Hyers-Ulam stability of ternary \((\sigma, \tau, \xi)\)-derivations on \(C^*\)-ternary algebras (Q2912570)

A \(\mathbb{C^{*}}\)-ternary algebra is a complex Banach space \(A\) equipped with a ternary product \((x,y,z) \to [xyz]\) of \(A^{3}\) into \(A\), which is \(\mathbb{C}\)-linear in the outer variables, conjugate \(\mathbb{C}\)-linear in the middle variable and associative, i.e., NEWLINE\[NEWLINE [xy[zwv]] = [x[wzy]v] = [[xyz]wv] NEWLINE\]NEWLINE and, moreover, two conditions NEWLINE\[NEWLINE \|[xyz]\| \leq \|x\| \|y\| \|z\|, \;\;\|[xxx]\| = \|x\|^{3} NEWLINE\]NEWLINE hold true.NEWLINENEWLINELet \(\sigma, \tau, \xi\) be linear maps on \(A\). A linear mapping \(\delta: A \to A\) is called \(\mathbb{C^{*}}\)-Jordan ternary \((\sigma, \tau, \xi)\)-derivation if NEWLINE\[NEWLINE \delta([xxx]) = [\delta(x)\tau(x)\xi(x)] + [\sigma(x)\delta(x)\xi(x)] + [\sigma(x)\tau(x)\delta(x)] NEWLINE\]NEWLINE for all \(x \in A\).NEWLINENEWLINELet \(q\) be a positive rational number. For a given mapping \(f: A \to A\) and a given \(\mu \in \mathbb{C}\) we define \(D_{\mu}f: A^{n} \to A\) by NEWLINE\[NEWLINE D_{\mu}f(x_{1},\dots,x_{n}) := \left( \sum_{i=1}^{n}f \left( \sum_{j=1}^{n}q \mu(x_{i} - x_{j})\right)\right) + nf\left( \sum_{i=1}^{n}q \mu x_{i}\right) - nq \mu \sum_{i=1}^{n}f(x_{i}) NEWLINE\]NEWLINE for all \(x_{1},\dots,x_{n} \in A\).NEWLINENEWLINEThe authors prove the generalized Hyers-Ulam stability of \(\mathbb{C^{*}}\)-Jordan ternary \((\sigma, \tau, \xi)\)-derivations in \(\mathbb{C^{*}}\)-ternary algebras. A result of this type is the following.NEWLINENEWLINE{ Theorem 2.1} Let \(n\) be a positive integer. Assume that \(r > 3\) if \(nq > 1\) and that \(0 < r < 1\) if \(nq < 1\). Let \(\theta\) be a positive real number and let \(f: A \to A\) be an odd mapping for which there exists mappings \(g, h, k : A \to A\) with \(g(0) = h(0) = k(0) = 0\) such that NEWLINE\[NEWLINE \| D_{\mu}f(x_{1},\dots,x_{n})\| \leq \theta \sum_{j=1}^{n}\|x_{j}\|^{r}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \|f([xxx]) - [f(x)h(x)k(x)] - [g(x)f(x)k(x)] - [g(x)h(x)f(x)]\| \leq 3\theta\|x\|^{r}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \|g(q \mu x_{1} +\dots+q \mu x_{n}) - q \mu g(x_{1}) -\dots-q \mu g(x_{n})\| \leq \theta(\|x_{1}\|^{r} +\dots + \|x_{1}\|^{r}), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \|h(q \mu x_{1} +\dots+q \mu x_{n}) - q \mu h(x_{1}) -\dots-q \mu h(x_{n})\| \leq \theta(\|x_{1}\|^{r} +\dots + \|x_{1}\|^{r}), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \|k(q \mu x_{1} +\dots+q \mu x_{n}) - q \mu k(x_{1}) -\dots-q \mu k(x_{n})\| \leq \theta(\|x_{1}\|^{r} +\dots + \|x_{1}\|^{r}), NEWLINE\]NEWLINE for all \(\mu, |\mu| = 1\) and all \(x_{1},\dots,x_{n}, x \in A\). Then there exist unique linear mappings \(\sigma, \tau\) and \(\xi\) from \(A\) to \(A\) and a unique \(\mathbb{C^{*}}\)-Jordan ternary \((\sigma, \tau, \xi)\)-derivation \(\delta: A \to A\) such that NEWLINE\[NEWLINE \|g(x) - \sigma(x) \| \leq \frac{n\theta}{(nq)^{r} - nq} \|x\|^{r}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \|h(x) - \tau(x) \| \leq \frac{n\theta}{(nq)^{r} - nq} \|x\|^{r}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \|k(x) - \xi(x) \| \leq \frac{n\theta}{(nq)^{r} - nq} \|x\|^{r}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \|f(x) - \delta(x) \| \leq \frac{\theta}{(nq)^{r} - nq} \|x\|^{r} NEWLINE\]NEWLINE for all \(x \in A\).NEWLINENEWLINEMoreover, \(\delta\) is an additive mapping of Euler-Lagrange type i.e. NEWLINE\[NEWLINE \left( \sum_{i=1}^{n}\delta \left( \sum_{j=1}^{n}q (x_{i} - x_{j})\right)\right) + n\delta \left( \sum_{i=1}^{n}q x_{i}\right) = nq \sum_{i=1}^{n}\delta(x_{i}). NEWLINE\]NEWLINENEWLINENEWLINEFurther, the generalized Hyers-Ulam stability of \(\mathbb{C^{*}}\)-ternary \((\sigma, \tau, \xi)\)-derivations in \(\mathbb{C^{*}}\)-ternary algebras and the generalized Hyers-Ulam stability of \(\mathbb{C^{*}}\)-Lie ternary \((\sigma, \tau, \xi)\)-derivations in \(\mathbb{C^{*}}\)-ternary algebras for the Euler-Lagrange type additive mappings are considered.