# Multi-$$C^{*}$$-ternary algebras and applications

## Abstract

In this paper, we look at the concept of multi-$$C^{*}$$-ternary algebras and consider some properties. As an application we approximate multi-$$C^{*}$$-ternary algebra homomorphisms and derivations in these spaces.

## Introduction and preliminaries

Ternary algebraic structures arise naturally in theoretical and mathematical physics, for example, the quark model inspired a particular brand of ternary algebraic system. We also refer the reader to ‘Nambu mechanics’  (see also [2, 3] and ).

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 C-linear in the outer variables, conjugate C-linear in the middle variable, and associative in the sense that $$[x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v]$$, and satisfies $$\|[x, y, z]\| \le\|x\| \cdot\|y\| \cdot \|z\|$$ and $$\|[x, x, x]\| = \|x\|^{3}$$ (see ).

If a $$C^{*}$$-ternary algebra $$(A, [\cdot, \cdot, \cdot] )$$ has the identity, i.e., the element $$e\in A$$ such that $$x = [x, e, e] = [e, e, x]$$ for all $$x\in A$$, then it is routine to verify that A, endowed with $$x\circ y : = [x, e, y]$$ and $$x^{*}:=[e, x, e]$$, is a unital $$C^{*}$$-algebra. Conversely, if $$(A, \circ)$$ is a unital $$C^{*}$$-algebra, then $$[x, y, z] : = x \circ y^{*} \circ z$$ makes A into a $$C^{*}$$-ternary algebra.

A C-linear mapping $$H: A \rightarrow B$$ is called a $$C^{*}$$-ternary algebra homomorphism if

$$H\bigl([x, y, z]\bigr) = \bigl[H(x), H(y), H(z)\bigr]$$

for all $$x, y, z \in A$$. A C-linear mapping $$\delta: A \rightarrow A$$ is called a $$C^{*}$$-ternary derivation if

$$\delta\bigl([x, y, z]\bigr) = \bigl[\delta(x), y, z\bigr] + \bigl[x, \delta(y), z\bigr] + \bigl[x, y, \delta(z)\bigr]$$

for all $$x, y, z \in A$$ (see ).

Ternary structures and their generalization, the so-called n-ary structures, are important in view of their applications in physics (see ).

Let X be a set. A function $$d: X \times X \rightarrow[0, \infty]$$ is called a generalized metric on X if d satisfies the following conditions:

1. (1)

$$d(x, y) = 0$$ if and only if $$x=y$$;

2. (2)

$$d(x, y) = d(y, x)$$ for all $$x, y \in X$$;

3. (3)

$$d(x, z) \le d(x, y) + d(y, z)$$ for all $$x, y, z\in X$$.

### Theorem 1.1

()

Let $$(X, d)$$ be a complete generalized metric space and let $$J: X \rightarrow X$$ be a strictly contractive mapping with Lipschitz constant $$L<1$$. Then, for each $$x\in X$$, either

$$d\bigl(J^{n} x, J^{n+1} x\bigr) = \infty$$

for all non-negative integers n or there exists a positive integer $$n_{0}$$ such that

1. (1)

$$d(J^{n} x, J^{n+1}x) <\infty$$ for all $$n\ge n_{0}$$;

2. (2)

the sequence $$\{J^{n} x\}$$ converges to a fixed point $$y^{*}$$ of J;

3. (3)

$$y^{*}$$ is the unique fixed point of J in the set $$Y = \{y\in X \mid d(J^{n_{0}} x, y) <\infty\}$$;

4. (4)

$$d(y, y^{*}) \le\frac{1}{1-L} d(y, Jy)$$ for all $$y \in Y$$.

## Multi-normed spaces

The notion of a multi-normed space was introduced by Dales and Polyakov in  and many examples are given in .

