Generalized derivations with nilpotent values on Lie ideals in semiprime rings

Let R be a prime ring of characteristic different from 2, Q(r) its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R, n >= 1 a fixed integer, F and G two generalized derivations of R. If (F(xy) - G(x)G(y))(n) = 0, for any x, y is an element of L, then there exists lambda is an element of C such that F(x) = lambda(2)x and G( x) = lambda x, for any x is an element of R. Moreover, we analyze the semiprime case.