Oscillation theorems for second order neutral differential equations

In this paper new oscillation criteria for the second order neutral differential equations of the form \begin{equation*} \left(r(t)\left[x(t)+p(t)x(\tau(t))\right]'\right)'+q(t)x(\sigma(t))+v(t)x(\eta(t))=0 \tag{$E$}\end{equation*} are presented. Gained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. Obtained comparison principles essentially simplify the examination of the studied equations. We cover all possible cases when arguments are delayed, advanced or mixed

Oscillation Criteria for Certain Even Order Neutral Delay Differential Equations with Mixed Nonlinearities

We establish some oscillation criteria for the following certain even order neutral delay differential equations with mixed nonlinearities: rtzn-1tα-1zn-1t'+q0(t)(xτ0tα-1x(τ0(t))+q1t(x(τ1(t))β-1x(τ1(t))+q2t(x(τ2(t))γ-1x(τ2(t))=0, t≥t0, where z(t)=x(t)+p(t)x(σ(t)),n is even integer, and γ>α>β>0. Our results generalize and improve some known results for oscillation of certain even order neutral delay differential equations with mixed nonlinearities