On eigenfunctions of Sturm-Liouville problem for a functional differential equation (Q1897955)

The Sturm-Liouville problem for a functional differential equation \[ - (pu')'+ qu- \int^1_0 (u(y)- u(x)) d_y r(x, y)= \lambda u,\quad x\in [0, l],\tag{1} \] \[ \alpha_0(pu')_{x= 0}- \beta_0 u(0)= \alpha_1(pu')_{x= 1}+ \beta_1 u(l)= 0\tag{2} \] is considered. The study of the Sturm-Liouville problems relies on the Hilbert-Schmidt theorem and therefore it is natural to take the Hilbert space \(L_2(0, l)\) as a ``working'' space. But there are no reasons to consider functions in \(L_2\), because a number of its properties (differentiability, satisfaction of boundary value conditions, and so on) define certain, nonclosed in \(L_2\) subspace, and considerations in this subspace yield undesirable technical difficulties. That is why the author uses a substitution \(u= Gz\), where \(G\) is a linear isomorphism between two spaces [see \textit{S. G. Krejn}, Linear equations in Banach space, Nauka, Moscow (1971; Zbl 0233.47001)]. This isomorphism enables to pass from (1)--(2) to an equation in \(L_2\) and to apply the Hilbert- Schmidt theorem immediately.