Spectral Analysis of Discontinuous
Boundary-Value Problems with Retarded
Argument

Erdoğan Şen

Namik Kemal University, Department of Mathematics, Faculty of Arts and Science,
Tekirdağ, 59030, Turkey
E-mail: erdogan.math@gmail.com

Received July 14, 2016, revised June 6, 2017.

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.

Mathematics Subject Classification 2010: 34L15, 34L20, 35R10 Key words: differential equation with retarded argument, eigenparameter, transmission conditions, asymptotics of eigenvalues, bounds for eigenvalues.