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...

We study asymptotic properties of solutions of the system of differential equations of neutral type.

This paper establishes existence of nonoscillatory solutions with specific asymptotic behaviors of second order quasilinear functional differential equations of neutral type. Then sufficient, sufficient and necessary conditions are proved under which every solution of the equation is either oscillatory or tends to zero as $t\to \infty$.