Cofiniteness and Artinianness of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring and $Isubseteq J$ be ideals of $R$. ##Let $M$ and $N$ be finitely generated $R$-modules such that ${rm pd}_{R}(M) infty$. ##The notion $tilde{{rm q}}_{J}(M,N)$ is the greatest integer $i$ such that $H^{i}_{J}(M,N)$ is not Artinian and $J$-cofinite. ##In this paper, we give a bound for $tilde{{rm q}}_{J}(M,N)$ by using $tilde{{rm q}}_{I}(M,N)$. We show that $tilde{{rm q}}_{J}(M,N)leq tilde{{rm q}}_{I}(M,N)+{rm cd}_{J}(M,N/IN)$.