Oscillation of third-order nonlinear delay dynamic equation with damping term on time scales

Abstract

In this article, we consider the oscillation of a class of third-order nonlinear damped delay dynamic equation on time scales of the form $$\begin{aligned} \left( r_2(r_1(y^\Delta )^\alpha )^\Delta \right) ^\Delta (t)+p(t)(y^\Delta )^\alpha (\sigma (t))+q(t)f(y(g(t)))=0. \end{aligned}$$ We offer a new description of oscillation of the third-order equation in terms of the nonoscillation of a related well studied second-order dynamic equation $$\begin{aligned} \left( r_2z^\Delta \right) ^\Delta (t)+\frac{p(t)}{r_1(\sigma (t))}z(\sigma (t))=0. \end{aligned}$$ Using generalized Riccati transformation and integral averaging technique, some new sufficient conditions which insure that any solution of the equation oscillates are established.