Action of three generalized derivations on Lie ideals in prime rings with n-Engel condition

Let R be a prime ring with extended centroid C and characteristic of R be different from 2. Suppose that U is the Utumi quotient ring of R and \({\mathcal {L}} \) $\mathcal{L}$ a noncentral Lie ideal of R. Assuming \({\mathcal {T}} _1, {\mathcal {T}} _2, {\mathcal {T}} _3\) ${\mathcal{T}}_1,{\mathcal{T}}_2,{\mathcal{T}}_3$ three generalized derivations of R, we investigate the identity \(\begin{aligned}{}[{\mathcal {T}} _2(u){\mathcal {T}} _1(u)-{\mathcal {T}} _1(u)u+u{\mathcal {T}} _3(u),u]_n=0 \end{aligned}\) $$\begin{array}{c}\hfill {[{\mathcal{T}}_2(u){\mathcal{T}}_1(u)-{\mathcal{T}}_1(u)u+u{\mathcal{T}}_3(u),u]}_n=0\end{array}$$ for all \(u\in {\mathcal {L}} \) $u\in \mathcal{L}$ , and then we describe the forms of the maps, where \(n\ge 1\) $n\ge 1$ is a fixed integer.