generalized sigma-derivation on banach algebras

let $mathcal{a}$ be a banach algebra and $mathcal{m}$ be a banach $mathcal{a}$-bimodule. we say that a linear mapping $delta:mathcal{a} rightarrow mathcal{m}$ is a generalized $sigma$-derivation whenever there exists a $sigma$-derivation $d:mathcal{a} rightarrow mathcal{m}$ such that $delta(ab) = delta(a)sigma(b) + sigma(a)d(b)$, for all $a,b in mathcal{a}$. giving some facts concerning generalized $sigma$-derivations, we prove that if $mathcal{a}$ is unital and if $delta:mathcal{a} rightarrow mathcal{a}$ is a generalized $sigma$-derivation and there exists an element $a in mathcal{a}$ such that emph{d(a)} is invertible, then $delta$ is continuous if and only if emph{d} is continuous. we also show that if $mathcal{m}$ is unital, has no zero divisor and $delta:mathcal{a} rightarrow mathcal{m}$ is a generalized $sigma$-derivation such that $d(textbf{1}) neq 0$, then $ker(delta)$ is a bi-ideal of $mathcal{a}$ and $ker(delta) = ker(sigma) = ker(d)$, where textbf{1} denotes the unit element of $mathcal{a}$.