Consider the nonlinear difference equation \[ x_{n+1}-x_{n}+a_{n}x_{n-k}=b_{n}f(x_{n-l})~\;\;\;,~\;\;n\geq 0,\tag{\(*\)} \] where \(l,~k\) are nonnegative integers, \(\{a_{n}\}\) \ and \ \(\{b_{n}\}\) \(\;\)\ are two sequences of real numbers, and \(a_{n}\geq 0\) for \ \(n=0,1,2,\dots,f\in C((-\infty ,\infty )~,~(-\infty ,\infty )).\) The author established the following sufficient conditions for the asymptotic stability of solutions of (\(*\)). Theorem. Assume that there exists a constant \(c\in [0,1)\) for large \(n\) such that \(\left| b_{n}f(u)\right| \leq ca_{n}\left| u\right| ,~\sum_{s=0}^{\infty }a_{s}=\infty,\) and \[ \underset{n\rightarrow \infty }{\lim \sup }\sum_{s=n-k}^{n}a_{s}<\begin{cases} \frac{1-c}{1+c}+\frac{k+2}{2(k+1)}&\text{if }0\leq c\leq 1/3,\\ \sqrt{\frac{2(1-c)(k+1)}{(1+c)k}}&\text{if }1/3\leq c\leq 1,\end{cases} \] then every solution of Eq. (\(*\)) tends to zero as \(n\rightarrow \infty.\)