Effective criteria for the strong sign-regularity and the oscillation property of the Green's functions of two-point boundary-value problems

Necessary and sufficient conditions for strong sign-regularity and the oscillation property (in the sense of Gantmakher and Krein) of the Green's function of a two-point boundary eigenvalue problem are obtained. These conditions guarantee that even in a non-self-adjoint case the eigenvalues are real and have several other spectral properties similar to those of the classical Sturm–Liouville problem. The conditions are formulated in terms of the properties of a uniquely defined fundamental system of solutions of the differential equation. This makes it possible to verify them effectively using a computer and to establish, as the final result, the oscillation property of the Green's function and the corresponding spectral properties of the boundary-value problem in a large number of cases in which these properties could not be detected on the basis of previously known sufficient conditions.