Oscillation for Third-Order Nonlinear Differential Equations with Deviating Argument

Abstract

We study necessary and sufficient conditions for the oscillation of the third-order nonlinear ordinary differential equation with damping term and deviating argument $x^{‴}\left(t\right)+q\left(t\right)x^{\prime }\left(t\right)+r\left(t\right)f\left(x\left(\phi \left(t\right)\right)\right)=0$. Motivated by the work of Kiguradze (1992), the existence and asymptotic properties of nonoscillatory solutions are investigated in case when the differential operator $ℒx=x^{‴}+q\left(t\right)x^{\prime }$ is oscillatory.