LAUSR

We establish some oscillation criteria for the third-order Emden–Fowler neutral delay dynamic equations of the form: $$\begin{aligned} (a(t)(x(t)+r(t)x(\tau (t)))^{\Delta \Delta })^\Delta +p(t)x^\gamma (\delta (t))=0 \end{aligned}$$(a(t)(x(t)+r(t)x(τ(t)))ΔΔ)Δ+p(t)xγ(δ(t))=0on a time scale $$\mathbb {T}$$T, where $$\gamma >0$$γ>0 is a quotient of odd positive integers, and a and p are real-valued positive rd-continuous functions defined on $$\mathbb {T}$$T. Due to the different values of $$\gamma $$γ, we give not only the oscillation criteria for superlinear neutral delay dynamic equations, but also the oscillation criteria for sublinear neutral delay dynamic equations based on the Hille and Nehari-type oscillation criteria. Our results extend and improve some known results in the literature and are new even for the corresponding third-order differential equations and difference equations as our special cases.