On the Sturm-Liouville problem for a linear singular functional-differential equation (Q1390435)

The author deals with the Sturm-Liouville problem for the linear integro-differential equation with singularity \[ -(pu')'+ qu- \int^\ell_0 (u(y)- u(x)) d_yr(x,y)= \lambda\rho u\quad\text{on} \quad [0,\ell], \] under the boundary conditions \(\alpha_0(pu')_{x= 0}- \beta_0u(0)= 0\), \(\alpha_1(pu')_{x= \ell}+ \beta_1u(\ell)= 0\). Using the variational method and a suitable solution space, the author adapts to this case some classical results. Thus, one proves the existence of a sequence of positive eigenvalues and of an orthonormal basis of eigenfunctions. A maximum principle is proved.