Permuting triderivations and permuting trihomomorphisms in complex Banach algebras

In this paper, we solve the following tri-additive $s$-functional inequalities \begin{eqnarray}\label{0.1} && \nonumber \| f(x+y, z-w, a+b) + f(x-y, z+w, a-b) \\ && \nonumber\qquad -2 f(x, z, a) + 2 f(x, w, b) -2f(y, z, b) +2 f(y, w, a)\| \\ && \quad \le \left \|s \left(2f\left(\frac{x+y}{2}, z-w, a+b \right) + 2f\left(\frac{x-y}{2}, z+w, a-b\right) \right. \right. \\ && \qquad \left. \left. -2 f(x, z, a) + 2 f(x, w, b) -2f(y, z, b) +2 f(y, w, a)\right)\right\| , \nonumber \end{eqnarray} \begin{eqnarray}\label{0.2} && \nonumber \left\|2f\left(\frac{x+y}{2}, z-w, a+b \right) + 2f\left(\frac{x-y}{2}, z+w, a-b\right) \right. \\ && \nonumber \qquad \left. -2 f(x, z, a) + 2 f(x, w, b) -2f(y, z, b) +2 f(y, w, a)\right\| \\ && \quad \le \|s ( f(x+y, z-w, a+b) + f(x-y, z+w, a-b) \\ && \nonumber\qquad -2 f(x, z, a) + 2 f(x, w, b) -2f(y, z, b) +2 f(y, w, a) )\| , \end{eqnarray} where $s$ is a fixed nonzero complex number with $|s |< 1$. Moreover, we prove the Hyers-Ulam stability and hyperstability of permuting triderivations and permuting trihomomorphisms in Banach algebras and unital $C^*$-algebras, associated with the tri-additive $s$-functional inequalities {\rm (\ref{0.1})} and {\rm (\ref{0.2})}.