On Delay-Independent Criteria for Oscillation of Higher-Order Functional Differential Equations

Abstract

We investigate the oscillation of the following higher-order functional differential equation: $x^{\left(n\right)}\left(t\right)+q\left(t\right)|x(t-\tau ){|}^{\lambda -1}x(t-\tau )=e\left(t\right),$ where $q\left(t\right)$ and $e\left(t\right)$ are continuous functions on $[t_0,\mathrm{\infty })$, $1>\lambda >0$ and $\tau \ne 0$ are constants. Unlike most of delay-dependent oscillation results in the literature, two delay-independent oscillation criteria for the equation are established in both the case $\tau >0$ and the case $\tau <0$ under the assumption that the potentials $q\left(t\right)$ and $e\left(t\right)$ change signs on $[t_0,\mathrm{\infty })$.