Oscillation results for third order nonlinear neutral differential equations of mixed type

Abstract

Some oscillation results are obtained for the third order nonlinear mixed type neutral differential equations of the form$$\left(\left(x(t)+b(t) x\left(t-\tau_1\right)+c(t) x\left(t+\tau_2\right)\right)^\alpha\right)^{\prime \prime \prime}=q(t) x^\beta\left(t-\sigma_1\right)+p(t) x^\gamma\left(t+\sigma_2\right), t \geq t_0$$where $\alpha, \beta$ and $\gamma$ are ratios of odd positive integers $\tau_1, \tau_2, \sigma_1$ and $\sigma_2$ are positive constants.