Generalized skew derivations on Lie ideals in prime rings

Let R be prime ring with characteristic different from 2, C denotes the extended centroid, L a Lie ideal of R and \(Q_r\) the right Martindale quotient of the ring R. Let \(\Delta _1\) and \(\Delta _2\) represents two generalized skew derivations of R associated with \((\psi ,l_1)\) and \((\psi , l_2)\), respectively, such that \(\psi .l_1=l_1.\psi \) and \(\psi . l_2= l_2.\psi \). If, for every \(r \in L\), \(\Delta _1^2(r)r=\Delta _2(r^2)\), then we characterize the maps \(\Delta _1\) and \(\Delta _2\). As an application of this generalization, we proved that if \(\Delta _1(\tau ^2)=0\) for all \(\tau \in R\), then R contains a non-zero central ideal.