Generalized Hyers–Ulam Stability of Bi-Homomorphisms, Bi-Derivations, and Bi-Isomorphisms in C*-Ternary Algebras

Abstract

In this paper, we investigate the generalized Hyers–Ulam stability of bi-homomorphisms, bi-derivations, and bi-isomorphisms in $C^{*}$-ternary algebras. The study of functional equations with a sufficient number of variables can be helpful in solving real-world problems such as artificial intelligence. In this paper, we build on previous research on functional equations with four variables to study functional equations with as many variables as desired. We introduce new bounds for the stability of mappings satisfying generalized bi-additive conditions and demonstrate the uniqueness of approximating bi-isomorphisms. The results contribute to the deeper understanding of ternary algebraic structures and related functional equations, relevant to both pure mathematics and quantum information science.

1. Introduction

The study of stability problems for functional equations began with a question posed by Ulam [1] in 1940 and partially answered by Hyers [2] in 1941. Since then, generalized forms—particularly those introduced by Rassias [3] and Găvruţa [4]—have greatly expanded the field. Brzdęk [5] provided a good overview of the stability problem. Ciepliński [6] obtained generalized stability results for the vector-variable functional equations

$$\begin{array}{c}\hfill f(a_{11}{x}_{11}+a_{12}{x}_{12},\cdots ,a_{n1}{x}_{n1}+a_{n2}{x}_{n2})=\sum _{i_1,\cdots ,i_n\in \{1,2\}}{A}_{i_1,\cdots ,i_n}f(x_{1i_1},\cdots ,x_{n{i}_n})\end{array}$$

and

$$f(x_{11}+x_{12},\cdots ,x_{n1}+x_{n2})=\sum _{i_1,\cdots ,i_n\in \{1,2\}}f(x_{1i_1},\cdots ,x_{n{i}_n})$$

in generalized distance spaces. Khodaei [7] obtained generalized stability for the multi-variable functional equations

$$f\left(\sum _{i=1}^n\mu s_i{x}_i\right)=\sum _{i=1}^n\mu s_{i}f\left(x_i\right)$$

and

$$f\left(\varsigma (x_1,\cdots ,x_n)\right)=\sigma (f\left(x_1\right),\cdots ,f\left(x_n\right))$$

in $C^{*}$-algebras, Banach Lie algebras, etc. However, even in recent research on stability theory, generalized stability for multi-variable bi-additive functional equations in $C^{*}$-ternary algebras has not yet been obtained.

In recent years, algebraic structures such as ternary and n-ary algebras have become critical in modeling complex systems in physics.

In this work, we obtain stability results for multi-variable bi-additive functional equations over $C^{*}$-ternary algebras, focusing on bi-homomorphisms, bi-derivations, and bi-isomorphisms.

2. Preliminaries

The following statements are well known to specialists in the field and are included for completeness.

A $C^{*}$-ternary algebra is defined as a complex Banach space with a ternary product that is linear in the outer variables, conjugate-linear in the middle, and satisfies associativity and norm conditions. Bi-homomorphisms and bi-derivations extend the notion of linear maps to this ternary setting.

Let m and n be positive integers greater than 1 and let X and Y be real or complex linear spaces. For a mapping $f:X\times X\to Y$, we denote the functional equation of interest as

$$f\left(\sum _{i=1}^m{x}_i,\sum _{j=1}^n{y}_j\right)=\sum _{i=1}^m\sum _{j=1}^{n}f(x_i,y_j).$$

(1)

This equation characterizes bi-additive mappings. The aim is to study mappings that approximately satisfy this condition and determine under what conditions they approximate a true bi-homomorphism or bi-isomorphism.

Note that a mapping $f:X\times X\to Y$ is bi-additive if and only if it satisfies (1).

Ternary algebraic operations are a topic explored by several mathematicians, including Cayley [8], who introduced the concept of cubic matrices in the 19th century.

A fundamental example of a nontrivial ternary operation can be illustrated by the following composition rule:

$${\{a,b,c\}}_{ijk}=\sum _{1\le l,m,n\le N}{a}_{nil}{b}_{ljm}{c}_{mkn}(i,j,k=1,2,\cdots ,N).$$

Ternary $C^{*}$-rings and ternary operations were introduced in [9]. A $C^{*}$-ternary algebra is a complex Banach space A, equipped with a ternary product $(x,y,z)\mapsto [x,y,z]$ of $A^3$ into A, which is $\mathbb{C}$-linear in the outer variables, conjugate $\mathbb{C}$-linear in the middle variable, and associative in the sense that $[x,y,[z,w,v\left]\right]=[x,[w,z,y],v]=\left[\right[x,y,z],w,v]$, and satisfies $\parallel [x,y,z]\parallel \le \parallel x\parallel ·\parallel y\parallel ·\parallel z\parallel$ and $\parallel [x,x,x]\parallel ={\parallel x\parallel }^3$ (see [9,10]). Every left Hilbert $C^{*}$-module is a $C^{*}$-ternary algebra with the ternary product $[x,y,z]:=\langle x,y\rangle z$.

If $(A,[·,·,·\left]\right)$ is a $C^{*}$-ternary algebra with an identity e, i.e., $x=[x,e,e]=[e,e,x]$ for all $x\in A$, then A, endowed with $x\circ y:=[x,e,y]$ and $x^{*}:=[e,x,e]$, is a unital $C^{*}$-algebra. If $(A,\circ )$ is a unital $C^{*}$-algebra, then it is also a $C^{*}$-ternary algebra with the ternary product $[x,y,z]:=x\circ y^{*}\circ z$.

Let A and B be two $C^{*}$-ternary algebras. A $\mathbb{C}$-linear mapping $H:A\to B$ is called a $C^{*}$-ternary algebra homomorphism (see [11]) if

$$H\left(\right[x,y,z\left]\right)=\left[H\right(x),H(y),H(z\left)\right]$$

for all $x,y,z\in A$. In addition, if the mapping H is bijective, then it is called a $C^{*}$-ternary algebra isomorphism. A $\mathbb{C}$-linear mapping $\delta :A\to A$ is called a $C^{*}$-ternary derivation (see [12]) if

$$\delta \left(\right[x,y,z\left]\right)=\left[\delta \right(x),y,z]+[x,\delta (y),z]+[x,y,\delta (z\left)\right]$$

for all $x,y,z\in A$.