Let $$( {\mathcal{E}},\|\cdot\|)$$ be a complex normed space and let $$k\in\mathbf{N}$$. We denote by $$\mathcal{E}^{k}$$ the linear space $$\mathcal{E}\oplus\cdots\oplus\mathcal{E}$$ consisting of k-tuples $$(x_{1}, \ldots, x_{k})$$, where $$x_{1}, \ldots, x_{k}\in\mathcal{E}$$. The linear operations on $$\mathcal{E}^{k}$$ are defined coordinate-wise. The zero element of either $$\mathcal{E}$$ or $$\mathcal{E}^{k}$$ is denoted by 0. We denote by $$\mathbf{N}_{k}$$ the set $$\{1, 2, \ldots ,k\}$$ and by $$\Sigma_{k}$$ the group of permutations on k symbols.

### Definition 2.1

A multi-norm on $$\{ {\mathcal{E}}^{k}: k\in\mathbf{N}\}$$ is a sequence

$$\bigl(\Vert \cdot \Vert _{k}\bigr)=\bigl(\Vert \cdot \Vert _{k}:k\in\mathbf{N}\bigr)$$

such that $$\|\cdot\|_{k}$$ is a norm on $${\mathcal{E}}^{k}$$ for each $$k\in\mathbf{N}$$ with $$k\geq2$$:

1. (A1)

$$\|(x_{\sigma(1)},\ldots,x_{\sigma(k)})\|_{k}=\|(x_{1},\ldots,x_{k})\|_{k}$$ for any $$\sigma\in\Sigma_{k}$$ and $$x_{1},\ldots,x_{k}\in\mathcal{E}$$;

2. (A2)

$$\|(\alpha_{1}x_{1},\ldots,\alpha_{k}x_{k})\|_{k}\leq (\max_{i\in{\mathbf{N}}_{k}}|\alpha_{i}| ) \|(x_{1},\ldots,x_{k})\| _{k}$$ for any $$\alpha_{1},\ldots,\alpha_{k} \in\mathbf{C}$$ and $$x_{1}, \ldots, x_{k}\in\mathcal{E}$$;

3. (A3)

$$\|(x_{1},\ldots,x_{k-1},0)\|_{k}=\|(x_{1},\ldots,x_{k-1})\|_{k-1}$$ for any $$x_{1}, \ldots, x_{k-1}\in\mathcal{E}$$;

4. (A4)

$$\|(x_{1},\ldots,x_{k-1},x_{k-1})\|_{k}=\|(x_{1},\ldots,x_{k-1})\| _{k-1}$$ for any $$x_{1},\ldots, x_{k-1}\in\mathcal{E}$$.

In this case, we say that $$((\mathcal{E}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ is a multi-normed space.

### Lemma 2.2

()

Suppose that $$((\mathcal{E}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ is a multi-normed space and let $$k\in\mathbf{N}$$. Then

1. (1)

$$\|(x,\ldots,x)\|_{k}=\|x\|$$ for any $$x\in\mathcal{E}$$;

2. (2)

$$\max_{i\in\mathbf{N}_{k}}\|x_{i}\|\leq \|x_{1},\ldots,x_{k}\|_{k}\leq\sum_{i=1}^{k}\|x_{i}\|\leq k \max_{i\in {\mathbf{N}}_{k}}\|x_{i}\|$$ for any $$x_{1},\ldots, x_{k}\in\mathcal{E}$$.

It follows from (2) that, if $$( \mathcal{E},\|\cdot\|)$$ is a Banach space, then $$( \mathcal{E}^{k},\|\cdot\|_{k})$$ is a Banach space for each $$k\in\mathbf{N}$$. In this case, $$((\mathcal{E}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ is a multi-Banach space.

Now, we present two examples (see ).

### Example 2.3

The sequence $$(\|\cdot\|_{k}: k\in\mathbf{N})$$ on $$\{\mathcal{E}^{k}: k\in\mathbf{N}\}$$ defined by

$$\bigl\Vert (x_{1},\ldots,x_{k})\bigr\Vert _{k}:=\max_{i\in\mathbf{N}_{k}}\|x_{i}\|$$

for any $$x_{1}, \ldots, x_{k}\in\mathcal{E}$$ is a multi-norm, which is called the minimum multi-norm.

