On the spectral properties and positivity of solutions of a periodic boundary value problem for a second-order functional differential equation

For a functional-differential operator
\begin{equation*} \mathcal{L} u = (1/\rho)\left(-(pu')'+\int_0^l u(s)d_s r(x,s)\right) \end{equation*}
with symmetry, the completeness and orthogonality of the eigenfunctions is shown. The positivity conditions of the Green function of the periodic boundary value problem are obtained.