Oscillation Theory on a Non-Compact Interval

Abstract

We shall consider in this chapter the behavior of non-identically vanishing real solutions of a linear second order differential equation of the form (II.1.1), (II.1.1⁰) or (II.1.1^#) on a non-compact interval which for the major portion of the discussion will be taken to be of the form I = [a,∞), where a is a finite value. Such an equation is said to be oscillatory in case one non-identically vanishing real solution, and hence all such solutions, have infinitely many zeros on I; clearly an equivalent statement is that the equation is not disconjugate on any non-degenerate subinterval I₀ = [a₀,∞) of I. It is to be remarked that alternate terminologies for this concept are “oscillatory in a neighborhood of ∞”, or “oscillatory for large t”. If an equation fails to be oscillatory it is said to be “non-oscillatory”, with the corresponding qualifications “in a neighborhood of ∞” or “for large t”.