Generalized Skew-Derivations Acting on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2, \(Q_r\) be its right Martindale quotient ring, C be its extended centroid and \(f(r_1,\ldots ,r_n)\) be a noncentral multilinear polynomial over C. Suppose that \(T_1\) , \(T_2\) are two generalized skew derivations on R. If \(\begin{aligned} T_1(u)T_2(u)=T_1(u)u-uT_2(u)\end{aligned}\) for all \(u\in f(R)=\{f(x_1,\ldots ,x_n) | x_1,\ldots ,x_n\in R\}\) , then we determine all possible forms of the maps.