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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.

Abstract

We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member 𝑓, a Δ-Carathéodory function. First, we consider the case where the nonlinearity 𝑓 does not depend on the Δ-derivative, 𝑥Δ(𝑡). We obtain existence results for Strum-Liouville and for periodic boundary conditions. Finally, we consider more general systems in which the nonlinearity 𝑓 depends on the Δ-derivative and satisfies a linear growth condition with respect to 𝑥Δ(𝑡). Our existence results rely on notions of solution-tube that are introduced in this paper.