Consider the second order superlinear dynamic equation \begin{equation*} (*)\qquad x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$} \end{equation*} where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale, $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for $x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly known as the Emden--Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for arbitrarily large values of $t$. In addition to extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we obtain analogues in the difference equation and $q$-difference equation cases.

We establish a flexible oscillation criterion based on an averaging technique that improves upon a result due to C.~G. Philos.

A new oscillation criterion is given for the delay differential equation $x'(t)+p(t)x \left(t-\tau(t)\right)=0$, where $p$, $\tau \in \C \left([0,\infty),[0,\infty)\right)$ and the function $T$ defined by $T(t)=t-\tau(t)$, $t\ge 0$ is increasing and such that $\lim_{t\to\infty}T(t)=\infty$. This criterion concerns the case where $\liminf_{t\to\infty} \int_{T(t)}^{t}p(s)\,ds\le \frac{1}{e}$.