Oscillation criteria for a class of even-order neutral delay differential equations

Abstract

In this work, we study the oscillatory behavior of the nth order neutral equation

$$\begin{aligned} \left( a\left( t\right) \vartheta ^{\left( n-1\right) }\left( t\right) \right) ^{\prime }+\sum _{i=1}^{k}q_{i}\left( t\right) \phi \left( u\left( g_{i}\left( t\right) \right) \right) =0,\quad t\ge t_{0}, \end{aligned}$$

where n, k are positive integers, n is even, $n\ge 2,$p is the p-Laplace operator (constant), $p>1$ and

$$\begin{aligned}\vartheta \left( t\right) :=\left| u\left( t\right) \right| ^{p-2}u\left( t\right) +h\left( t\right) u\left( \tau \left( t\right) \right) . \end{aligned}$$

New oscillation criteria are obtained by employing a refinement of the Riccati transformations, comparison principles and integral averaging technique. This new theorem complements and improves a number of results reported in the literature. One example is provided to illustrate the main results.