Permuting triderivations and permuting trihomomorphisms in complex Banach algebras (Q6649404)

Let \(X\) be a complex normed space and \(Y\) be a complex Banach space. Assume that \(s\) is a fixed nonzero complex number with \(|s|< 1\).\N\NAmong other results, the authors prove what follows.\N\N{Lemma.} If a mapping \(f: X^3\to Y\) satisfies \(f(x, 0, a)= f(0,z, a)= f(x,z, 0)= 0\) and\N\[\N\begin{multlined} \Big\|2f\Big(\frac{x+y}{2},z-w,a+b\Big) +2f\Big(\frac{x-y}{2},z+w,a-b\Big) -2f(x, z, a) \\\N+2f(x,w,b)-2f(y,z,b)+2f(y,w,a)\Big\| \\\N\leq \Big\| s\Big(f(x+y,z -w, a + b)+f(x-y,z +w, a - b) -2f(x, z, a) \\\N+2f(x,w,b)-2f(y,z,b)+2f(y,w,a)\Big)\Big\| \end{multlined}\tag{1}\N\]\Nfor all \(x, y,z, w, a, b\in X\), then \(f : X^3\to Y\) is tri-additive.\N\N{Theorem.} Let \(r>1\) and \(\theta\) be nonnegative real numbers and \(f: X^3\to Y\) be a mapping satisfying \(f(x, 0, a)= f(0,z, a)= f(x,z, 0)= 0\) and\N\[\N\begin{multlined}\N\Big\|2f\Big(\frac{x+y}{2},z-w,a+b\Big) +2f\Big(\frac{x-y}{2},z+w,a-b\Big) -2f(x, z, a) \\\N+2f(x,w,b) -2f(y,z,b)+2f(y,w,a)\Big\| \\\N\leq \Big\| s\Big(f(x+y,z -w, a + b)+f(x-y,z +w, a - b) -2f(x, z, a) \\\N+2f(x,w,b) -2f(y,z,b)+2f(y,w,a)\Big)\Big\| \\\N+\theta(\|x\|^r+\|y\|^r)(\|z\|^r+\|w\|^r)(\|a\|^r+\|b\|^r)\N\end{multlined}\N\]\Nfor all \(x, y,z, w, a, b\in X\). Then there exists a unique tri-additive mapping \(L : X^3\to Y\) such that\N\[\N\|f(x, z, a)-L(x, z, a)\|\leq \frac{2^r\theta}{2(2^r-2)}\|x\|^r\|y\|^r\|z\|^r.\N\]\Nfor all \(x,z, a\in X\).\N\NThis is the so-called the Hyers-Ulam stability of the tri-additive \(s\)-functional inequality (1).\N\NThe authors also investigate the Hyers-Ulam stability and hyperstability of permuting triderivations and permuting trihomomorphisms in Banach algebras and unital \(C^{\ast}\)-algebras, associated with tri-additive \(s\)-functional inequalities.