Certain differential identities on prime rings and Banach algebras

Let R be a prime ring and L a Lie ideal of R. The purpose of this paper is to classify generalized derivations \(F_1,\) $F_1,$ \(F_2,\) $F_2,$ \(F_3\) $F_3$ of R satisfying the following differential identity \(\begin{aligned} F_1(x)\bot F_2(y)=F_3(x)\bot y \;\; for\, all \, x,y \in L, \end{aligned}\) $$\begin{array}{c}\hfill F_1(x)\perp F_2(y)=F_3(x)\perp yforallx,y\in L,\end{array}$$ where \(\bot\) $\perp$ represents either the Lie product [., .], or the Jordan product \(\circ\) $\circ$ . Furthermore, as an application, the same identities are studied locally on nonvoid open subsets of prime Banach algebras.