Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

Abstract

In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients $\delta =\gamma =\mathrm{1}$ and retarded argument $\mathrm{\Delta }\equiv \mathrm{0}$ in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.