Definition 1.

For two $\mathbb{C}$-linear spaces V and W, a mapping $L:V\times V\to W$ is called a $\mathbb{C}$-bilinear mapping if it satisfies

$$L(\lambda x+y,u)=\lambda L(x,u)+L(y,u)$$

and

$$L(x,\mu u+v)=\mu L(x,u)+L(x,v)$$

for all $\lambda ,\mu \in \mathbb{C}$ and $x,y,u,v\in V$. Let A and B be $C^{*}$-ternary algebras. A $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ is called a $C^{*}$-ternary algebra bi-homomorphism if it satisfies

$$H\left(\right[x,y,z],[u,v,w\left]\right)=\left[H\right(x,u),H(y,v),H(z,w\left)\right]$$

(2)

for all $x,y,z,u,v,w\in A$. A $C^{*}$-ternary algebra bi-isomomorphism is a bijective $C^{*}$-ternary algebra bi-homomorphism. Let $\delta :A\times A\to A$ be a $\mathbb{C}$-bilinear mapping. If

$$\delta ([x,y,z],w)=[\delta (x,w),y,z]+[x,\delta (y,w^{*}),z]+[x,y,\delta (z,w)]$$

and

$$\delta (x,[y,z,w])=[\delta (x,y),z,w]+[y,\delta (x^{*},z),w]+[y,z,\delta (x,w)]$$

for all $x,y,z,w\in A$, δ is called a $C^{*}$-ternary bi-derivation.

3. Main Results

From now on, assume that A is a $C^{*}$-ternary algebra with the norm ${\parallel ·\parallel }_A$ and that B is a $C^{*}$-ternary algebra with the norm ${\parallel ·\parallel }_B$.

Let V and W be $\mathbb{C}$-linear spaces and let $f:V\times V\to W$ be a bi-additive mapping such that $f(\lambda x,\mu y)=\lambda \mu f(x,y)$ for all $\lambda ,\mu \in \mathbb{C}$ with $\left|\lambda \right|=\left|\mu \right|=1$ and all $x,y\in V$. According to [13], f is $\mathbb{C}$-bilinear.

Let V and W be $\mathbb{C}$-linear spaces and let $f:V\times V\to W$ be a mapping such that

$$f\left(\lambda \sum _{i=1}^m{x}_i,\mu \sum _{j=1}^n{y}_j\right)=\lambda \mu \sum _{i=1}^m\sum _{j=1}^{n}f(x_i,y_j)$$

(3)

for all $\lambda ,\mu \in \mathbb{C}$ with $\left|\lambda \right|=\left|\mu \right|=1$ and all $x_1,\cdots ,x_m,y_1,\cdots ,y_n\in V$. Then, we also know that f is $\mathbb{C}$-bilinear.

For a given mapping $f:A\times A\to B$, we define

$$D_{\lambda ,\mu }f(x_1,\cdots ,x_m,y_1,\cdots ,y_n):=f\left(\lambda \sum _{i=1}^m{x}_i,\mu \sum _{j=1}^n{y}_j\right)-\lambda \mu \sum _{i=1}^m\sum _{j=1}^{n}f(x_i,y_j)$$

for all $\lambda ,\mu \in \mathbb{C}$ with $\left|\lambda \right|=\left|\mu \right|=1$ and all $x_1,\cdots ,x_m,y_1,\cdots ,y_n\in A$.

We prove the generalized Hyers–Ulam stability of the functional Equation (3) in $C^{*}$-ternary algebras for $p<1$.

Theorem 1.

Let $p,\theta$ be positive real numbers with $p<1$, and let $f:A\times A\to B$ be a mapping such that

$$\parallel D_{\lambda ,\mu }f(x_1,\cdots ,x_m,y_1,\cdots ,y_n){\parallel }_B\le \theta \left(\sum _{i=1}^m\parallel x_i{\parallel }_A^p+\sum _{j=1}^n{\parallel y_j\parallel }_A^p\right),$$

(4)

$$\begin{array}{c}\parallel f([x,y,z],[u,v,w])-[f(x,u),f(y,v),f(z,w)]{\parallel }_B\hfill \\ \le \theta (\parallel x{\parallel }_A^p+\parallel y{\parallel }_A^p+\parallel z{\parallel }_A^p+\parallel u{\parallel }_A^p+\parallel v{\parallel }_A^p+\parallel w{\parallel }_A^p)\hfill \end{array}$$

(5)

for all $\lambda ,\mu \in \mathbb{C}$ with $\left|\lambda \right|=\left|\mu \right|=1$ and all $x_1,\cdots ,x_m,y_1,\cdots ,y_n,x,y,z,u,v,w\in A$. Then, there exists a unique $C^{*}$-ternary algebra bi-homomorphism $H:A\times A\to B$ such that

$${\parallel f(x,y)-H(x,y)\parallel }_B\le \theta \left({\displaystyle \frac{m}{mn-m^p}}{\parallel x\parallel }_A^p+{\displaystyle \frac{n}{mn-n^p}}{\parallel y\parallel }_A^p\right)$$

(6)

for all $x,y\in A$.

Proof. 

