Lower bounds of certain general local cohomology modules

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$, $\fa \in \Phi$, $M$ an arbitrary $R$-module and $t$ a non-negative integer. Let $\mathcal{S}$ be a Melkersson subcategory of $R$-modules. Among other things, we prove that if $\lc^{i}_\Phi(M)$ is in $\mathcal{S}$ for all $i < t$ then $\lc^{i}_\fa(M)$ is in $\mathcal{S}$ for all $i < t$ and for all $\fa \in \Phi$. If $\mathcal{S}$ is the class of $R$-modules $N$ with $\dim N \leq k$ where $k \geq -1$, is an integer, then $\lc^{i}_\Phi(M)$ is in $\mathcal{S}$ for all $i < t$ (if and only if) $\lc^{i}_\fa(M)$ is in $\mathcal{S}$ for all $i < t$ and for all $\fa \in \Phi$. As consequences we study and compare vanishing, Artinianness and support of general local cohomology and ordinary local cohomology supported at ideals of its system of ideals at initial points $i <t$. We show that $\Supp_{R}(\lc^{\dim M-1}_{\Phi}(M))$ is not necessarily finite whenever $(R,\fm)$ is local and $M$ a finitely generated $R$-module.