Construction of separating functions for the quasi—linear differential equation (py′)′+qy=0

Abstract

By using Sturrn's Comparison theorem on the quasi-differential equation

we generate a sequence of functions which has the property that its members separate the mutually exclusive behaviour of oscillation and non-oscillation of the given equation. It is shown that this ‘separation property’ is an increasing function of the order of members in the sequence generated i.e. F _k+l is a ‘sharper’ and ‘better’ separating function than F _k whenever is a suitably constructed sequence for the differential equation. Using several examples we establish necessary and sufficient conditions for this trend to hold.