Letting $\lambda =\mu =1$, $x_1=\cdots =x_m=x$ and $y_1=\cdots =y_n=y$ in (4), we gain

$${\parallel f(mx,ny)-mnf(x,y)\parallel }_B\le {\theta (m\parallel x\parallel }_A^p+{n\parallel y\parallel }_A^p)$$

(7)

for all $x,y\in A$. Replacing x with $m^{j}x$ and y with $n^{j}y$ and dividing ${\left(mn\right)}^{j+1}$ in the above inequality, we obtain that

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{1}{{\left(mn\right)}^j}}f(m^{j}x,n^{j}y)& -{\displaystyle \frac{1}{{\left(mn\right)}^{j+1}}}f(m^{j+1}x,n^{j+1}y){\parallel }_B\hfill \\ & \le {\displaystyle \frac{\theta }{{\left(mn\right)}^{j+1}}}(m^{1+jp}{\parallel x\parallel }_A^p+n^{1+jp}{\parallel y\parallel }_A^p)\hfill \end{array}$$

for all $x,y\in A$ and all $j=0,1,2,\cdots$. For given integers $l,k(0\le l<k)$, we obtain that

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{1}{{\left(mn\right)}^l}}f(m^{l}x,n^{l}y)& -{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y){\parallel }_B\hfill \\ & \le \sum _{j=l}^{k-1}{\displaystyle \frac{\theta }{{\left(mn\right)}^{j+1}}}(m^{1+jp}{\parallel x\parallel }_A^p+n^{1+jp}{\parallel y\parallel }_A^p)\hfill \end{array}$$

(8)

for all $x,y\in A$. According to the above inequality, the sequence $\left\{{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y)\right\}$ is a Cauchy sequence for all $x,y\in A$. Since B is complete, the sequence $\left\{{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y)\right\}$ converges for all $x,y\in A$. Define $H:A\times A\to B$ as

$$H(x,y):=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y)$$

for all $x,y\in A$. According to (4), we have

$$\begin{array}{cc}& \parallel D_{\lambda ,\mu }H(x_1,\cdots ,x_m,y_1,\cdots ,y_n){\parallel }_B\hfill \\ & =\parallel H\left(\lambda \sum _{i=1}^m{x}_i,\mu \sum _{j=1}^n{y}_j\right)-\lambda \mu \sum _{i=1}^m\sum _{j=1}^{n}H(x_i,y_j){\parallel }_B\hfill \\ & =\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}\parallel f\left(m^k\lambda \sum _{i=1}^m{x}_i,n^k\mu \sum _{j=1}^n{y}_j\right)-\lambda \mu \sum _{i=1}^m\sum _{j=1}^{n}f(m^k{x}_i,n^k{y}_j){\parallel }_B\hfill \\ & \le \underset{k\to \infty }{\lim }{\displaystyle \frac{\theta }{{\left(mn\right)}^k}}\left(\sum _{i=1}^m\parallel m^k{x}_i{\parallel }_A^p+\sum _{j=1}^n{\parallel n^k{y}_j\parallel }_A^p\right)\hfill \\ & =\theta \underset{k\to \infty }{\lim }\left[{\left({\displaystyle \frac{m^p}{mn}}\right)}^k\sum _{i=1}^m\parallel x_i{\parallel }_A^p+{\left({\displaystyle \frac{n^p}{mn}}\right)}^k\sum _{j=1}^n{\parallel y_j\parallel }_A^p\right]=0\hfill \end{array}$$

for all $\lambda ,\mu \in \mathbb{C}$ with $\left|\lambda \right|=\left|\mu \right|=1$ and all $x_1,\cdots ,x_m,y_1,\cdots ,y_n\in A$. Thus, the mapping $H:A\times A\to B$ is $\mathbb{C}$-bilinear. Setting $l=0$ and taking $k\to \infty$ in (8), one can obtain the inequality (6).

It follows from (5) that

$$\begin{array}{cc}& {\parallel H([x,y,z],[u,v,w])-[H(x,u),H(y,v),H(z,w)]\parallel }_B\hfill \\ \hfill =& \underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}\parallel f(m^k[x,y,z],n^k[u,v,w])\hfill \\ & -[f(m^{k}x,n^{k}u),f(m^{k}y,n^{k}v),f(m^{k}z,n^{k}w)]{\parallel }_B\hfill \\ \hfill \le & \underset{k\to \infty }{\lim }{\displaystyle \frac{\theta }{{\left(mn\right)}^k}}\left[m^{kp}{(\parallel x\parallel }_A^p+{\parallel y\parallel }_A^p+{\parallel z\parallel }_A^p)+n^{kp}{(\parallel u\parallel }_A^p+{\parallel v\parallel }_A^p+{\parallel w\parallel }_A^p)\right]=0\hfill \end{array}$$

for all $x,y,z,u,v,w\in A$. Thus, H satisfies the Equation (2).

Now, let $T:A\times A\to B$ be another $C^{*}$-ternary algebra bi-homomorphism satisfying (6). Then, we have

$$\begin{array}{cc}& {\parallel H(x,y)-T(x,y)\parallel }_B={\displaystyle \frac{1}{{\left(mn\right)}^k}}\parallel H\left(m^{k}x,n^{k}y\right)-T\left(m^{k}x,n^{k}y\right){\parallel }_B\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\left(mn\right)}^k}}\left[\parallel H\left(m^{k}x,n^{k}y\right)-f\left(m^{k}x,n^{k}y\right){\parallel }_B+\parallel f\left(m^{k}x,n^{k}y\right)-T\left(m^{k}x,n^{k}y\right){\parallel }_B\right]\hfill \\ \hfill \le & {\displaystyle \frac{2\theta }{{\left(mn\right)}^k}}\left({\displaystyle \frac{m^{1+kp}}{mn-m^p}}{\parallel x\parallel }_A^p+{\displaystyle \frac{n^{1+kp}}{mn-n^p}}{\parallel y\parallel }_A^p\right),\hfill \end{array}$$

which tends to zero as $k\to \infty$ for all $x,y\in A$. So, we can conclude that $H(x,y)=T(x,y)$ for all $x,y\in A$. This proves the uniqueness of H.

Thus, the mapping H is a unique $C^{*}$-ternary algebra bi-homomorphism satisfying (6). □

Remark 1.

