Boundary value problems for systems of second-order dynamic equations on time scales with \(\Delta \)-Carathéodory functions (Q624471)

Summary: We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right hand side \(f\), a \(\Delta \)-Carathéodory function. First, we consider the case where the nonlinearity \(f\) does not depend on the \(\Delta \)-derivative, (\(\chi^{\Delta}(t)\)). 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 \(\Delta \)-derivative and satisfies a linear growth condition with respect to \(\chi^{\Delta}(t)\). Our existence results rely on notions of solution-tube that are introduced in this paper.