On the expansion theorem for a certain boundary value problem for a functional differential equation

Abstract

The boundary value problem $$-y''+q(x)y=\lambda y+ınt_0^{\pi}y\,d\sigma(x),\quad y(0)=y(\pi)=0,$$ is concerned, where $qın C[0,\pi]$ and $\sigma$ is a function of bounded variation. It is proved that the system of eigenfunctions of the given problem is complete and minimal in $L^2(0,\pi)$, and also that functions of a certain class can be expanded into uniformly convergent series with respect to the mentioned system.