In Theorem 1, instead of the assumptions (3) and (4), under the conditions

$${∥D_{\lambda ,\mu }f(x_1,\dots ,x_m,y_1,\dots ,y_n)∥}_B\le \theta \left(\sum _{i=1}^m\parallel x_i{\parallel }_A^p+\sum _{j=1}^n{\parallel y_j\parallel }_A^q\right),$$

and

$$\begin{array}{cc}\hfill \parallel f\left(\right[x,y,z],[u,v,w\left]\right)& -[f(x,u),f(y,v),f(z,w)]{\parallel }_B\hfill \\ & \le \theta \left({\parallel x\parallel }_A^p+{\parallel y\parallel }_A^p+{\parallel z\parallel }_A^p+{\parallel u\parallel }_A^q+{\parallel v\parallel }_A^q+{\parallel w\parallel }_A^q\right),\hfill \end{array}$$

where $p<{\log }_{m}n+1$ and $q<{\log }_{n}m+1$ are positive real numbers, we have a similar result.

Example 1.

Let $A=B={\mathbb{R}}^2$, and define the ternary product as

$$[x,y,z]:=(x·y)z,$$

where $x·y$ is the usual Euclidean inner product. We define a function

$$f(x,y):=\sigma (x·y)(1,0),$$

which maps any two vectors to a 2D vector whose first component reflects a bounded “score” depending on how aligned the inputs are, where $\sigma \left(x\right):={\displaystyle \frac{1}{1+e^{-x}}}$ is the sigmoid function. This mimics user–item scoring in recommender systems with saturation.

The sigmoid function is Lipschitz continuous and bounded by 1, and hence, both additive and ternary deviations of f are bounded in terms of ${\parallel x\parallel }^p+{\parallel y\parallel }^p$ for $p<1$. Therefore, f satisfies the hypotheses of Theorem 1 and can be approximated by a unique bi-homomorphism H.

Example 2.

In quantum mechanics, a ternary product structure naturally arises in the form

$$[x,y,z]:=\langle x,y\rangle z,$$

which can represent a ternary state interaction where x and y describe incoming quantum states, and z is a resultant state scaled by transition amplitude. Let $A=B={\mathbb{C}}^2$, representing a single-qubit Hilbert space.

We define a nonlinear mapping

$$f(x,y):=\left({\displaystyle \frac{{\left|\langle x,y\rangle \right|}^2}{{1+\parallel x+y\parallel }^2}}\right)(1,0),$$

which reflects a normalized transition probability between states x and y. This function is smooth and bounded, and it satisfies the assumptions of Theorem 1 due to the boundedness of the numerator and the regularizing denominator.

Therefore, despite the nonlinear quantum nature of f, it can be approximated by a unique bi-homomorphism H, indicating that complex ternary couplings in quantum systems can be robustly modeled within a stable algebraic framework.

Next, we prove the generalized Hyers–Ulam stability of the functional Equation (3) in $C^{*}$-ternary algebras for $p>1+\max \{{\log }_{m}n,{\log }_{n}m\}$.

Theorem 2.

Let $p,\theta$ be positive real numbers with $p>1+\max \{{\log }_{m}n,{\log }_{n}m\}$, and let $f:A\times A\to B$ be a mapping satisfying (4) and (5). Then there exists a unique $C^{*}$-ternary algebra bi-homomorphism $H:A\times A\to B$ such that

$${\parallel f(x,y)-H(x,y)\parallel }_B\le \theta \left({\displaystyle \frac{m}{m^p-mn}}{\parallel x\parallel }_A^p+{\displaystyle \frac{n}{n^p-mn}}{\parallel y\parallel }_A^p\right)$$

for all $x,y\in A$.

Proof. 

By the same argument as in the proof of Theorem 1, one can obtain the inequality (7). It follows from (7) that

$$\parallel f(x,y)-mnf\left({\displaystyle \frac{x}{m}},{\displaystyle \frac{y}{n}}\right){\parallel }_B\le \theta (m^{1-p}{\parallel x\parallel }_A^p+n^{1-p}{\parallel y\parallel }_A^p)$$

for all $x,y\in A$. So

$$\begin{array}{cc}\hfill \parallel {\left(mn\right)}^{l}f\left({\displaystyle \frac{x}{m^l}},{\displaystyle \frac{y}{n^l}}\right)& -{\left(mn\right)}^{k}f\left({\displaystyle \frac{x}{m^k}},{\displaystyle \frac{y}{n^k}}\right){\parallel }_B\hfill \\ & \le \sum _{j=l}^{k-1}\parallel {\left(mn\right)}^{j}f\left({\displaystyle \frac{x}{m^j}},{\displaystyle \frac{y}{n^j}}\right)-{\left(mn\right)}^{j+1}f\left({\displaystyle \frac{x}{m^{j+1}}},{\displaystyle \frac{y}{n^{j+1}}}\right){\parallel }_B\hfill \\ & \le \theta \sum _{j=l}^{k-1}{\left(mn\right)}^j\left(m^{1-p(j+1)}{\parallel x\parallel }_A^p+n^{1-p(j+1)}{\parallel y\parallel }_A^p\right)\hfill \end{array}$$

(9)

for all nonnegative integers l and k with $l<k$ and all $x,y\in A$. It follows from (9) that the sequence $\left\{{\left(mn\right)}^{k}f({\displaystyle \frac{x}{m^k}},{\displaystyle \frac{y}{n^k}})\right\}$ is a Cauchy sequence for all $x,y\in A$. Since B is complete, the sequence $\left\{{\left(mn\right)}^{k}f({\displaystyle \frac{x}{m^k}},{\displaystyle \frac{y}{n^k}})\right\}$ converges for all $x,y\in A$. So, one can define the mapping $H:A\times A\to B$ as

$$H(x,y):=\underset{k\to \infty }{\lim }{\left(mn\right)}^{k}f\left({\displaystyle \frac{x}{m^k}},{\displaystyle \frac{y}{n^k}}\right)$$

for all $x,y\in A$. Moreover, letting $l=0$ and passing the limit $k\to \infty$ in (9), we get the desired inequality for f and H.

The rest of the proof is similar to the proof of Theorem 1. □

We prove the generalized Hyers–Ulam stability of the functional Equation (3) in $C^{*}$-ternary algebras for $p<\min \left\{{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{n}}\right\}$.

Theorem 3.

