Oscillation of Second-Order Neutral Functional Differential Equations with MixedNonlinearities

Abstract

We study the following second-order neutral functional differential equation with mixed nonlinearities ${\left(r\right(t\left)\right|\left(u\right(t)+p(t\left)u\right(t-\sigma \left)\right)\text{'}{|}^{\alpha -1}{\left(u\right(t)+p(t\left)u\right(t-\sigma \left)\right)}^{\prime })}^{\prime }+q_0\left(t\right)|u\left({\tau }_0\right(t\left)\right){|}^{\alpha -1}u\left({\tau }_0\right(t\left)\right)+q_1\left(t\right)|u\left({\tau }_1\right(t\left)\right){|}^{\beta -1}u\left({\tau }_1\right(t\left)\right)+q_2\left(t\right)|u\left({\tau }_2\right(t\left)\right){|}^{\gamma -1}u\left({\tau }_2\right(t\left)\right)=0$, where , ${{\displaystyle \int }}_{t_0}^{\infty }(1/r^{1/\alpha }(t\left)\right)\text{d}t<\infty$. Oscillation results for the equation are established which improve the results obtained by Sun and Meng (2006), Xu and Meng (2006), Sun and Meng (2009), and Han et al. (2010).