New Oscillation Criteria for Second Order Quasilinear Neutral Delay Differential Equations

The authors present some new sufficient conditions for the oscillation of second order quasilinear neutral delay differential equation $$\begin{aligned} (a(t)(z'(t))^{\beta })'+q(t)x^{\gamma }(\sigma (t))=0,\;t\ge t_0>0, \end{aligned}$$ where $$z(t)=x(t)+p(t)x(\tau (t))$$ . Our oscillation results depend on only one condition and essentially improve, complement and simplify many related ones in the literature. Examples are provided to illustrate the value of the main results.