Let $p,\theta$ be positive real numbers with $p<\min \left\{{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{n}}\right\}$, and let $f:A\times A\to B$ be a mapping such that

$$\parallel D_{\lambda ,\mu }f(x_1,\cdots ,x_m,y_1,\cdots ,y_n){\parallel }_B\le \theta \parallel x_1{\parallel }_A^p\cdots \parallel x_m{\parallel }_A^p\parallel y_1{\parallel }_A^p\cdots {\parallel y_n\parallel }_A^p,$$

(10)

$$\begin{array}{c}\parallel f([x,y,z],[u,v,w])-[f(x,u),f(y,v),f(z,w)]{\parallel }_B\hfill \\ \le {\theta \parallel x\parallel }_A^p{\parallel y\parallel }_A^p{\parallel z\parallel }_A^p{\parallel u\parallel }_A^p{\parallel v\parallel }_A^p{\parallel w\parallel }_A^p\hfill \end{array}$$

(11)

for all $\lambda ,\mu \in \mathbb{C}$ with $\left|\lambda \right|=\left|\mu \right|=1$ and all $x_1,\cdots ,x_m,y_1,\cdots ,y_n,x,y,z,u,v,w\in A$. Then there exists a unique $C^{*}$-ternary algebra bi-homomorphism $H:A\times A\to B$ such that

$${\parallel f(x,y)-H(x,y)\parallel }_B\le {\displaystyle \frac{\theta }{mn-{\left(m^m{n}^n\right)}^p}}{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p$$

(12)

for all $x,y\in A$.

Proof. 

Letting $\lambda =\mu =1$, $x_1=\cdots =x_m=x$ and $y_1=\cdots =y_n=y$ in (11), we gain

$${\parallel f(mx,ny)-mnf(x,y)\parallel }_B\le \theta {\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p$$

(13)

for all $x,y\in A$. Replacing x with $m^{j}x$ and y with $n^{j}y$ and dividing ${\left(mn\right)}^{j+1}$ in the above inequality, we obtain that

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{1}{{\left(mn\right)}^j}}f(m^{j}x,n^{j}y)& -{\displaystyle \frac{1}{{\left(mn\right)}^{j+1}}}f(m^{j+1}x,n^{j+1}y){\parallel }_B\hfill \\ & \le {\displaystyle \frac{\theta }{mn}}{\left(m^{mp-1}{n}^{np-1}\right)}^j{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p\hfill \end{array}$$

for all $x,y\in A$ and all $j=0,1,2,\cdots$. For the given integers $l,k(0\le l<k)$, we obtain that

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{1}{{\left(mn\right)}^l}}f(m^{l}x,n^{l}y)& -{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y){\parallel }_B\hfill \\ & \le \sum _{j=l}^{k-1}{\displaystyle \frac{\theta }{mn}}{\left(m^{mp-1}{n}^{np-1}\right)}^j{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p\hfill \end{array}$$

(14)

for all $x,y\in A$. According to the above inequality, the sequence $\left\{{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y)\right\}$ is a Cauchy sequence for all $x,y\in A$. Since B is complete, the sequence $\left\{{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y)\right\}$ converges for all $x,y\in A$. Define $H:A\times A\to B$ as

$$H(x,y):=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y)$$

for all $x,y\in A$. According to (10), we have

$$\begin{array}{cc}\hfill \parallel D_{\lambda ,\mu }& H(x_1,\cdots ,x_m,y_1,\cdots ,y_n){\parallel }_B\hfill \\ & =\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}\parallel f\left(m^k\lambda \sum _{i=1}^m{x}_i,n^k\mu \sum _{j=1}^n{y}_j\right)-\lambda \mu \sum _{i=1}^m\sum _{j=1}^{n}f(m^k{x}_i,n^k{y}_j){\parallel }_B\hfill \\ & \le \theta \parallel x_1{\parallel }_A^p\cdots \parallel x_m{\parallel }_A^p\parallel y_1{\parallel }_A^p\cdots {\parallel y_n\parallel }_A^p\underset{k\to \infty }{\lim }{\left(m^{mp-1}{n}^{np-1}\right)}^k=0\hfill \end{array}$$

for all $\lambda ,\mu \in \mathbb{C}$ with $\left|\lambda \right|=\left|\mu \right|=1$ and all $x_1,\cdots ,x_m,y_1,\cdots ,y_n\in A$. Thus the mapping $H:A\times A\to B$ is $\mathbb{C}$-bilinear. Setting $l=0$ and taking $k\to \infty$ in (14), one can obtain the inequality (12). It follows from (11) that

$$\begin{array}{cc}\hfill \parallel H\left(\right[x,y,& {z],[u,v,w])-[H(x,u),H(y,v),H(z,w)]\parallel }_B\hfill \\ & =\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}\parallel f(m^k[x,y,z],n^k[u,v,w])\hfill \\ & -[f(m^{k}x,n^{k}u),f(m^{k}y,n^{k}v),f(m^{k}z,n^{k}w)]{\parallel }_B\hfill \\ & \le {\theta \parallel x\parallel }_A^p{\parallel y\parallel }_A^p{\parallel z\parallel }_A^p{\parallel u\parallel }_A^p{\parallel v\parallel }_A^p{\parallel w\parallel }_A^p\underset{k\to \infty }{\lim }{\left(mn\right)}^{(3p-1)k}=0\hfill \end{array}$$

for all $x,y,z,u,v,w\in A$. Thus, H satisfies the Equation (2).

Now, let $T:A\times A\to B$ be another $C^{*}$-ternary algebra bi-homomorphism satisfying (12). Then, we have

$$\begin{array}{cc}\hfill {\parallel H(x,y)-T(x,y)\parallel }_B& ={\displaystyle \frac{1}{{\left(mn\right)}^k}}\parallel H\left(m^{k}x,n^{k}y\right)-T\left(m^{k}x,n^{k}y\right){\parallel }_B\hfill \\ & \le {\displaystyle \frac{2\theta {\left(m^{mp-1}{n}^{np-1}\right)}^k}{mn-{\left(m^m{n}^n\right)}^p}}{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p,\hfill \end{array}$$

which tends to zero as $k\to \infty$ for all $x,y\in A$. Hence $H(x,y)=T(x,y)$ for all $x,y\in A$.

