About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 398681, 6 pages
http://dx.doi.org/10.1155/2013/398681
Research Article

Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems

Department of Mathematics, Xiangnan College, Chenzhou, Hunan 423000, China

Received 3 September 2012; Accepted 27 November 2012

Academic Editor: Jinde Cao

Copyright © 2013 Xiaoping Wang. 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.

Linked References

  1. A. M. Lyapunov, “Probléme général de la stabilité du mouvement,” Annde La Faculté, vol. 2, no. 9, pp. 203–474, 1907.
  2. M. Bohner, S. Clark, and J. Ridenhour, “Lyapunov inequalities for time scales,” Journal of Inequalities and Applications, vol. 7, no. 1, pp. 61–77, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. S. Cheng, “A discrete analogue of the inequality of Lyapunov,” Hokkaido Mathematical Journal, vol. 12, no. 1, pp. 105–112, 1983. View at Zentralblatt MATH
  4. S.-S. Cheng, “Lyapunov inequalities for differential and difference equations,” Polytechnica Posnaniensis, no. 23, pp. 25-–41, 1991. View at Zentralblatt MATH
  5. S. Clark and D. Hinton, “A Liapunov inequality for linear Hamiltonian systems,” Mathematical Inequalities & Applications, vol. 1, no. 2, pp. 201–209, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. S. Clark and D. Hinton, “Discrete Lyapunov inequalities,” Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 369–380, 1999. View at Zentralblatt MATH
  7. G. Sh. Guseinov and B. Kaymakçalan, “Lyapunov inequalities for discrete linear Hamiltonian systems,” Computers & Mathematics with Applications, vol. 45, no. 6–9, pp. 1399–1416, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. G. Sh. Guseinov and A. Zafer, “Stability criteria for linear periodic impulsive Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1195–1206, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. P. Hartman, “Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity,” Transactions of the American Mathematical Society, vol. 246, pp. 1–30, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. X. He and Q.-M. Zhang, “A discrete analogue of Lyapunov-type inequalities for nonlinear difference systems,” Computers & Mathematics with Applications, vol. 62, no. 2, pp. 677–684, 2011. View at Publisher · View at Google Scholar
  11. L. Jiang and Z. Zhou, “Lyapunov inequality for linear Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 310, no. 2, pp. 579–593, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. S. H. Lin and G. S. Yang, “On discrete analogue of Lyapunov inequality,” Tamkang Journal of Mathematics, vol. 20, no. 2, pp. 169–186, 1989. View at Zentralblatt MATH
  13. X. Wang, “Stability criteria for linear periodic Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 329–336, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. X.-H. Tang and M. Zhang, “Lyapunov inequalities and stability for linear Hamiltonian systems,” Journal of Differential Equations, vol. 252, no. 1, pp. 358–381, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. X. H. Tang, Q.-M. Zhang, and M. Zhang, “Lyapunov-type inequalities for the first-order nonlinear Hamiltonian systems,” Computers & Mathematics with Applications, vol. 62, no. 9, pp. 3603–3613, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. A. Tiryaki, M. Ünal, and D. Çakmak, “Lyapunov-type inequalities for nonlinear systems,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 497–511, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. Ünal, D. Çakmak, and A. Tiryaki, “A discrete analogue of Lyapunov-type inequalities for nonlinear systems,” Computers & Mathematics with Applications, vol. 55, no. 11, pp. 2631–2642, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. Ünal and D. Çakmak, “Lyapunov-type inequalities for certain nonlinear systems on time scales,” Turkish Journal of Mathematics, vol. 32, no. 3, pp. 255–275, 2008. View at Zentralblatt MATH
  19. Q.-M. Zhang and X. H. Tang, “Lyapunov inequalities and stability for discrete linear Hamiltonian systems,” Applied Mathematics and Computation, vol. 218, no. 2, pp. 574–582, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Q.-M. Zhang and X. H. Tang, “Lyapunov inequalities and stability for discrete linear Hamiltonian systems,” Journal of Difference Equations and Applications, vol. 18, no. 9, pp. 1467–1484, 2012. View at Publisher · View at Google Scholar
  21. R. Agarwal, C. Ahlbrandt, M. Bohner, and A. Peterson, “Discrete linear Hamiltonian systems: a survey,” Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 307–333, 1999. View at Zentralblatt MATH
  22. C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic, Dordrecht, The Netherlands, 1996.
  23. S. N. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 3rd edition, 2004.