Discontinuous boundary value problems with retarded argument and eigenparameter-dependent boundary conditions (Q1943579)

The differential equation with retarded argument \[ y''(x)+\lambda\,y(x)+q(x)\,y(x-\Delta(x))=0,\quad x\in\left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right] \] is considered with eigenparameter-dependent boundary conditions \[ \lambda\,y(0)+y'(0)=0, \] \[ \lambda^ 2\,y(\pi)+y'(\pi)=0, \] and transmission conditions \[ y\left(\frac{\pi}{2}-0\right)-\delta\,y\left(\frac{\pi}{2}+0\right)=0, \] \[ y'\left(\frac{\pi}{2}-0\right)-\delta\,y'\left(\frac{\pi}{2}+0\right)=0. \] It assumed that \(\lambda\) is a real eigenparameter, \(q\) and \(\Delta\) are real-valued functions continuous in \([0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi]\) whose sided-limits at the point \(\frac{\pi}{2}\) exist and are finite. The authors study the properties of the eigenvalues and corresponding eigenfunctions of the boundary value problem aiming at its asymptotic representation. Asymptotic formulas, which take into account retardation, are proved under conditions of existence and boundedness of the derivatives of the functions \(q\) and \(\Delta\).