Therefore, H is a unique $C^{*}$-ternary algebra bi-homomorphism satisfying (12). □

Remark 2.

In Theorem 3, instead of the assumptions (10) and (11), under the conditions

$$\begin{array}{cc}& \parallel D_{\lambda ,\mu }f(x_1,\cdots ,x_m,y_1,\cdots ,y_n){\parallel }_B\le \theta \parallel x_1{\parallel }_A^p\cdots \parallel x_m{\parallel }_A^p\parallel y_1{\parallel }_A^q\cdots {\parallel y_n\parallel }_A^p,\hfill \\ & \parallel f([x,y,z],[u,v,w])-[f(x,u),f(y,v),f(z,w)]{\parallel }_B\hfill \\ & \le {\theta \parallel x\parallel }_A^s{\parallel y\parallel }_A^s{\parallel z\parallel }_A^s{\parallel u\parallel }_A^s{\parallel v\parallel }_A^s{\parallel w\parallel }_A^s,\hfill \end{array}$$

where $p<{\displaystyle \frac{1}{m}}$, $q<{\displaystyle \frac{1}{n}}$, and $s<{\displaystyle \frac{1}{3}}$ are positive real numbers, we have a similar result.

Next, we prove the generalized Hyers–Ulam stability of the functional Equation (3) in $C^{*}$-ternary algebras for $p>{\displaystyle \frac{1+{\log }_{m}n}{m+n{\log }_{m}n}}$.

Theorem 4.

Let $p,\theta$ be positive real numbers with $p>{\displaystyle \frac{1+{\log }_{m}n}{m+n{\log }_{m}n}}$, and let $f:A\times A\to B$ be a mapping satisfying (10) and (11). Then there exists a unique $C^{*}$-ternary algebra bi-homomorphism $H:A\times A\to B$ such that

$${\parallel f(x,y)-H(x,y)\parallel }_B\le {\displaystyle \frac{\theta }{{\left(m^m{n}^n\right)}^p-mn}}{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p$$

for all $x,y\in A$.

Proof. 

It follows from (13) that

$$\parallel f(x,y)-mnf\left({\displaystyle \frac{x}{m}},{\displaystyle \frac{y}{n}}\right){\parallel }_B\le {\displaystyle \frac{\theta }{{\left(m^m{n}^n\right)}^p}}{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p$$

for all $x,y\in A$. For the given integers $l,k(0\le l<k)$, we obtain that

$$\begin{array}{cc}\hfill \parallel {\left(mn\right)}^{l}f\left({\displaystyle \frac{x}{m^l}},{\displaystyle \frac{y}{n^l}}\right)& -{\left(mn\right)}^{k}f\left({\displaystyle \frac{x}{m^k}},{\displaystyle \frac{y}{n^k}}\right){\parallel }_B\hfill \\ & \le \sum _{j=l}^{k-1}\parallel {\left(mn\right)}^{j}f\left({\displaystyle \frac{x}{m^j}},{\displaystyle \frac{y}{n^j}}\right)-{\left(mn\right)}^{j+1}f\left({\displaystyle \frac{x}{m^{j+1}}},{\displaystyle \frac{y}{n^{j+1}}}\right){\parallel }_B\hfill \\ & \le {\displaystyle \frac{\theta }{{\left(m^m{n}^n\right)}^p}}\sum _{j=l}^{k-1}{\left(m^{1-mp}{n}^{1-np}\right)}^j{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p\hfill \end{array}$$

(15)

for all $x,y\in A$. According to the above inequality, the sequence $\left\{{\left(mn\right)}^{k}f\left({\displaystyle \frac{x}{m^k}},{\displaystyle \frac{y}{n^k}}\right)\right\}$ is a Cauchy sequence for all $x,y\in A$. Since B is complete, the sequence $\left\{{\left(mn\right)}^{k}f\left({\displaystyle \frac{x}{m^k}},{\displaystyle \frac{y}{n^k}}\right)\right\}$ converges for all $x,y\in A$. Define $H:A\times A\to B$ as

$$H(x,y):=\underset{k\to \infty }{\lim }{\left(mn\right)}^{k}f\left({\displaystyle \frac{x}{m^k}},{\displaystyle \frac{y}{n^k}}\right)$$

for all $x,y\in A$.

Setting $l=0$ and taking $k\to \infty$ in (15), one can obtain the inequality for f and H. The rest of the proof is similar to the proof of Theorem 3. □

We prove the stability of bi-derivations and bi-isomorphisms in $C^{*}$-ternary algebras. We established the existence and uniqueness of these mappings, which approximate given mappings within specified bounds.

Theorem 5.

Let $p,\theta$ be positive real numbers with $p<1$, and let $f:A\times A\to A$ be a mapping satisfying (4) such that

$$\begin{array}{cc}\hfill \parallel f([x,y,z],w)-[f(x,w),y,z]-[x,f(y,w^{*}),& z]-[x,y,f(z,w)]{\parallel }_A\hfill \\ & \le \theta \left({\parallel x\parallel }_A^p+{\parallel y\parallel }_A^p+{\parallel z\parallel }_A^p+{\parallel w\parallel }_A^p\right)\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\hfill \parallel f(x,[y,z,w])-[f(x,y),z,w]-[y,f(x^{*},z),& w]-[y,z,f(x,w)]{\parallel }_A\hfill \\ & \le \theta \left({\parallel x\parallel }_A^p+{\parallel y\parallel }_A^p+{\parallel z\parallel }_A^p+{\parallel w\parallel }_A^p\right)\hfill \end{array}$$

(17)

for all $x,y,z,w\in A$. If f satisfies

$$\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{k}x,n^{k}y\right)=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{3k}x,n^{k}y\right)=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{k}x,n^{3k}y\right)$$

(18)

for all $x,y\in A$, then there is a unique $C^{*}$-ternary bi-derivation $\delta :A\times A\to A$ such that

$${\parallel f(x,y)-\delta (x,y)\parallel }_A\le \theta \left({\displaystyle \frac{m}{mn-m^p}}{\parallel x\parallel }_A^p+{\displaystyle \frac{n}{mn-n^p}}{\parallel y\parallel }_A^p\right)$$

for all $x,y\in A$.

