Some results on prime rings with multiplicative derivations

Let $R$ be a prime ring with center $Z R $ and an automorphism $\alpha.$ A mapping $\delta:R\to R$ is called multiplicative skew derivation if $\delta xy =\delta x y+ \alpha x \delta y $ for all $x,y\in R$ and a mapping $F:R\to R$ is said to be multiplicative generalized -skew derivation if there exists a unique multiplicative skew derivation $\delta$ such that $F xy =F x y+\alpha x \delta y $ for all $x,y\in R.$ In this paper, our intent is to examine the commutativity of $R$ involving multiplicative generalized -skew derivations that satisfy the following conditions: i $F x^{2} +x\delta x =\delta x^{2} +xF x $, ii $F x\circ y =\delta x\circ y \pm x\circ y$, iii $F [x,y] =\delta [x,y] \pm [x,y]$, iv $F x^{2} =\delta x^{2} $, v $F [x,y] =\pm x^{k}[x,\delta y ]x^{m}$, vi $F x\circ y =\pm x^{k} x\circ\delta y x^{m}$, vii $F [x,y] =\pm x^{k}[\delta x ,y]x^{m}$, viii $F x\circ y =\pm x \delta x \circ y x^{m}$ for all $x,y\in R.$