Spectral Analysis of Discontinuous Boundary-Value Problems with Retarded Argument

Abstract

In the paper, we are concerned with spectral properties of discontinuous Sturm-Liouville type problems with retarded argument. We extend and generalize some approaches and results of the classical regular and discontinuous Sturm-Liouville problems. First, we study the spectral properties of a Sturm-Liouville problem on the half-axis and obtain lower bounds for the eigenvalues of this problem. Then we study spectral properties of a Sturm-Liouville problem with discontinuous weight function which contains a spectral parameter in the boundary conditions. We also obtain asymptotic formulas for eigenvalues and eigenfunctions of this problem and bounds for the distance between eigenvalues.