### Example 2.4

Let $$\{(\|\cdot\|_{k}^{\alpha}: k\in\mathbf{N}):\alpha\in A\}$$ be the (non-empty) family of all multi-norms on $$\{\mathcal{E}^{k}:k\in\mathbf{N}\}$$. For each $$k\in\mathbf{N}$$, set

$$\bigl\Vert (x_{1},\ldots,x_{k})\bigr\Vert _{k}:=\sup_{\alpha\in A}\bigl\Vert (x_{1}, \ldots,x_{k})\bigr\Vert _{k}^{\alpha}$$

for any $$x_{1}, \ldots, x_{k}\in \mathcal{E}$$. Then $$( \|\cdot\|_{k} : k\in\mathbf{N})$$ is a multi-norm on $$\{\mathcal{E}^{k}: k\in\mathbf{N}\}$$, which is called the maximum multi-norm.

Now, we need the following observation which can easily be deduced from Lemma 2.2(2) of multi-norms.

### Lemma 2.5

Suppose that $$k\in\mathbf{N}$$ and $$(x_{1},\ldots, x_{k})\in \mathcal{E}^{k}$$. For each $$j\in\{1,\ldots,k\}$$, let $$(x_{n}^{j})$$ be a sequence in $$\mathcal{E}$$ such that $$\lim_{n\to\infty}x_{n}^{j}=x_{j}$$. Then, for each $$(y_{1},\ldots,y_{k})\in\mathcal{E}^{k}$$,

$$\lim_{n\to \infty}\bigl(x_{n}^{1}-y_{1}, \ldots,x_{n}^{k}-y_{k}\bigr)=(x_{1}-y_{1}, \ldots,x_{k}-y_{k}).$$

### Definition 2.6

Let $$((\mathcal{E}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ be a multi-normed space. A sequence $$(x_{n})$$ in $$\mathcal{E}$$ is a multi-null sequence if, for each $$\epsilon>0$$, there exists $$n_{0}\in\mathbf{N}$$ such that

$$\sup_{k\in\mathbf{N}}\bigl\Vert (x_{n},\ldots,x_{n+k-1}) \bigr\Vert _{k}< \epsilon$$

for any $$n\geq n_{0}$$. Let $$x\in\mathcal{E}$$. We say that the sequence $$(x_{n})$$ is multi-convergent to $$x\in\mathcal{E}$$ and write

$$\lim_{n\to\infty}x_{n}=x$$

if $$(x_{n}-x)$$ is a multi-null sequence.

### Definition 2.7

([8, 11])

Let $$({A},\|\cdot\|)$$ be a normed algebra such that $$(({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ is a multi-normed space. Then $$(({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ is called a multi-normed algebra if

$$\bigl\Vert (a_{1}b_{1},\ldots, a_{k}b_{k}) \bigr\Vert _{k}\leq\bigl\Vert (a_{1},\ldots, a_{k})\bigr\Vert _{k} \cdot \bigl\Vert (b_{1}, \ldots,b_{k})\bigr\Vert _{k}$$

for all $$k\in\mathbf{N}$$ and $$a_{1},\ldots,a_{k},b_{1},\ldots,b_{k}\in {A}$$. Further, the multi-normed algebra $$(({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ is a multi-Banach algebra if $$(({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ is a multi-Banach space.

### Example 2.8

([8, 11])

Let p, q with $$1 \leq p \leq q < \infty$$ and let $${A}=\ell^{p}$$. The algebra A is a Banach sequence algebra with respect to a coordinate-wise multiplication of sequences (see ). Let $$(\|\cdot\|_{k}: k\in\mathbf{N})$$ be the standard $$(p, q)$$-multi-norm on $$\{{A}^{k}: k\in\mathbf{N}\}$$. Then $$(({A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ is a multi-Banach algebra.

### Definition 2.9

Let $$(({ A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ be a multi-Banach algebra. A multi-$$C^{*}$$-algebra is a complex multi-Banach algebra $$(({ A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ with an involution satisfying

$$\bigl\Vert \bigl(a_{1}^{*}a_{1},\ldots, a_{k}^{*}a_{k} \bigr)\bigr\Vert _{k}= \bigl\Vert (a_{1},\ldots, a_{k})\bigr\Vert _{k} ^{2}$$

for all $$k\in\mathbf{N}$$ and $$a_{1},\ldots,a_{k}\in {A}$$.

