Open Access

Multiple Periodic Solutions to Nonlinear Discrete Hamiltonian Systems

Advances in Difference Equations20072007:041830

DOI: 10.1155/2007/41830

Received: 15 April 2007

Accepted: 19 August 2007

Published: 25 October 2007


An existence result of multiple periodic solutions to the asymptotically linear discrete Hamiltonian systems is obtained by using the Morse index theory.


Authors’ Affiliations

College of Mathematics and Econometrics, Hunan University


  1. Chang K-C: Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications. Volume 6. Birkhäuser, Boston, Mass, USA; 1993.View ArticleGoogle Scholar
  2. Ekeland I: Convexity Methods in Hamiltonian Mechanics, Results in Mathematics and Related Areas (3). Volume 19. Springer, Berlin, Germany; 1990.View ArticleGoogle Scholar
  3. Long Y: Index Theory for Symplectic Paths with Applications, Progress in Mathematics. Volume 207. Birkhäuser, Basel, Switzerland; 2002.View ArticleGoogle Scholar
  4. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989.View ArticleGoogle Scholar
  5. Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics. Volume 65. American Mathematical Society, Providence, RI, USA; 1986.Google Scholar
  6. Ahlbrandt CD, Peterson AC: Discrete Hamiltonian Systems, Kluwer Texts in the Mathematical Sciences. Volume 16. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996.View ArticleGoogle Scholar
  7. Bohner M: Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions. Journal of Mathematical Analysis and Applications 1996,199(3):804–826. 10.1006/jmaa.1996.0177MATHMathSciNetView ArticleGoogle Scholar
  8. Erbe LH, Yan PX: Disconjugacy for linear Hamiltonian difference systems. Journal of Mathematical Analysis and Applications 1992,167(2):355–367. 10.1016/0022-247X(92)90212-VMATHMathSciNetView ArticleGoogle Scholar
  9. Hartman P: Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity. Transactions of the American Mathematical Society 1978, 246: 1–30.MATHMathSciNetGoogle Scholar
  10. Guo Z, Yu J: Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. Nonlinear Analysis: Theory, Methods & Applications 2003,55(7–8):969–983. 10.1016/ ArticleGoogle Scholar
  11. Zhou Z, Yu J, Guo Z: The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems. The ANZIAM Journal 2005,47(1):89–102. 10.1017/S1446181100009792MATHMathSciNetView ArticleGoogle Scholar
  12. Guo Z, Yu J: Multiplicity results for periodic solutions to second-order difference equations. Journal of Dynamics and Differential Equations 2006,18(4):943–960. 10.1007/s10884-006-9042-1MATHMathSciNetView ArticleGoogle Scholar
  13. Guo Z, Yu J: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Science in China A 2003,46(4):506–515.MATHMathSciNetView ArticleGoogle Scholar
  14. Guo Z, Yu J: The existence of periodic and subharmonic solutions of subquadratic second order difference equations. Journal of the London Mathematical Society 2003,68(2):419–430. 10.1112/S0024610703004563MATHMathSciNetView ArticleGoogle Scholar
  15. Zhou Z, Yu J, Guo Z: Periodic solutions of higher-dimensional discrete systems. Proceedings of the Royal Society of Edinburgh 2004,134(5):1013–1022. 10.1017/S0308210500003607MATHMathSciNetView ArticleGoogle Scholar
  16. Chang K-C, Lin YQ: Functional Analysis(I). Peking University Press, Peking, China; 1987.Google Scholar


© Bo Zheng 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.