Advertisement

Frontiers of Mathematics in China

, Volume 13, Issue 6, pp 1313–1323 | Cite as

Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions

  • Xingfan Chen
  • Fei Guo
  • Peng Liu
Research Article
  • 10 Downloads

Abstract

We consider a class of asymptotically linear nonautonomous second-order Hamiltonian systems. Using the Saddle Point Theorem, we obtain the existence result, which extends some previously known results.

Keywords

Existence periodic solution second-order Hamiltonian systems Saddle Point Theorem 

MSC

34C25 47E05 49J35 58E50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors would like to express their deep gratitude to the referees for giving many valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11371276, 10901118) and the Elite Scholar Program in Tianjin University, China.

References

  1. 1.
    Bartolo P, Benci V, Fortunato D. Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal, 1983, 7: 241–273MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cerami G. An existence criterion for critical points on unbounded mainfolds. Istit Lombardo Accad Sci Lett Rend A, 1978, 112: 332–336 (in Italian)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chen X F, Guo F. Existence and multiplicity of periodic solutions for nonautonomous second order Hamiltonian systems. Bound Value Probl, 2016, 138: 1–10MathSciNetGoogle Scholar
  4. 4.
    Izydorek M, Janczewska J. Homoclinic solutions for a class of second-order Hamiltonian systems. J Differential Equations, 2005, 219: 375–389MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jiang Q, Tang C L. Periodic and subharmonic solutions of a class subquadratic second order Hamiltonian systems. J Math Anal Appl, 2007, 328: 380–389MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li L, Schechter M. Existence solutions for second order Hamiltonian systems. Nonlinear Anal, 2016, 27: 283–296MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Long Y M. Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials. Nonlinear Anal, 1995, 24: 1665–1671MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Long Y M. Index Theory for Symplectic Paths with Applications. Basel: Birkhaser, 2002CrossRefzbMATHGoogle Scholar
  9. 9.
    Luan S, Mao A. Periodic solutions for a class of non-autonomous Hamiltonian systems. Nonlinear Anal, 2005, 61: 1413–1426MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mawhin J, Willem M. Critical Point Theory and Hamiltonian Systems. Berlin-New York: Springer-Verlag, 1989CrossRefzbMATHGoogle Scholar
  11. 11.
    Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg Conf Ser Math, No 65. Providence: Amer Math Soc, 1986CrossRefzbMATHGoogle Scholar
  12. 12.
    Tang C L. Existence and multiplicity of periodic solutions for nonautonomous second order systems. Nonlinear Anal, 1998, 32: 299–304MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Tang C L, Wu X P. Periodic solutions of a class of new superquadratic second order Hamiltonian systems. Appl Math Lett, 2014, 34: 65–71MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Tang X H, Jiang J C. Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems. Comput Math Appl, 2010, 59: 3646–3655MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tang X H, Xiao L. Homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal, 2009, 71: 1140–1152MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tao Z L, Tang C L. Periodic and subharmonic solutions of second-order Hamiltonian systems. J Math Anal Appl, 2004, 293: 435–445MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tao Z L, Yan S A, Wu S L. Periodic solutions for a class of superquadratic Hamiltonian systems. J Math Anal Appl, 2007, 331: 152–158MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang Z Y, Xiao J Z. On periodic solutions of subquadratic second order nonautonomous Hamiltonian systems. Appl Math Lett, 2015, 40: 71–72CrossRefGoogle Scholar
  19. 19.
    Zhang Q Y, Liu C G. Infinitely many periodic solutions for second order Hamiltonian systems. J Differential Equations, 2011, 251: 816–833MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhao F K, Chen J, Yang M B. A periodic solution for a second order asymptotically linear Hamiltonian systems. Nonlinear Anal, 2009, 70: 4021–4026MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinChina

Personalised recommendations