Characterization of ($\alpha$,$\alpha$)-derivation on $B(X)$

Let $X$ be a Banach algebra and $B(X)$ be the set of all bounded linear operators on $X$. Suppose that $\alpha :B(X)\to B(X)$ is an automorphism. We say that a mapping $\delta$ from $B(X)$ into itself is derivable at $G\in B(X)$ if $\delta (G)=\alpha (A)\delta (B)+\delta (A)\alpha (B)$ for all $A,B\in B(X)$ with $AB=G$. We say that an element $G\in B(X)$ is an $(\alpha ,\alpha )$-all derivable point of $B(X)$ if every $(\alpha ,\alpha )$-derivable mapping $\delta$ at G is an $(\alpha ,\alpha )$-derivation. In this paper, we show that every $(\alpha ,\alpha )$-derivable mapping at a nonzero element in $B(X)$ is an $(\alpha ,\alpha )$-derivation.