Monotone and oscillatory solution of $y^{(n)}+py=0$

Abstract:

Monotone and oscillatory behaviors of the solutions with the property that $y(x)/{x^2} \to 0$ as $x \to \infty$ or $y(x)/x \to 0$ as $x \to \infty$ are discussed. For example, it is shown that every nonoscillatory solution y, such that $y(x)/x \to 0$ as $x \to \infty$, monotonically tends to zero as $x \to \infty$, provided n is odd, $p \geqq 0$, and ${\smallint ^\infty }{x^{n - 1}}p(x)dx = \infty$.