Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals

Abstract

Let $R$ be a prime ring with extended centroid $C$, $L$ anon-central Lie ideal of $R$ and $n\geq 1$ a fixed integer. If $R$admits the generalized derivations $H$ and $G$ such that$H(u^2)^n=G(u)^{2n}$ for all $uın L$, then one of the followingholds: {(1)} $H(x)=ax$ and $G(x)=bx$ for all $xınR$, with $a,bın C$ and $a^n=b^{2n}$; {(2)} char$(R)\neq 2$, $R$ satisfies $s_4$, $H(x)=ax+[p,x]$ and $G(x)=bx$ for all $xın R$, with$bın C$ and $a^n=b^{2n}$; {(3)} char$(R)=2$ and $R$ satisfies $s_4$.As an application we also obtain some range inclusion results ofcontinuous generalized derivations on Banach algebras.