Oscillation Criteria for Second-Order Superlinear Neutral Differential Equations

Abstract

Some oscillation criteria are established for the second-order superlinear neutral differential equations $\left(r\right(t\left)\right|z\text{'}\left(t\right){|}^{\alpha -1}z\text{'}\left(t\right))\text{'}+f(t,x(\sigma \left(t\right)\left)\right)=0$, $t\ge t_0$, where $z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)$, $\tau \left(t\right)\ge t$, $\sigma \left(t\right)\ge t$, $p\in C\left(\right[t_0,\infty ),[0,p_0])$, and $\alpha \ge 1$. Our results are based on the cases ${{\displaystyle \int }}_{t_0}^{\infty }1/r^{1/\alpha }\left(t\right)\text{d}t=\infty$ or ${{\displaystyle \int }}_{t_0}^{\infty }1/r^{1/\alpha }\left(t\right)\text{d}t<\infty$. Two examples are also provided to illustrate these results.