Proof. 

According to (4), we have $f(0,0)=0$. According to the proof of Theorem 1, the sequence $\left\{{\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{k}x,n^{k}y\right)\right\}$ is a Cauchy sequence for all $x,y\in A$. Since A is complete, the sequence $\left\{{\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{k}x,n^{k}y\right)\right\}$ converges for all $x,y\in A$. So, one can define the mapping $\delta :A\times A\to A$ as

$$\delta (x,y):=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{k}x,n^{k}y\right)$$

for all $x,y\in A$. By the same reasoning as in the proof of Theorem 1, we get the desired inequality for f and $\delta$ and the fact that $\delta$ is $\mathbb{C}$-bilinear.

It follows from (16)–(18) that

$$\begin{array}{ccc}& & \parallel \delta ([x,y,z],w)-[\delta (x,w),y,z]-[x,\delta (y,w^{*}),z]-[x,y,\delta (z,w)]{\parallel }_A\hfill \\ & & =\underset{k\to \infty }{\lim }(\parallel {\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{3k}[x,y,z],n^{k}w\right)-\left[{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}w),m^{k}y,m^{k}z\right]\hfill \\ & & -\left[m^{k}x,{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}y,n^k{w}^{*}),m^{k}z\right]-\left[m^{k}x,m^{k}y,{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}z,n^{k}w)\right]{\parallel }_A)\hfill \\ & & =\underset{k\to \infty }{\lim }(\parallel {\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left([m^{k}x,m^{k}y,m^{k}z],n^{k}w\right)-{\displaystyle \frac{1}{{\left(mn\right)}^k}}\left[f\left(m^{k}x,n^{k}w\right),m^{k}y,m^{k}z\right]\hfill \\ & & -{\displaystyle \frac{1}{{\left(mn\right)}^k}}\left[m^{k}x,f\left(m^{k}y,n^{k}w\right),m^{k}z\right]-{\displaystyle \frac{1}{{\left(mn\right)}^k}}\left[m^{k}x,m^{k}y,f\left(m^{k}z,n^{k}w\right)\right]{\parallel }_A)\hfill \\ & & \le \underset{k\to \infty }{\lim }{\displaystyle \frac{\theta }{{\left(mn\right)}^k}}\left(m^{kp}{\parallel x\parallel }_A^p+m^{kp}{\parallel y\parallel }_A^p+m^{kp}{\parallel z\parallel }_A^p+n^{kp}{\parallel w\parallel }_A^p\right)=0\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \parallel \delta (x,[y,z,w])-[\delta (x,y),z,w]-[y,\delta (x^{*},z),w]-[y,z,\delta (x,w)]{\parallel }_A\hfill \\ & & =\underset{k\to \infty }{\lim }(\parallel {\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{k}x,n^{3k}[y,z,w]\right)-\left[{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y),n^{k}z,n^{k}w\right]\hfill \\ & & -\left[n^{k}y,{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^k{x}^{*},n^{k}z),n^{k}w\right]-\left[n^{k}y,n^{k}z,{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}w)\right]{\parallel }_A)\hfill \\ & & =\underset{k\to \infty }{\lim }(\parallel {\displaystyle \frac{1}{{\left(mn\right)}^k}}f\left(m^{k}x,[n^{k}y,n^{k}z,n^{k}w]\right)-{\displaystyle \frac{1}{{\left(mn\right)}^k}}\left[f\left(m^{k}x,n^{k}y\right),n^{k}z,n^{k}w\right]\hfill \\ & & -{\displaystyle \frac{1}{{\left(mn\right)}^k}}\left[n^{k}y,f\left(m^{k}x,n^{k}z\right),n^{k}w\right]-{\displaystyle \frac{1}{{\left(mn\right)}^k}}\left[n^{k}y,n^{k}z,f\left(m^{k}x,n^{k}w\right)\right]{\parallel }_A)\hfill \\ & & \le \underset{k\to \infty }{\lim }{\displaystyle \frac{\theta }{{\left(mn\right)}^k}}\left(m^{kp}{\parallel x\parallel }_A^p+n^{kp}{\parallel y\parallel }_A^p+n^{kp}{\parallel z\parallel }_A^p+n^{kp}{\parallel w\parallel }_A^p\right)=0\hfill \end{array}$$

for all $x,y,z,w\in A$. Thus, $\delta$ is a $C^{*}$-ternary bi-derivation.

By the same argument as in the proof of Theorem 1, the mapping $\delta$ is a unique $C^{*}$-ternary bi-derivation satisfying (16) and (17). □

In Theorem 5, for the case $p>1+\max \{{\log }_{m}n,{\log }_{n}m\}$, one can obtain a similar result.

Theorem 6.

Let $p,\theta$ be positive real numbers satisfying

$$p<\min \left\{{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{n}}\right\}\mathrm{or}p>\max \left\{{\displaystyle \frac{1+{\log }_{m}n}{m+n{\log }_{m}n}},{\displaystyle \frac{1+{\log }_{m}n}{3+{\log }_{m}n}},{\displaystyle \frac{1+{\log }_{n}m}{3+{\log }_{n}m}}\right\}$$

and let $f:A\times A\to A$ be a mapping satisfying (10) such that

$$\begin{array}{cc}\hfill \parallel f([x,y,z],w)-[f(x,w),y,z]-[x,f(y,w^{*}),& z]-[x,y,f(z,w)]{\parallel }_A\hfill \\ & \le {\theta \parallel x\parallel }_A^p{\parallel y\parallel }_A^p{\parallel z\parallel }_A^p{\parallel w\parallel }_A^p\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \parallel f(x,[y,z,w])-[f(x,y),z,w]-[y,f(x^{*},z),& w]-[y,z,f(x,w)]{\parallel }_A\hfill \\ & \le {\theta \parallel x\parallel }_A^p{\parallel y\parallel }_A^p{\parallel z\parallel }_A^p{\parallel w\parallel }_A^p\hfill \end{array}$$

for all $x,y,z,w\in A$.

