Traces and inverse nodal problems for a class of delay Sturm–Liouville operators

Abstract

In this study, we investigate the regularized sums of eigenvalues, oscillation of eigenfunctions and solutions of inverse nodal problems of discontinuous Sturm-Liouville operators with a delayed argument and with a finite number of transmission conditions. With this aim, we obtain asymptotic formulas for eigenvalues, eigenfunctions and nodal points of the problem. Moreover, some numerical examples are given to illustrate the results. The problem differs from the other discontinuous Sturm-Liouville problems with retarded argument in that it contains a spectral parameter in boundary conditions. If we take the delayed argument $\Delta\equiv0$, the coefficients $\alpha _{i}^{+}=\beta _{i}^{+}=0$ ($i=1,2$) in boundary conditions and the transmission coefficients $\delta_i=1$ ($i=\overline{1,m-1}$) the results obtained below coincide with corresponding results in the classical Sturm-Liouville operator.