Oscillation theory of ordinary linear differential equations

Abstract

The purpose of this article [a compilation of lectures originally presented at the Associated Western Universities Differential Equations Symposium, Baulder, Colorado in the summer of 19671 is to give motivating examples and ideas which ‘have influenced the author in his studies of oscillation properties of solutions of linear ordinary differential equations. This is a subject where the mathematical tools needed are relatively elementary but where it is easy to state an unsolved problem. For example, the familiar second-order equation y” + Q(X)JJ = 0 is still a valid subject for research, although it has a voluminous literature. As far as oscillation theory is concerned, most texts in Differential Equations, both elementary and advanced, deal only with second-order equations. A few deal with self-adjoint fourth-order equations and, perhaps, those of arbitrary even order and systems of first or second-order equations. Any discussion of oscillatory properties of third-order equations or other nonself-adjoint equations is hard to find and that is the lowest order where truly nonself-adjoint equations occur. In this article an attempt is made to give a self-contained inductive development from equations of one order to the next. Most of the discussion will deal with equations of second, third, and fourth orders, with linear systems of second-order equations and with generalizations of those results to equations of higher orders. No attempt is made to survey all of the oscillation theory of equations of orders higher than four. Considerable attention is devoted to equations of order three (Section II) and this is in line with the increased recent interest in these equations, as the Bibliography will show. Instead of the usual format where proofs follow the respective