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In the paper we consider the difference equation of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,n\in \mathbb{N}\left(n_0\right),$$ where $p,q:\mathbb{N}\left(n_0\right)\to {ℝ}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma$ is strictly increasing and $\underset{n\to \infty }{lim}\sigma \left(n\right)=\infty ;$ $\tau$ is nondecreasing and $\underset{n\to \infty }{lim}\tau \left(n\right)=\infty$, $f:ℝ\to ℝ$, $xf\left(x\right)>0$. We examine the following two cases: $$0<p\left(n\right)\le {\lambda }^{*}<1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l,$$ and $$1<{\lambda }_{*}\le p\left(n\right),\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l,$$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \infty$ with a weaker assumption on $q$ than the usual assumption $\sum _{i=n_0}^{\infty }q\left(i\right)=\infty$ that is used in literature.

In this note we consider the third order linear difference equations of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+\delta q\left(n\right)x\left(\tau \left(n\right)\right)=0,n\in N\left(n_0\right),\left(\mathrm{E}\right)$$ where $\delta =\pm 1$, $p,qN\left(n_0\right)\to {ℝ}_{+};$ $\sigma ,\tau N\left(n_0\right)\to \mathbb{N}$, ${lim}_{n\to \infty }\sigma \left(n\right)=\underset{n\to \infty }{lim}\tau \left(n\right)=\infty .$ We examine the following two cases: $$\begin{array}{ccc}\hfill \{0<p(n)& \le 1,\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l\},\{p\left(n\right)\hfill & \hfill >1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l\},\end{array}$$ where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.