\begin{abstract} Let $R$ be a commutative Noetherian ring and let $\fa$, $\fb$ be two ideals of $R$ such that $R/({\fa+\fb})$ is Artinian. Let $M$, $N$ be two finitely generated $R$-modules. We prove that $H_{\fb}^j(H_{\fa}^t(M,N))$ is Artinian for $j=0,1$, where $t=\inf\{i\in{\mathbb{N}_0}: H_{\fa}^i(M,N)$ is not finitely generated $\}$. Also, we prove that if $\Dim\Supp(H_{\fa}^i(M,N))\leq 2$, then $H_{\fb}^1(H_{\fa}^i(M,N))$ is Artinian for all $i$. Moreover, we show that if $\dim N=d$, then $H_{\fb}^j(H_{\fa}^{d-1}(N))$ is Artinian for all $j\geq 1$. \end{abstract}
