Some finiteness properties of generalized graded local cohomology modules

Let $R = \bigoplus_{n \in \mathbb{N}_0} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak{m}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in\mathbb{N}_0$. In this paper we will study Artinianess of $\Gamma_{\mathfrak m_0R}(H_{R_+}^i(M,N)), H_{\mathfrak m_0R}^1(H_{R_+}^i(M,N)), H_{R_+}^i(M,N)/{\mathfrak m_0}H_{R_+}^i(M,N), H_{R_+}^j(M,H_{\mathfrak m_0R}^i(N)), H_{\mathfrak m_0R}^j(M,H_{R_+}^i(N))$, where $R_+$ denotes the irrelevant ideal of $R$.