Let R be a 2-torsion-free prime ring with center Z(R), F a generalized derivation associated with a nonzero derivation d, L a Lie ideal of R. If ðdðuÞ l1 FðuÞ l2 dðuÞ l3 FðuÞ l4 . . . FðuÞ l k Þ n ¼ 0 for all u 2 L, where l 1 , l 2 , . . . , l k are fixed non-negative integers not all zero, and n is fixed positive integer, then L˝Z(R). We also examine the case when R is a semiprime ring. Finally, we apply the above result to Banach algebras.