An extension of Herstein's theorem on Banach algebra

Let $\mathcal{A}$ be a $(p+q)!$-torsion free semiprime ring. We proved that if $\mathcal{H},\mathcal{D}:\mathcal{A}\to \mathcal{A}$ are two additive mappings fulfilling the algebraic identity $2\mathcal{H}(a^{p+q})=\mathcal{H}(a^p)a^q+a^p\mathcal{D}(a^q)+\mathcal{H}(a^q)a^p+a^q\mathcal{D}(a^p)$ for all $a\in \mathcal{A}$, then $\mathcal{H}$ is a generalized derivation with $\mathcal{D}$ as an associated derivation on $\mathcal{A}$. In addition to that, it is also proved in this article that ${\mathcal{H}}_1$ is a generalized left derivation associated with a left derivation $\delta$ on $\mathcal{A}$ if they fulfilled the algebraic identity $2{\mathcal{H}}_1(a^{p+q})=a^p{\mathcal{H}}_1(a^q)+a^q\delta (a^p)+a^q{\mathcal{H}}_1(a^p)+a^p\delta (a^q)$ for all $a\in \mathcal{A}$. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.