On Eventually Positive Solutions of Quasilinear Second-Order Neutral Differential Equations

Abstract

We study the second-order neutral delay differential equation ${[r(t){\Phi }_{\gamma }(z^{\prime }(t))]}^{\prime }+q(t){\Phi }_{\beta }(x(\sigma (t)))=0$, where ${\mathrm{\Phi }}_{\alpha }(t)=|t{|}^{\alpha -\mathrm{1}}t$, $\alpha \ge \mathrm{1}$ and $z(t)=x(t)+p(t)x(\tau (t))$. Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ${\int \infty r}^{-\mathrm{1}/\gamma }(t)\text{d}t$. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.