Oscillation of two-term differential equations through asymptotics
Oscillation theory is developed for differential equations of type (1) \(y^{(n)} = p(x)y\) for a positive integer \(n \geq 2\) and a smooth real or complex-valued function \(p\) on an interval \([a, \infty)\) such that \(x^ n p(x)\) grows sufficiently fast (in a technical way) as \(x \to \infty\). The asymptotic behavior of solutions of (1) is obtained and applied to generate oscillation theorems. A typical result, for suitable growth conditions on \(x^ n p(x)\), is that (1) has Property A if \(p(x) > 0\) and Property B if \(p(x) < 0\). The definitions of Properties A and B are given in the book by \textit{I. T. Kiguradze} and \textit{T. A. Chanturia} [Asymptotic properties of solutions of nonautonomous ordinary differential equations, Mathematics and its Applications (Soviet Series) 89, Dordrecht (1993; Zbl 0782.34002)]. In particular, the stated result implies that all nontrivial solutions of (1) are oscillatory if \(p > 0\) and \(n\) is even. Previous asymptotic analysis has been summarized by \textit{M. S. P. Eastham} [The asymptotic solution of linear differential systems, Oxford (1989; Zbl 0674.34045)].
scientific article; zbMATH DE number 706455