Science in China Series A: Mathematics

, Volume 46, Issue 4, pp 506–515 | Cite as

Existence of periodic and subharmonic solutions for second-order superlinear difference equations

  • Zhiming Guo
  • Jianshe Yu


By critical point theory, a new approach is provided to study the existence and multiplicity results of periodic and subharmonic solutions for difference equations. For secord-order difference equations
$$\Delta ^2 x_{n - 1} + f(n, x_n ) = 0,$$
some new results are obtained for the above problems when f(t, z) has superlinear growth at zero and at infinity in z.


superlinear difference equation periodic solution subharmonic solution critical point linking 


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  1. 1.
    Agarwal, R. P., Difference Equations and Inequalities: Theory, Methods and Applications, New York: Marcel Dekker, 1992.zbMATHGoogle Scholar
  2. 2.
    Erbe, L. H., Xia, H., Yu, J. S., Global stability of a linear nonautonomous delay difference equations, J. Diff. Equations Appl., 1995, 1: 151–161.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Györi, I., Ladas, G., Oscillation Theory of Delay Differential Equations with Applications, Oxford: Oxford University Press, 1991.zbMATHGoogle Scholar
  4. 4.
    Kocic, V. L., Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Boston: Kluwer Academic Publishers, 1993.zbMATHGoogle Scholar
  5. 5.
    Matsunaga, H., Hara, T., Sakata, S., Global attractivity for a nonlinear difference equation with variable delay, Computers Math. Applic., 2001, 41: 543–551.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Tang, X. H., Yu, J. S., Oscillation of nonlinear delay difference equations, J. Math. Anal. Appl., 2000, 249: 476–490.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Yu, J. S., Asymptotic stability for a linear difference equation with variable delay, Computers Math. Applic., 1998, 36: 203–210.zbMATHCrossRefGoogle Scholar
  8. 8.
    Zhou, Z., Zhang, Q., Uniform stability of nonlinear difference systems, J. Math. Anal. Appl., 1998, 225: 486–500.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Elaydi, S. N., Zhang, S., Stability and periodicity of difference equations with finite delay, Funkcialaj Ekvac., 1994, 37: 401–413.zbMATHMathSciNetGoogle Scholar
  10. 10.
    Chang, K. C., Critical Point Theory and Its Applications (in Chinese), Shanghai: Shanghai Science and Technical Press, 1986.Google Scholar
  11. 11.
    Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems, Boston: Birkhäuser, 1993.zbMATHGoogle Scholar
  12. 12.
    Kaplanm, J. L., Yorke, J. A., Ordinary differential equations which yield periodic solution of delay differential equations, J. Math. Anal. Appl., 1974, 48: 317–324.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Liu, J. Q., Wang, Z. Q., Remarks on subharmonics with minimal periods of Hamiltonian systems, Nonlinear. Anal. T. M. A., 1993, 7: 803–821.CrossRefGoogle Scholar
  14. 14.
    Mawhin, J., Willem, M., Critical Point Theory and Hamiltonian Systems, New York: Spinger-Verlag Inc., 1989.zbMATHGoogle Scholar
  15. 15.
    Michalek, R., Tarantello, G., Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems, J. Differential Equations, 1988, 72: 28–55.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS. AMS 65, 1986.Google Scholar
  17. 17.
    Capietto, A., Mawhin, J., Zanolin, F., A continuation approach to superlinear periodic boundary value problems, J. Differential Equations, 1990, 88: 347–395.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hale, J. K., Mawhin, J., Coincidence degree and periodic solutions of neutral equations, J. Differential Equations, 1974, 15: 295–307.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Elaydi, S. N., An Introduction to Difference Equations, New York: Springer-Verlag, 1999.zbMATHGoogle Scholar
  20. 20.
    Pielou, E. C., An Introduction to Mathematical Ecology, New York: Willey Interscience, 1969.zbMATHGoogle Scholar
  21. 21.
    Palais, R. S., Smale, S., A generalized Morse theory, Bull. Amer. Math. Soc., 1964, 70: 165–171.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press 2003

Authors and Affiliations

  1. 1.Department of Applied MathematicsHunan UniversityChangshaChina

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