A nonlocal Sturm–Liouville eigenvalue problem

A nonlocal eigenvalue problem of the form u ″ + a ( x ) u + Bu = λu with homogeneous Dirichlet boundary conditions is considered, where B is a rank-one bounded linear operator and x belongs to some bounded interval on the real line. The behaviour of the eigenvalues is studied using methods of linear perturbation theory. In particular, some results are given which ensure that the spectrum remains real. A Sturm-type comparison result is obtained. Finally, these results are applied to the study of some nonlocal reaction–diffusion equations.