71–72 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, Z., Zhang, J.: New existence results on periodic solutions of non-autonomous second order Hamiltonian systems. Appl. Math. Lett. 79, 43–50 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)

    Book  MATH  Google Scholar 

  24. Yin, Q., Liu, D.: Periodic solutions of a class of superquadratic second order Hamiltonian systems. Appl. Math. J. Chin. Univ. Ser. B 15(3), 259–266 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, Q., Liu, C.: Infinitely many periodic solutions for second order Hamiltonian systems. J. Differ. Equ. 251(4–5), 816–833 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhao, F., Chen, J., Yang, M.: A periodic solution for a second-order asymptotically linear Hamiltonian system. Nonlinear Anal. 70(11), 4021–4026 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhao, F.K., Chen, J., Yang, M.B.: A periodic solution for a second order asymptotically linear Hamiltonian systems. Nonlinear Anal. 70, 4021–4026 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to express their sincere thanks to the referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute Preparatory for Engineering Studies, University of Kairouan, Kairouan, Tunisia

    Khachnaoui Khaled

Authors
  1. Khachnaoui Khaled
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Khachnaoui Khaled.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khaled, K. New results on periodic solutions for second order damped vibration systems. Ricerche mat 72, 709–721 (2023). https://doi.org/10.1007/s11587-021-00567-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-021-00567-3

Keywords

Mathematics Subject Classification

Search

Navigation