Generalized skew derivations and generalization of commuting maps on prime rings

Let R be a prime ring of characteristic different from 2, $$Q_r$$ be the right Martindale quotient ring of R and $$C=Z(Q_r)$$ be the extended centroid of R. Suppose that $$f(x_1,\ldots ,x_n)$$ is a noncentral multilinear polynomial over C and F, G are two nonzero generalized skew-derivations of R associated to the same automorphism of R. If $$\begin{aligned} F(u)u-G(u)F(u)=0 \end{aligned}$$ for all $$u\in f(R)$$ , then one of the following holds: