Dependence of eigenvalues of a class of fourth-order Sturm–Liouville problems on the boundary

Abstract

In this paper, we consider the dependence of eigenvalues of a class of fourth-order Sturm–Liouville problems on the boundary. We show that the eigenvalues depend not only continuously but smoothly on boundary points, and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher fourth-order Dirichlet eigenvalues march off to plus infinity, this is also true for the first (i.e., lowest) eigenvalue.