Then there exists a unique $C^{*}$-ternary bi-derivation $\delta :A\times A\to A$ such that

$$\begin{array}{cc}\hfill \parallel f& {(x,y)-\delta (x,y)\parallel }_B\hfill \\ & \le \left\{\begin{array}{cc}{\displaystyle \frac{\theta }{mn-{\left(m^m{n}^n\right)}^p}}{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p\hfill & \left(p<\min \left\{{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{n}}\right\}\right)\hfill \\ {\displaystyle \frac{\theta }{{\left(m^m{n}^n\right)}^p-mn}}{\left({\parallel x\parallel }_A^m{\parallel y\parallel }_A^n\right)}^p\hfill & \left(p>\max \left\{{\displaystyle \frac{1+{\log }_{m}n}{m+n{\log }_{m}n}},{\displaystyle \frac{1+{\log }_{m}n}{3+{\log }_{m}n}},{\displaystyle \frac{1+{\log }_{n}m}{3+{\log }_{n}m}}\right\}\right)\hfill \end{array}\right.\hfill \end{array}$$

for all $x,y\in A$.

Proof. 

Let $p<\min \{{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{n}}\}$. By the same argument as in the proof of Theorem 3, one can define the mapping $\delta :A\times A\to A$ as

$$\delta (x,y):=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y)$$

for all $x,y\in A$ satisfying the desired inequality for f and $\delta$. The rest of the proof is similar to the proof of Theorem 5.

If $p>\max \left\{{\displaystyle \frac{1+{\log }_{m}n}{m+n{\log }_{m}n}},{\displaystyle \frac{1+{\log }_{m}n}{3+{\log }_{m}n}},{\displaystyle \frac{1+{\log }_{n}m}{3+{\log }_{n}m}}\right\}$, the proof is similar to the proofs of Theorems 4 and 5 and the case $p<\min \{{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{n}}\}$. □

From now on, assume that A is a unital $C^{*}$-ternary algebra with the norm ${\parallel ·\parallel }_A$ and unit e, and that B is a unital $C^{*}$-ternary algebra with the norm ${\parallel ·\parallel }_B$ and unit $e^{\prime }$.

We examine bi-isomorphisms between $C^{*}$-ternary algebras related to the equation $D_{\lambda ,\mu }f(x,y,z,w)=0$.

Theorem 7.

Let $p,\theta$ be positive real numbers satisfying $p<1$ or $p>1+\max \{{\log }_{m}n,{\log }_{n}m\}$. If a mapping $f:A\times A\to B$ is bijective and satisfies (4), (5) and $f(e,e)=e^{\prime }$, where e and $e^{\prime }$ are units in A and B, respectively, then the mapping f is a $C^{*}$-ternary algebra bi-isomorphism.

Proof. 

Define $H:A\times A\to B$ as

$$H(x,y):=\left\{\begin{array}{cc}{\lim }_{k\to \infty }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,m^{k}y)\hfill & (p<1)\hfill \\ {\lim }_{k\to \infty }{\left(mn\right)}^{k}f\left({\displaystyle \frac{1}{m^k}}x,{\displaystyle \frac{1}{m^k}}y\right)\hfill & (p>1+\max \{{\log }_{m}n,{\log }_{n}m\})\hfill \end{array}\right.$$

for all $x,y\in A$. By the same reasoning as in the proof of Theorems 1 and 2, the mapping H is a $C^{*}$-ternary algebra bi-homomorphism.

For the case $p<1$, it follows from (5) that

$$\begin{array}{cc}\hfill H(x,y)& =H([e,e,x],[e,e,y])=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^k[e,e,x],n^k[e,e,y])\hfill \\ & =\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}f([e,e,m^{k}x],[e,e,n^{k}y])\hfill \\ & =\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(mn\right)}^k}}[f(e,e),f(e,e),f(m^{k}x,n^{k}y)]\hfill \\ & =\underset{k\to \infty }{\lim }\left[f(e,e),f(e,e),{\displaystyle \frac{1}{{\left(mn\right)}^k}}f(m^{k}x,n^{k}y)\right]\hfill \\ & =[e^{\prime },e^{\prime },f(x,y)]=f(x,y)\hfill \end{array}$$

for all $x,y\in A$. Hence, the bijective mapping $f:A\times A\to B$ is a $C^{*}$-ternary algebra bi-isomorphism.

Similarly, the bijective mapping $f:A\times A\to B$ is also a $C^{*}$-ternary algebra bi-isomorphism for the case $p>1+\max \{{\log }_{m}n,{\log }_{n}m\}$. □

Theorem 8.

Let $p,\theta$ be positive real numbers satisfying $p<\min \{{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{m}},{\displaystyle \frac{1}{n}}\}$ or $p>{\displaystyle \frac{1+{\log }_{m}n}{m+n{\log }_{m}n}}$. If a mapping $f:A\times A\to B$ is bijective and satisfies (10), (11) and $f(e,e)=e^{\prime }$, where e and $e^{\prime }$ are units in A and B, respectively, then the mapping f is a $C^{*}$-ternary algebra bi-isomorphism.

Proof. 

The proof is similar to the proof of Theorem 7. □

4. Applications and Discussion

Our results are relevant to the analysis of functional structures in operator theory and quantum algebra. In particular, the stability of bi-isomorphisms provides mathematical justification for approximations in physical models involving ternary operations, such as those found in Nambu mechanics and ternary quantum systems.

5. Conclusions

In this study, we presented a refined analysis of Hyers–Ulam stability for functional equations in two dimensional $C^{*}$-ternary algebras. Through a generalized norm inequality approach, we ensured the existence and uniqueness of bi-homomorphic and bi-isomorphic approximations. The significance of our work is treating functional equations with arbitrary numbers of variables and applying them to bi-homomorphisms and bi-derivations in $C^{*}$-ternary algebraic settings. This application of functional equations with arbitrary numbers of variables has not been previously examined. Directions for future work include extending these results to topological and categorical settings.