Generalized Skew Derivation of Order 2 in Prime Ring with Multilinear Polynomial

Let \(\mathcal {R}\) be a prime ring of characteristic not equal to 2, \(\mathcal {U}\) be the Utumi quotient ring of \(\mathcal {R}\) , \(\mathcal {C}\) be the extended centroid of \(\mathcal {R}\) and \(f(\zeta _1, \ldots ,\zeta _n)\) be a non-central polynomial identity of \(\mathcal {R}\) . Let \(\mathcal {H}\) , \(\mathcal {F}\) are generalized skew derivations on \(\mathcal {R}\) such that \(\mathcal {F}(\mathcal {F}(u)u) = \mathcal {H}(u^2)\) for all \(u =f(\zeta _1,\ldots ,\zeta _n)\) , \(\zeta _1,\zeta _2,\dots ,\zeta _n\in \mathcal {R}\) . In this article, we study the complete structure of \(\mathcal {F}\) and \(\mathcal {H}\) .