Communications of the
Korean Mathematical Society CKMS

By using the Riccati transformation technique, we study the oscillation and asymptotic behavior for the third-order nonlinear delay dynamic equations $$\left(c(t)\left(p(t)x^\Delta(t)\right)^\Delta\right)^\Delta+q(t)f(x(\tau(t)))=0$$ on a time scale $\mathbb{T}$, where $c(t)$, $p(t)$ and $q(t)$ are real-valued positive rd-continuous functions defined on $\mathbb{T}$. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our oscillation results are essentially new. Some examples are considered to illustrate the main results.

Keywords: oscillation behavior, third order delay dynamic equations, time scales

MSC numbers: 39A21, 34C10, 34K11, 34N05