On a functional equation related to generalized inner derivations

The purpose of this paper is to prove the following result. Let $X$ be a real or complex Banach space, let $ \mathcal{L}(X)$ be the algebra of all bounded linear operators on $X$, and let $\mathcal{A}(X)\subseteq\mathcal{L}(X)$ be a standard operator algebra, which posses the identity operator. Suppose there exists a linear mapping $F:\mathcal{A}(X)\rightarrow\mathcal{L}(X)$ satisfying the relation $F(A^{n})=F(A^{n-1})A-AF(A^{n-2})A+AF(A^{n-1})$ for all $A\in\mathcal{A}(X)$ and some fixed integer $n\geq3$. In this case, $F$ is of the form $F(A)=AB_{1}+B_{2}A$ for all $A\in\mathcal{A}(X)$ and some fixed $B_{1},B_{2}\in\mathcal{L}(X)$. In particular, $F$ is continuous.