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

Abstract

The boundary value problem $$-y^{″}+q(x)y=\lambda y+ın{t}_0^{\pi }yd\sigma (x),y(0)=y(\pi )=0,$$ is concerned, where $qınC[0,\pi ]$ 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.