Sturm–Liouville Problems on Time Scales with Separated Boundary Conditions

Abstract

We consider the general regular Sturm–Liouville problems on time scales with separated boundary conditions. By extending the result by Agarwal, Bohner, and Wong [2] on the existence of eigenvalues, we study the dependence of the eigenvalues on the boundary condition. We show that the n-th eigenvalue λ n depends continuously on the boundary condition except at the generalized “Dirichlet” conditions, where certain jump-discontinuities may occur. Furthermore, λ n as a function of the boundary condition angles is continuously differentiable wherever it is continuous. Formulas for such derivatives are obtained which reveal the monotone properties of λ n in terms of the boundary condition angles. Our results not only unify the Sturm–Liouville problems for differential equations and difference equations, but are also new for the discrete case.