On the number of non-real eigenvalues of the Sturm–Liouville problem

In this paper we consider a spectral problem for the Sturm–Liouville equation with a spectral parameter in a boundary conditions. It is shown that under certain assumptions on the coefficients of boundary conditions, problems of this type cannot have more than two non-real eigenvalues. Note that, in some special cases of boundary conditions, this kind of results have usually been obtained by using the results of the theory of Pontryagin spaces. The aim of this paper is to prove this result in a more general setting. Since the result was fairly predictable and could also be proved by using Pontryagin space methods, the author does not claim the absolute novelty of the obtained result but aims to provide an elementary proof, using only some facts of mathematical analysis and theory of ordinary differential equations, which, probably, will make the proof more accessible to a wide audience, especially to students.