Generalized derivations acting on Lie ideals in prime rings and Banach algebras

Let $R$ be a prime ring and $L$ a non-central Lie ideal of $R.$ The purpose of this paper is to describe generalized derivations of $R$ satisfying some algebraic identities locally on $L.$ More precisely, we consider two generalized derivations $F_1$ and $F_2$ of a prime ring $R$ satisfying one of the following identities:1. $F_1(x)\circ y +x \circ F_2(y) =0,$2. $[F_1(x),y] + F_2([x,y]) =0,$for all $x,y$ in a non-central Lie ideal $L$ of $R.$ Furthermore, as an application, we study continuous generalized derivations satisfying similar algebraic identities with power values on nonvoidopen subsets of a prime Banach algebra $A$. Our topological approach is based on Baire'scategory theorem and some properties from functional analysis.