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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 234015, 26 pages
http://dx.doi.org/10.1155/2010/234015
Research Article

Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with Δ-Carathéodory Functions

1Département de Mathématiques et de Statistique, Université de Montréal, CP 6128, Succursale Centre-Ville, Montréal, QC, Canada H3C 3J7
2Département de Mathématiques, Collège Édouard-Montpetit, 945 Chemin de Chambly, Longueuil, QC, Canada J4H 3M6

Received 31 August 2010; Accepted 30 November 2010

Academic Editor: J. Mawhin

Copyright © 2010 M. Frigon and H. Gilbert. 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. E. Akın, “Boundary value problems for a differential equation on a measure chain,” Panamerican Mathematical Journal, vol. 10, no. 3, pp. 17–30, 2000. View at Google Scholar · View at Zentralblatt MATH
  2. P. Stehlík, “Periodic boundary value problems on time scales,” Advances in Difference Equations, no. 1, pp. 81–92, 2005. View at Google Scholar · View at Zentralblatt MATH
  3. A. C. Peterson, Y. N. Raffoul, and C. C. Tisdell, “Three point boundary value problems on time scales,” Journal of Difference Equations and Applications, vol. 10, no. 9, pp. 843–849, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. P. Stehlík, “On lower and upper solutions without ordering on time scales,” Advances in Difference Equations, Article ID 73860, 12 pages, 2006. View at Google Scholar · View at Zentralblatt MATH
  5. C. C. Tisdell and H. B. Thompson, “On the existence of solutions to boundary value problems on time scales,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis, vol. 12, no. 5, pp. 595–606, 2005. View at Google Scholar · View at Zentralblatt MATH
  6. J. Henderson, A. Peterson, and C. C. Tisdell, “On the existence and uniqueness of solutions to boundary value problems on time scales,” Advances in Difference Equations, no. 2, pp. 93–109, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. P. Amster, C. Rogers, and C. C. Tisdell, “Existence of solutions to boundary value problems for dynamic systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 308, no. 2, pp. 565–577, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. F. M. Atici, A. Cabada, C. J. Chyan, and B. Kaymakçalan, “Nagumo type existence results for second-order nonlinear dynamic BVPs,” Nonlinear Analysis. Theory, Methods & Applications, vol. 60, no. 2, pp. 209–220, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. Henderson and C. C. Tisdell, “Dynamic boundary value problems of the second-order: Bernstein-Nagumo conditions and solvability,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 5, pp. 1374–1386, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Frigon, “Boundary and periodic value problems for systems of nonlinear second order differential equations,” Topological Methods in Nonlinear Analysis, vol. 1, no. 2, pp. 259–274, 1993. View at Google Scholar · View at Zentralblatt MATH
  11. J. Zhou and Y. Li, “Sobolev's spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 5, pp. 1375–1388, 2010. View at Publisher · View at Google Scholar
  12. M. Bohner and G. Guseinov, “Riemann and Lebesgue integration,” in Advances in Dynamic Equations on Time Scales, pp. 117–163, Birkhäuser, Boston, Mass, USA, 2003. View at Google Scholar
  13. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkhäuser, Boston, Mass, USA, 2001.
  14. A. Cabada and D. R. Vivero, “Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral: application to the calculus of Δ-antiderivatives,” Mathematical and Computer Modelling, vol. 43, no. 1-2, pp. 194–207, 2006. View at Publisher · View at Google Scholar
  15. 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 Google Scholar · View at Zentralblatt MATH
  16. 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
  17. H. Gilbert, “Existence Theorems for First-Order Equations on Time Scales with Δ-Carathéodory Functions,” Advances in Difference Equations, vol. 2010, Article ID 650827, 20 pages, 2010. View at Publisher · View at Google Scholar
  18. 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, Article ID 38121, 14 pages, 2006. View at Google Scholar · View at Zentralblatt MATH
  19. G. T. Bhaskar, “Comparison theorem for a nonlinear boundary value problem on time scales. Dynamic equations on time scale,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 117–122, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH