Let $\mathcal{A}$ be a subalgebra of $\mathcal{B}(\mathcal{X})$ containing the identity operator $I$ and an idempotent $P$. Suppose that $\alpha,\beta:\mathcal{A}\rightarrow \mathcal{A}$ are ring epimorphisms and there exists some nest $\mathcal{N}$ on $\mathcal{X}$ such that $\alpha(P)(\mathcal{X})$ and $\beta(P)(\mathcal{X})$ are non-trivial elements of $\mathcal{N}$. Let $\mathcal{A}$ contain all rank one operators in $Alg\mathcal{N}$ and $\delta: \mathcal{A}\rightarrow \mathcal{B}(\mathcal{X})$ be an additive mapping. It is shown that, if $\delta$ is $(\alpha,\beta)$-derivable at zero point, then there exists an additive $(\alpha,\beta)$-derivation $\tau:\mathcal{A}\rightarrow \mathcal{B}(\mathcal{X})$ such that $\delta(A)=\tau(A)+\alpha(A)\delta(I)$ for all $A\in \mathcal{A}$. It is also shown that if $\delta$ is generalized $(\alpha,\beta)$-derivable at zero point, then $\delta$ is an additive generalized $(\alpha,\beta)$-derivation. Moreover, by use of this result, the additive maps (generalized) $(\alpha,\beta)$-derivable at zero point on several nest algebras, are also characterized.
