Non-global nonlinear mixed skew Jordan Lie triple derivations on prime $\ast$-rings

Let $\mathcal{R}$ be a 2-torsion free unital prime $\ast$-ring containing a nontrivial symmetric idempotent and $Z_c(\mathcal{R})$ be the anti-symmetric center of $\mathcal{R}$. We prove that if a map $\phi :\mathcal{R}\to \mathcal{R}$ satisfies $\phi ([A\bullet B,C])=[\phi (A)\bullet B,C]+[A\bullet \phi (B),C]+[A\bullet B,\phi (C)]$ for any $A,B,C\in \mathcal{R}$ with $AB{C}^{\ast }=0$, then there exists an additive $\ast$-derivation $\mathrm{\Theta }$ of $\mathcal{R}$ and a nonlinear map $g:\mathcal{R}\to Z_c(\mathcal{R})$ such that $\phi (A)=\mathrm{\Theta }(A)+g(A)$ for any $A\in \mathcal{R}$.