Dependence of eigenvalues of Sturm–Liouville problems with eigenparameter dependent boundary conditions

Abstract

This paper is concerned with regular Sturm–Liouville problems with eigenparameter dependent boundary conditions. We obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, in particular, eigenparameter-dependent boundary condition matrix. Moreover the differential expression of the eigenvalues with respect to the data is given. The dependence of the nth eigenvalue as a function of the parameters of the problem is also investigated by Pru¨fer transformation and the inequalities of eigenvalues.