Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with -Carath\primoslcurlybercurlybodory Functions

We establish the existence of solutions to systems of second-orderdynamic equations on time scales with the right member , a Δ-Carathéodoryfunction. First, we consider the case where the nonlinearity does not dependon the Δ-derivative, Δ(). We obtain existence results for Strum-Liouville andfor periodic boundary conditions. Finally, we consider more general systemsin which the nonlinearity depends on the Δ-derivative and satisfies a lineargrowth condition with respect to Δ(). Our existence results rely on notionsof solution-tube that are introduced in this paper.