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 $ \varphi:\mathcal{R}\rightarrow \mathcal{R} $ satisfies $ \varphi([A\bullet B, C]) = [\varphi(A)\bullet B, C]+[A\bullet\varphi(B), C] +[A\bullet B, \varphi(C)] $ for any $ A, B, C\in \mathcal{R} $ with $ ABC^{\ast} = 0 $, then there exists an additive $ \ast $-derivation $ \Theta $ of $ \mathcal{R} $ and a nonlinear map $ g:\mathcal{R}\rightarrow Z_{c}(\mathcal {R}) $ such that $ \varphi(A) = \Theta(A)+g(A) $ for any $ A\in \mathcal{R} $.