On the oscillation of certain class of third-order nonlinear delay differential equations

Abstract

In this paper we consider the third-order nonlinear delay differential equation (*) $$ ( a(t)\left ( x''(t)\right ) ^{\gamma })' +q(t)x^{\gamma }(\tau (t))=0,\quad t\geq t_0, $$ where $a(t)$, $q(t)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau (t)\leq t$ satisfies $\lim _{t\rightarrow infty }\tau (t)=infty $. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.