Differential identities involving three generalized derivations on prime rings and Banach algebras

Let L be a non-central Lie ideal of a prime ring R. The primary objective of this paper is to characterize generalized derivations H, F and G of R satisfying the following algebraic identity: x perpendicular to H(y) = F(x) y + yG(x) for all x, y is an element of L, where perpendicular to represents either the Lie product [., .], or the Jordan product degrees. Furthermore, we extend our results to Banach algebras.