Oscillation Properties of Higher-Order Sublinear Differential Equations

Abstract

For higher-order sublinear nonautonomous ordinary differential equations, necessary and sufficient conditions for the existence of properties A and B are obtained. In particular, we prove that if n is even, a > 0, and the function p : [a, +∞) → (−∞, 0] is Lebesgue integrable on each finite interval, then, for the oscillation property of all proper solutions of the differential equation u⁽ⁿ) = p(t) ln(1+|u|) sgn(u), it is necessary and sufficient that \(\int_{a}^{+\infty}p(t)\rm{ln} \it{t} dt=-\infty\).