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In this work, oscillatory behaviour of solutions of a class of fourth-order neutral functional difference equations of the form $${\Delta }^2\left(r\left(n\right){\Delta }^2(y\left(n\right)+p\left(n\right)y(n-m))\right)+q\left(n\right)G\left(y(n-k)\right)=0$$ is studied under the assumption $$\sum _{n=0}^{\infty }\frac{n}{r\left(n\right)}<\infty .$$ New oscillation criteria have been established which generalize some of the existing results in the literature.

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $${\left(r\left(t\right){\left(z^{\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+\sum _{i=1}^m{q}_i\left(t\right)x^{{\alpha }_i}\left({\sigma }_i\left(t\right)\right)=0,t\ge t_0,$$ where $z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)$. Under the assumption ${\int }^{\infty }{\left(r\left(\eta \right)\right)}^{-1/\gamma }\mathrm{d}\eta =\infty$, we consider two cases when $\gamma >{\alpha }_i$ and $\gamma <{\alpha }_i$. Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.