On (m, n)-derivations of Some Algebras

Let $\mathcal{A}$ be a unital algebra, $\delta$ be a linear mapping from $\mathcal{A}$ into itself and $m$, $n$ be fixed integers. We call $\delta$ an ($\mathit{\text{m,}}\mathit{\text{n}}$)-derivable mapping at $Z$, if $m\delta (AB)+n\delta (BA)=m\delta (A)B+mA\delta (B)+n\delta (B)A+nB\delta (A)$ for all $A,B\in \mathcal{A}$ with $AB=Z$. In this paper, ($\mathit{\text{m,}}\mathit{\text{n}}$)-derivable mappings at 0 (resp. $I_{\mathcal{A}}\oplus 0$, $I$) on generalized matrix algebras are characterized. We also study ($\mathit{\text{m,}}\mathit{\text{n}}$)-derivable mappings at 0 on CSL algebras. We reveal the relationship between this kind of mappings with Lie derivations, Jordan derivations and derivations.