Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals

Let $R$ be a non-commutative prime ring, with center $Z(R)$ , extended centroid $C$ and let $F$ be a non-zero generalized derivation of $R$ . Denote by $L$ a non-central Lie ideal of $R$ . If there exists $0ne ain R$ such that $a[F(x),x]_kin Z(R)$ for all $xin L$ , where $k$ is a fixed integer, then one of the followings holds: (1) either there exists $lambda in C$ such that $F(x)=lambda x$ for all $xin R$ , (2) or $R$ satisfies $s_4$ , the standard identity in $4$ variables, and $char(R)=2$ ; (3) or $R$ satisfies $s_4$ and there exist $qin U, gamma in C$ such that $F(x)=qx+xq+gamma x$ .