Nonlinear mixed Jordan and bi-skew Jordan derivations on prime $\ast$-algebras

Abstract

Let $\mathcal{A}$ be a prime $\ast$-algebra. In this paper, we suppose that $\Phi:\mathcal{A}\to\mathcal{A}$ satisfies\begin{eqnarray*}\Phi[[A,B],C]_{\dagger}=[[\Phi(A),B],C]_{\dagger}+[[A,\Phi(B)],C]_{\dagger}+[[A,B],\Phi(C)]_{\dagger}\end{eqnarray*}where $[A,B]=AB+BA$ and $[A,B]_{\dagger}=A^{*}B+B^{*}A$ for all $A,B\in\mathcal{A}$. Then, $\Phi$ is additive $\ast$-derivation.