In this article, we investigate the behaviour of a non-linear map ? on *-algebra A, which satisfies ?([X?Y,Z]?) = [?(X)?Y,Z]? +[X??(Y),Z]? +[X?Y,?(Z)]?, where X?Y = XY+YX and [X,Y]? = XY* ?YX* (namely, Jordan and bi-skewLie product, respectively), for all X,Y,Z ? A. Furthermore, we apply the above mentioned result to several distinct algebras.

Let A be aprime *-algebra. For any A, B ? A, define a new product A ? B = AB* + BA*. Let ? be a non-linear map satisfying ?(A ? B ? C) = ?(A) ? B ? C + A ? ?(B) ? C + A ? B ? ?(C) for all A, B, C ? A. In this article, we show that ? is an additive *-derivation. Furthermore, we also discuss above result for higher derivable maps on A.

Let $\mathfrak{A}$ be a prime *-algebra. A product defined as $U•V=U{V}^{\ast }+V{U}^{\ast }$ for any $U,V\in \mathfrak{A}$, is called a bi-skew Jordan product. A map $\xi :\mathfrak{A}\to \mathfrak{A}$, defined as $\xi \left(p_n\left(U_1,U_2,\cdots ,U_n\right)\right)={\sum }_{k=1}^n{p}_n\left(U_1,U_2,...,U_{k-1},\xi \left(U_k\right),U_{k+1},\cdots ,U_n\right)$ for all $U_1,U_2,...,U_n\in \mathfrak{A}$, is called a non-linear bi-skew Jordan n-derivation. In this article, it is shown that $\xi$ is an additive ∗-derivation.

Let $\Delta =\{\Ta_n\}_{n\in \mathbb{N}}$ be a non-linear bi-skew Jordan triple higher derivation on a prime $*$-algebra $\mathfrak{A}$. In this article, we prove that $\Delta$ is an additive $*$-higher derivation on $\mathfrak{A}$.