Higher order separation and comparison theorems,with applications to solution space problems

Summary General separation and comparison theorems for n-th order ordinary linear equations are established, which reduce to the Sturm theorems when the operator is self-adjoint and n is an even integer. The study of solution spaces of third order equations and their adjoints, as begun by Dolan (J.D.E.,7 (1970),pp. 367–388), is completed here, with structure theorems for special kinds of solution spaces, and the presentation of two examples: 1) an equation Lu=0 of order three with all solutions of Lu=0 and L*v=0 oscillatory, 2) an equation Lu=0 of order three such that all two-dimensional subspaces of the solution space contain both oscillatory and non-oscillatory solutions. The general separation theorem is used to obtain example 1), and the structure theorems for solution spaces are used to obtain example 2). Entrata in Redazione il 6 febbraio 1972.