Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 328479, 17 pages
doi:10.1155/2009/328479
Research Article

Periodic Solution of Second-Order Hamiltonian Systems with a Change Sign Potential on Time Scales

You-Hui Su1,2 and Wan-Tong Li2

1School of Mathematics and Physical Sciences, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221008, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 26 November 2008; Accepted 4 April 2009

Academic Editor: Yong Zhou

Copyright © 2009 You-Hui Su and Wan-Tong Li. 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. S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität Würzburg, Würzburg, Germany, 1988.
  2. R. P. Agarwal, M. Bohner, and W.-T. Li, Nonoscillation and Oscillation Theory for Functional Differential Equations, vol. 267 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2004. View at Zentralblatt MATH · View at MathSciNet
  3. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. View at Zentralblatt MATH · View at MathSciNet
  4. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkhäuser, Boston, Mass, USA, 2001. View at Zentralblatt MATH · View at MathSciNet
  5. S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Zentralblatt MATH · View at MathSciNet
  6. V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol. 370 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996. View at Zentralblatt MATH · View at MathSciNet
  7. M. A. Jones, B. Song, and D. M. Thomas, “Controlling wound healing through debridement,” Mathematical and Computer Modelling, vol. 40, no. 9-10, pp. 1057–1064, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. V. Spedding, “Taming nature's numbers,” New Scientist, vol. 179, no. 2404, pp. 28–31, 2003.
  9. D. M. Thomas, L. Vandemuelebroeke, and K. Yamaguchi, “A mathematical evolution model for phytoremediation of metals,” Discrete and Continuous Dynamical Systems. Series B, vol. 5, no. 2, pp. 411–422, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. W.-T. Li and X.-L. Liu, “Eigenvalue problems for second-order nonlinear dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 578–592, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W.-T. Li and H.-R. Sun, “Multiple positive solutions for nonlinear dynamical systems on a measure chain,” Journal of Computational and Applied Mathematics, vol. 162, no. 2, pp. 421–430, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Q. Wei, Y.-H. Su, S. Li, and X.-X. Yan, “Existence of positive solutions to a singular p-Laplacian general dirichlet BVPs with sign changing nonlinearity,” Abstract and Applied Analysis, vol. 2009, Article ID 512402, 21 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Y.-H. Su, S. Li, and C.-Y. Huang, “Positive solution to a singular p-Laplacian BVPs with sign-changing nonlinearity involving derivative on time scales,” Advances in Difference Equations, vol. 2009, Article ID 623932, 21 pages, 2009.
  14. Y.-H. Su, “Multiple positive pseudo-symmetric solutions of p-Laplacian dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49, no. 7-8, pp. 1664–1681, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  15. Y.-H. Su, W.-T. Li, and H.-R. Sun, “Triple positive pseudo-symmetric solutions of three-point BVPs for p-Laplacian dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1442–1452, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y.-H. Su, W.-T. Li, and H.-R. Sun, “Positive solutions of singular p-Laplacian BVPs with sign changing nonlinearity on time scales,” Mathematical and Computer Modelling, vol. 48, no. 5-6, pp. 845–858, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y.-H. Su, W.-T. Li, and H.-R. Sun, “Positive solutions of singular p-Laplacian dynamic equations with sign changing nonlinearity,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 352–368, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y.-H. Su and W.-T. Li, “Triple positive solutions of m-point BVPs for p-Laplacian dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3811–3820, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J.-P. Sun and W.-T. Li, “Existence of solutions to nonlinear first-order PBVPs on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 3, pp. 883–888, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J.-P. Sun and W.-T. Li, “Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales,” Computers & Mathematics with Applications, vol. 54, no. 6, pp. 861–871, 2007. View at Zentralblatt MATH · View at MathSciNet
  21. H.-R. Sun and W.-T. Li, “Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales,” Journal of Differential Equations, vol. 240, no. 2, pp. 217–248, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Multiple positive solutions of singular Dirichlet problems on time scales via variational methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 2, pp. 368–381, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1263–1274, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations,” Advances in Difference Equations, vol. 2008, Article ID 796851, 13 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. G. Sh. Guseinov, “Integration on time scales,” Journal of Mathematical Analysis and Applications, vol. 285, no. 1, pp. 107–127, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at Zentralblatt MATH · View at MathSciNet
  27. C.-L. Tang and X.-P. Wu, “Periodic solutions for second order Hamiltonian systems with a change sign potential,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 506–516, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. B. Aulbach and L. Neidhart, “Integration on measure chains,” in Proceedings of the Sixth International Conference on Difference Equations, pp. 239–252, CRC Press, Boca Raton, Fla, USA, 2004. View at Zentralblatt MATH · View at MathSciNet
  29. A. Cabada and D. R. Vivero, “Criterions for absolute continuity on time scales,” Journal of Difference Equations and Applications, vol. 11, no. 11, pp. 1013–1028, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. B. P. Rynne, “L2 spaces and boundary value problems on time-scales,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1217–1236, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. F. A. Davidson and B. P. Rynne, “Eigenfunction expansions in L2 spaces for boundary value problems on time-scales,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1038–1051, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Basic properties of Sobolev's spaces on time scales,” Advances in Difference Equations, vol. 2006, Article ID 38121, 14 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, A Pergamon Press Book, Macmillan, New York, NY, USA, 1964. View at Zentralblatt MATH · View at MathSciNet
  34. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986. View at Zentralblatt MATH · View at MathSciNet