### $(\phi ,\varphi )$-derivations on semiprime rings and Banach algebras

Let $\mathcal{R}$ be a semiprime ring with unity $e$ and $\phi $, $\varphi $ be automorphisms of $\mathcal{R}$. In this paper it is shown that if $\mathcal{R}$ satisfies $$2\mathcal{D}\left({x}^{n}\right)=\mathcal{D}\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)\mathcal{D}\left(x\right)+\mathcal{D}\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)\mathcal{D}\left({x}^{n-1}\right)$$ for all $x\in \mathcal{R}$ and some fixed integer $n\ge 2$, then $\mathcal{D}$ is an ($\phi $, $\varphi $)-derivation. Moreover, this result makes it possible to prove that if $\mathcal{R}$ admits an additive mappings $\mathcal{D},\mathcal{G}:\mathcal{R}\to \mathcal{R}$ satisfying the relations $$\begin{array}{c}2\mathcal{D}\left({x}^{n}\right)=\mathcal{D}\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)\mathcal{G}\left(x\right)+\mathcal{G}\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)\mathcal{G}\left({x}^{n-1}\right)\phantom{\rule{0.166667em}{0ex}},\\ 2\mathcal{G}\left({x}^{n}\right)=\mathcal{G}\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)\mathcal{D}\left(x\right)+\mathcal{D}\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)\mathcal{D}\left({x}^{n-1}\right)\phantom{\rule{0.166667em}{0ex}},\end{array}$$ for all $x\in \mathcal{R}$ and some fixed integer $n\ge 2$, then $\mathcal{D}$ and $\mathcal{G}$ are ($\phi $, $\varphi $)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.