Permuting triderivations and permuting trihomomorphisms in Banach algebras

Abstract

Using the direct method, we prove the Hyers–Ulam stability of permuting triderivations and permuting trihomomorphisms in Banach algebras and unital $C^{\ast }$-algebras associated with the triadditive $s$-functional inequality $\parallel f\left(x+y,z-w,a+b\right)+f\left(x-y,z+w,a-b\right)-2f\left(x,z,a\right)+2f\left(x,w,b\right)-2f\left(y,z,b\right)+2f\left(y,w,a\right)\parallel \le \parallel s\left(2f\left(\left(x+y\right)∕2,z-w,a+b\right)+2f\left(\left(x-y\right)∕2,z+w,a-b\right)-2f\left(x,z,a\right)+2f\left(x,w,b\right)-2f\left(y,z,b\right)+2f\left(y,w,a\right)\right)\parallel$, where $s$ is a fixed nonzero complex number with $\left|s\right|<1$.