Open Access

Periodic Solutions for Subquadratic Discrete Hamiltonian Systems

Advances in Difference Equations20072007:013916

DOI: 10.1155/2007/13916

Received: 4 February 2007

Accepted: 26 April 2007

Published: 17 June 2007

Abstract

Some existence conditions of periodic solutions are obtained for a class of nonautono mous subquadratic first-order discrete Hamiltonian systems by the minimax methods in the critical point theory.

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Authors’ Affiliations

(1)
College of Mathematics and Econometrics, Hunan University
(2)
Department of Information, Hunan Business College

References

  1. Daouas A, Timoumi M: Subharmonics for not uniformly coercive Hamiltonian systems. Nonlinear Analysis: Theory, Methods & Applications 2007,66(3):571–581. 10.1016/j.na.2005.12.002MATHMathSciNetView ArticleGoogle Scholar
  2. Rabinowitz PH: On subharmonic solutions of Hamiltonian systems. Communications on Pure and Applied Mathematics 1980,33(5):609–633. 10.1002/cpa.3160330504MATHMathSciNetView ArticleGoogle Scholar
  3. Silva EA: Subharmonic solutions for subquadratic Hamiltonian systems. Journal of Differential Equations 1995,115(1):120–145. 10.1006/jdeq.1995.1007MATHMathSciNetView ArticleGoogle Scholar
  4. Jiang Q, Tang C-L: Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems. Journal of Mathematical Analysis and Applications 2007,328(1):380–389. 10.1016/j.jmaa.2006.05.064MATHMathSciNetView ArticleGoogle Scholar
  5. Tang C-L: Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proceedings of the American Mathematical Society 1998,126(11):3263–3270. 10.1090/S0002-9939-98-04706-6MATHMathSciNetView ArticleGoogle Scholar
  6. Tang C-L, Wu X-P: Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems. Journal of Mathematical Analysis and Applications 2002,275(2):870–882. 10.1016/S0022-247X(02)00442-0MATHMathSciNetView ArticleGoogle Scholar
  7. Tang C-L, Wu X-P: Notes on periodic solutions of subquadratic second order systems. Journal of Mathematical Analysis and Applications 2003,285(1):8–16. 10.1016/S0022-247X(02)00417-1MATHMathSciNetView ArticleGoogle Scholar
  8. Tang C-L, Wu X-P: Subharmonic solutions for nonautonomous sublinear second order Hamiltonian systems. Journal of Mathematical Analysis and Applications 2005,304(1):383–393. 10.1016/j.jmaa.2004.09.032MATHMathSciNetView ArticleGoogle Scholar
  9. 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/j.na.2003.07.019MATHMathSciNetView ArticleGoogle Scholar
  10. Yu J, Bin H, Guo Z: Multiple periodic solutions for discrete Hamiltonian systems. Nonlinear Analysis: Theory, Methods & Applications 2007,66(7):1498–1512. 10.1016/j.na.2006.01.029MATHMathSciNetView ArticleGoogle Scholar
  11. Guo Z, Yu J: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Science in China. Series A 2003,46(4):506–515.MATHMathSciNetView ArticleGoogle Scholar
  12. Yu J, Deng X, Guo Z: Periodic solutions of a discrete Hamiltonian system with a change of sign in the potential. Journal of Mathematical Analysis and Applications 2006,324(2):1140–1151. 10.1016/j.jmaa.2006.01.013MATHMathSciNetView ArticleGoogle Scholar
  13. 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
  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. Rodriguez J, Etheridge DL: Periodic solutions of nonlinear second-order difference equations. Advances in Difference Equations 2005,2005(2):173–192. 10.1155/ADE.2005.173MATHMathSciNetView ArticleGoogle Scholar
  16. Zhou Z, Yu J, Guo Z: Periodic solutions of higher-dimensional discrete systems. Proceedings of the Royal Society of Edinburgh. Section A 2004,134(5):1013–1022. 10.1017/S0308210500003607MATHMathSciNetView ArticleGoogle Scholar
  17. Agarwal RP, Perera K, O'Regan D: Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Analysis: Theory, Methods & Applications 2004,58(1–2):69–73. 10.1016/j.na.2003.11.012MATHMathSciNetView ArticleGoogle Scholar
  18. Agarwal RP, Perera K, O'Regan D: Multiple positive solutions of singular discrete p -Laplacian problems via variational methods. Advances in Difference Equations 2005,2005(2):93–99. 10.1155/ADE.2005.93MATHMathSciNetView ArticleGoogle Scholar
  19. Xue Y-F, Tang C-L: Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system. Nonlinear Analysis: Theory, Methods & Applications 2007,67(7):2072–2080. 10.1016/j.na.2006.08.038MATHMathSciNetView ArticleGoogle Scholar
  20. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.View ArticleGoogle Scholar
  21. Cerami G: An existence criterion for the critical points on unbounded manifolds. Istituto Lombardo. Accademia di Scienze e Lettere. Rendiconti. A 1978,112(2):332–336.MATHMathSciNetGoogle Scholar

Copyright

© Xiaoqing Deng. 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.