The First Nonlinear mixed Jordan triple derivation on $\ast$-algebras

Abstract

Let $\mathcal{A}$ be a unital $\ast$-algebra containing a non-trivial projection.  We prove that if a  map $\Pi$ : $\mathcal{A}$ $\to$ $\mathcal{A}$ such that $\Pi$ : $\mathcal{A}$ $\to$ $\mathcal{A}$ such that $ \Pi ( S \bullet T \circ U) = \Pi( S) \bullet T \circ U +  S \bullet \Pi (T) \circ U +  S \bullet T \circ \Pi (U)$ for all $ S, T, U \in \mathcal{A}, $  then $\Pi$ is an additive $\ast$-derivation.