Spectral analysis of discontinuous boundary-value problems with retarded argument

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.