We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member , a -Carathéodory function. First, we consider the case where the nonlinearity does not depend on the -derivative, (). We obtain existence results for Strum-Liouville and for periodic boundary conditions. Finally, we consider more general systems in which the nonlinearity depends on the -derivative and satisfies a linear growth condition with respect to (). Our existence results rely on notions of solution-tube that are introduced in this paper.
M. Frigon. H. Gilbert. "Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with -Carathéodory Functions." Abstr. Appl. Anal. 2010 1 - 26, 2010. https://doi.org/10.1155/2010/234015