Some Oscillation Criteria for a Class of Third-Order Delay Differential Equations

Abstract

In this paper, we study oscillatory properties of solutions to a class of third-order differential equation ${\left(a\left(t\right){\left({\left(r\left(t\right)x^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\alpha }\right)}^{\prime }+{\left(p\left(t\right){\left(\left(r\left(t\right)x^{\prime }\left(t\right)\right)\right)}^{\prime }\right)}^{\alpha }+q\left(t\right)f\left(x\left(\sigma \left(t\right)\right)\right)=0,$ where $f\left(x\right)/x^{\beta }\ge k>0$ and $\alpha$ and $\beta$ are quotients of odd positive integers. By using the generalized Riccati technique, we obtain some oscillation and asymptotic criteria when $\alpha \ge \beta$ and $\alpha <\beta$ . Finally, some examples are given to show the effectiveness of the criteria obtained.