A Sturm-Liouville Type Problem with Retarded Argument Which Contains a Spectral Parameter in the Boundary Condition

Abstract

The aim of this study is to investigate a discontinuous problem of the Sturm-Liouville type with a retarded argument containing a spectral parameter in the boundary condition and two additional conditions of transmission at the two discontinuity points. Also, eigenparameter dependent boundary conditions and obtains asymptotic formulas for the eigenvalues and eigenfunctions. And the spectral parameter is real. And the real valued function is continuous on that interval and it has a finite limit. The goal of this article is to obtain asymptotic formulas for eigenvalues of eigenfunctions for problem of the form : with boundary conditions and transmission conditions To this aim, first, the principal term of asymptotic distribution of eigenvalues and eigenfunctions of given problem was obtained up to but, afterwards under some additional conditions, we improve these formulas up to Thus, when the number of points of discontinuity is more than one, we see how the asymptotic behaviour of eigenvalues and eigenfunctions of a boundary value problem with retarded argument which contains a spectral parameter in the boundary conditions change.