Let $\mathcal{A}$ be a unital *-algebra containing a non-trivial projection. In this paper, it is shown that a map Γ: $\mathcal{A}$ $\to$ $\mathcal{A}$ such that $\Gamma (U°V•W)=\Gamma (U)°V•W+U°\Gamma (V)•W+U°V•\Gamma (W)$ for all $U,V,W\in \mathcal{A}$. Then Γ is an additive *-derivation.

Let u be a standard operator algebra on an infinite dimensional complex Hilbert space H containing identity operator I, which is closed under the adjoint operation. Suppose that delta : u -> u is the nonlinear mixed Jordan triple *-derivation. Then delta is an additive *-derivation.

Let A be a unital *-algebra containing a non-trivial projection. We prove that if a map ?: A ? A such that ?(S ? T ? U) = ?(S) ? T ? U + S ? ?(T) ? U + S ? T ? ?(U) for all S, T, U ? A, then ? is an additive *-derivation.

The paper investigates nonadditive mappings \(\Omega: \Re \to \Re\) on a prime ring \(\Re\) with involution \( * \), characterized by satisfying one of the following conditions: \begin{enumerate} \item \( [\Omega(u), \Omega(v)]_\bullet = [u, v]_\bullet \) for all \( u, v \in \Re \). \item \( [\Omega(u), v]_\bullet = [u, \Omega(v)]_\bullet \) for all \( u, v \in \Re \). \item \( \Omega(u \bullet v) = \Omega(u) \bullet v \) for all \( u, v \in \Re \). \end{enumerate} Furthermore, the paper characterizes generalized bi-skew Jordan derivations within prime \( * \)-rings and examines the implications of these results in the context of various operator algebras.

This paper concentrates on examining the characterization of nonlinear mixed bi-skew Lie triple ∗- derivations within an ∗-algebra denoted by $\mathcal{A}$ which contains a nontrivial projection with a unit I. Additionally, we expand this investigation to applications by describing these derivations within prime ∗-algebras, von Neumann algebras, and standard operator algebras.

On an infinite dimensional complex Hilbert space $ \mathcal{H} $, we consider a standard operator algebra $ \mathcal{S} $ with an identity operator $ I $ that is closed with respect to adjoint operation. $ P_{n}\left(\mathcal{X}_{1}, \mathcal{X}_{2}, \mathcal{X}_{3}, \ldots, \mathcal{X}_{n}\right) $ is set of polynomials defined under indeterminates $ \mathcal{X}_1, \mathcal{X}_2, \cdots, \mathcal{X}_n $ by $ n $ with multiplicative Lie products with set of positive integers $ \mathbb{N}. $ It is shown that a map $ \Theta: \mathcal{S} \rightarrow \mathcal{S} $ satisfying \begin{document}$ \begin{eqnarray*} \Theta\left(P_{n}\left(\mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n}\right)\right) = \sum\limits_{i = 1}^{n} P_{n}\left(\mathcal{D}_{1}, \ldots, \mathcal{D}_{i-1}, \Theta\left(\mathcal{D}_{i}\right), \mathcal{D}_{i+1}, \ldots, \mathcal{D}_{n}\right), \end{eqnarray*} $\end{document} for any $ \mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n} \in \mathcal{S} $ with $ \mathcal{D}_{1} \mathcal{D}_{2} \mathcal{D}_{3}\ldots \mathcal{D}_{n} = 0 $ can be represented as $ d(x)+\tau(x) $ for every $ x \in \mathcal{S} $, where $ d: \mathcal{S} \rightarrow \mathcal{S} $ is an additive derivation with another map $ \tau: \mathcal{S} \rightarrow \mathcal{Z}(\mathcal{S}) $ that vanishes on each $ (n-1)^{th} $ commutator $ P_{n}\left(\mathcal{D}_{1}, \mathcal{D}_{2}, \mathcal{D}_{3}, \ldots, \mathcal{D}_{n}\right) $ with $ \mathcal{D}_{1} \mathcal{D}_{2} \mathcal{D}_{3}\cdots \mathcal{D}_{n} = $ 0.