A characterization of $b$-generalized skew derivations on a Lie ideal in a prime ring

This paper investigates the analysis of $\mathrm{b}$-generalized skew derivations, denoted as ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, within a prime ring $\mathcal{R}$ with characteristic different from 2. Here, ${\mathcal{Q}}_r$ represents the right Martindale quotient ring of $\mathcal{R}$, and $\mathcal{C}$ denoted its extended centroid. Additionally, $\mathcal{L}$ is a noncentral Lie ideal of $\mathcal{R}$. Assuming ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ are nontrivial $\mathrm{b}$-generalized skew derivations associated with the same automorphism $\alpha$, the paper aims to explore the detailed structure of these generalized derivations that satisfy the specific equation:

$$pu{\mathrm{\Delta }}_1(u)+{\mathrm{\Delta }}_1(u)uq={\mathrm{\Delta }}_2(u^2),\text{with}p+q\notin \mathcal{C},\text{for all}u\in \mathcal{L}.$$

The above-studied result generalized the already existing results ^[1,2] in the literature.