Oscillation of Half-Linear Differential Equations with Delay

Abstract

We study the half-linear delay differential equation $(r(t)\mathrm{\Phi }(x\mathrm{\text{'}}(t)))\mathrm{\text{'}}+c(t)\mathrm{\Phi }(x(\tau (t)))=0$, $\mathrm{\Phi }(x):=|x{|}^{p-2}x$, $p>1.$ We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.