Quadratic Functionals of nth Order

Abstract

It is well known that disconjugacy theorems for self-adjoint differential equations are very closely related to the study of the positivity of certain associated quadratic functionals. In this paper, this relationship is closely examined for quadratic functionals of nth order associated with selfadjoint differential equations of order 2n. The motivation of this work comes from a recent paper of W. Leighton [8] on quadratic functionals of second order. It is shown that the method used by W. Leighton extends in a straightforward manner to the study of quadratic functionals of nth order, provided one makes use of an identity due to G. Cimmino [1]. Several authors have considered related problems. In particular we wish to mention the recent papers of G. Ladas [6], W. Simons [13], D. B. Hinton [5]. The book by C. A. Swanson [14] also contains many other related results. As a consequence of our results we can obtain several known and some new criteria for oscillation of 2nth order self-adjoint differential equations.