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

Abstract

We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member $f$, a $\Delta$-Carathéodory function. First, we consider the case where the nonlinearity $f$ does not depend on the $\Delta$-derivative, $x^{\Delta }$($t$). 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 $x^{\Delta }$($t$). Our existence results rely on notions of solution-tube that are introduced in this paper.