\begin{abstract} We consider two finitely generated graded modules over a homogeneous Noetherian ring $R=\oplus_{n\in{\mathbb{N}}_0}R_n$ with local base ring $(R_0,{\fm}_0)$ and irrelevant ideal $R_{+}$ of $R$ and we study the generalized local cohomology modules $H_{\fb}^i(M,N)$ with respect to the ideal $\fb={\fb}_0+{R}_+$, where ${\fb}_0$ is an ideal of $R_0$. We prove that if $\Dim R_0/{{\fb}_0}\leq 1$, then the following cases hold: \begin{itemize} \item[(i)] for all $i\geq 0$ the $R$-module $H_{\fb}^i(M,N)/{{\fa}_0H_{\fb}^i(M,N)}$ is Artinian , where $ \sqrt{{\fa}_0+{\fb}_0}={\fm}_0$; \item[(ii)] for all $i\geq 0$ the set $\Ass_{R_0}(H_{\fb}^i(M,N)_n)$ is asymptotically stable , where $n\longrightarrow{-\infty}$. \end{itemize} Moreover, if $H_{\fb}^j(M,N)_n$ is a finitely generated $R_0$-module for all $n\leqslant n_0$ and all $j
