Journal of Dynamics and Differential Equations

, Volume 18, Issue 4, pp 943–960

Multiplicity Results for Periodic Solutions to Second-Order Difference Equations



By using the Zp geometrical index theory, some sufficient conditions on the multiplicity results of periodic solutions to the second-order difference equations
are obtained. By two examples, we show that our results are the best possible in the sense that the lower bound of the number of periodic solutions cannot be improved.


Second-order difference equations periodic solution critical point Zp index theory 

Mathematics Subject Classification



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  1. 1.
    Agarwal R.P., Perera K., O’Regan D. (2004). Multiple periodic solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Anal. TMA 58:69–73MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Benci V. (1982). On critical point theory for indefinite functionals in the presence of symmetries. Trans. Amer. Math. Soc. 274:533–572MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Chang, K. C. (1986). Critical Point Theory and its Applications, Science and Technical Press, Shanghai, China, 1986. (in Chinese).Google Scholar
  4. 4.
    Guo Z.M., Yu J.S. (2003). Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China (Ser. A) 46:506–515MathSciNetCrossRefGoogle Scholar
  5. 5.
    Guo Z.M., Yu J.S. (2003). The existence of periodic and subharmonic solutions for solutions of subquadatic second order difference equations. J. Lon. Math. Soc. 68(2):419–430MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Guo Z.M., Yu J.S. (2003). Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. Nonlinear Anal. TMA 55:669–683MathSciNetCrossRefGoogle Scholar
  7. 7.
    Liu J.Q. (1989). A geometrical index for the group Z p. Acta Math. Sinica, New Ser. 5(3):193–196MATHCrossRefGoogle Scholar
  8. 8.
    Mawhin J., Willen M. (1989). Critical Point Theory and Hamiltonian Systems. Springer Verlag, New YorkMATHGoogle Scholar
  9. 9.
    Yu J.S., Long Y.H., Guo Z.M. (2004). Subharmonic solutions with prescribed minimal period of discrete forced pendulum equation. J. Dyn. Diff. Equat. 12(2):575–586MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Zhang G., Cheng S.S. (2006). Existence of solutions for a nonlinear system with a parameter. J. Math. Anal. Appl. 314(1):311–319MATHMathSciNetGoogle Scholar
  11. 11.
    Zhang G., Yang Z.L. (2004). Existence of 2n nontrivial solutions for discrete two-point boundary value problems. Nonlinear Anal. TMA 59:1181–1187MATHCrossRefGoogle Scholar
  12. 12.
    Zhou Z., Yu J.S., Guo Z.M. (2004). Periodic solutions of higner-dimensional discrete systems. Proc. R. Soc. Edinburgh 134A:1013–1022MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.College of Mathematics and Information SciencesGuangzhou UniversityGuangzhouP. R. China

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