Generalized Derivations and Generalization of Co-commuting Maps in Prime Rings

Suppose that $R$ is a prime ring of characteristic different from $2$ with Utumi quotient ring $U$, $C = Z(U)$ the extended centroid of $R$, and $f(x_1,\ldots,x_n)$ a noncentral multilinear polynomial over $C$. If $F$, $G$ and $H$ are three nonzero generalized derivations of $R$ such that \[ F\big( G(f(X)) f(X) \big) = f(X) H(f(X)) \] for all $X = (x_{1},\ldots,x_{n}) \in R^n$, then we describe the nature of the maps $F$, $G$ and $H$.