Related oscillation criteria for higher order self-adjoint differential and integro-differential equations

Abstract

Within recent years considerable attention has been devoted to extensions of the classical Sturmian theory of real linear homogeneous differential equations of the second order. In particular, such extensions have included not only self-adjoint systems of differential equations, but also higher order self-adjoint differential and integro-differential equations. For problems in these latter categories, however, only limited attention has been given to detailed application of the general oscillation and comparison criteria. The present paper is devoted to this area, and, in particular, it is shown how existing criteria may be exploited to obtain comparison theorems between equations of different orders. Although the presented results have ready extensions to vector differential and integro-differential equations of higher order, [see, for example, 5,6,7], for simplicity attention is restricted to scalar equations. Section 2 is devoted to the statement of known general criteria of oscillation for self-adjoint equations of higher order, with special applications of these criteria presented in Section 3. Finally, Section 4 sketches the framework of corresponding applications for self-adjoint higher order integro-differential equations.