Asymptotic behavior of solutions for third-order half-linear delay dynamic equations on time scales

AbstractBy means of Riccati transformation technique and integral averaging technique, we establish some new sufficient conditions which guarantee that every solution of the third-order half-linear delay dynamic equations $$(r(t)(x^{\Delta\Delta}(t))^\gamma)^\Delta+p(t)x^\gamma(\tau(t))=0$$ oscillates or converges to zero on a time scale $\mathbb{T}$, here γ>0 is a quotient of odd positive integers with r and p real-valued positive rd-continuous functions defined on $\mathbb{T}$. Our results not only extend and improve the results given in literatures but also unify the asymptotic behavior results of the third order nonlinear delay differential equation and the third order nonlinear delay difference equation. Some examples are considered to illustrate the importance of our results.