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

Abstract

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