### Definition 2.10

Let $$(({ A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ be a multi-Banach space. A multi-$$C^{*}$$-ternary algebra is a complex multi-Banach space $$(({ A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ equipped with a ternary product.

## Approximation of homomorphisms in multi-Banach algebras

Throughout this paper, assume that A, B are $$C^{*}$$-ternary algebras.

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

$$C_{\mu}f(x_{1},\ldots,x_{p},y_{1}, \ldots,y_{d}):= 2f \Biggl(\frac{\sum_{j=1}^{p}\mu x_{j}}{2}+\sum _{j=1}^{d}\mu y_{j} \Biggr)-\sum _{j=1}^{p}\mu f(x_{j})-2\sum _{j=1}^{d}\mu f(y_{j})$$

for all $$\mu\in{\mathbf{T}}^{1}:=\{ \lambda\in\mathbf{C}: |\lambda|=1\}$$ and $$x_{1},\ldots,x_{p},y_{1},\ldots,y_{d}\in A$$.

One can easily show that a mapping $$f:A \rightarrow B$$ satisfies

$$C_{\mu}f(x_{1}, \ldots, x_{p}, y_{1}, \ldots, y_{d}) =0$$

for all $$\mu\in {\mathbf{T}}^{1}$$ and all $$x_{1},\ldots,x_{p},y_{1},\ldots,y_{d}\in A$$ if and only if

$$f(\mu x+\lambda y)=\mu f(x)+\lambda f(y)$$

for all $$\mu, \lambda\in {{\mathbf{T}}}^{1}$$ and $$x, y \in A$$.

### Lemma 3.1

()

Let $$f:A \rightarrow B$$ be an additive mapping such that $$f(\mu x) = \mu f(x)$$ for all $$x\in A$$ and $$\mu\in{\mathbf{T}}^{1}$$. Then the mapping f is C-linear.

### Lemma 3.2

Let $$\{x_{n}\}$$, $$\{y_{n}\}$$, and $$\{z_{n}\}$$ be the convergent sequences in A. Then the sequence $$\{[x_{n},y_{n},z_{n}]\}$$ is convergent in A.

### Proof

Let $$x,y,z\in A$$ be such that

$$\lim_{n\to\infty}x_{n}=x,\qquad \lim_{n\to\infty}y_{n}=y, \qquad \lim_{n\to\infty}z_{n}=z.$$

Since

\begin{aligned}& [x_{n},y_{n},z_{n}]-[x,y,z] \\& \quad = [x_{n}-x,y_{n}-y,z_{n},z]+[x_{n},y_{n},z] +[x,y_{n}-y,z_{n}]+[x_{n},y,z_{n}-z] \end{aligned}

for all $$n\geq1$$, we get

\begin{aligned} \bigl\Vert [x_{n},y_{n},z_{n}]-[x,y,z]\bigr\Vert =&\|x_{n}-x\|\|y_{n}-y\|\|z_{n}-z\|+ \|x_{n}-x\|\|y_{n}\| \|z\| \\ &{}+\|x\|\|y_{n}-y\|\|z_{n}\|+\|x_{n}\|\|y\| \|z_{n}-z\| \end{aligned}

for all $$n\geq1$$, and so

$$\lim_{n\to\infty}[x_{n},y_{n},z_{n}]=[x,y,z].$$

This completes the proof. □

Using Theorem 1.1, we approximate homomorphisms in multi-$$C^{*}$$-ternary algebras for the functional equation $$C_{\mu}f(x_{1},\ldots,x_{m}) =0$$.

### Theorem 3.3

Let $$(( {B}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ be a multi-$$C^{*}$$-ternary algebra. Let $$f: A \rightarrow B$$ be a mapping for which there are functions $$\varphi: A^{(p+d)k} \rightarrow[0, \infty)$$ and $$\psi: A^{3k} \rightarrow[0, \infty)$$ such that

\begin{aligned}& \lim_{n\to\infty} {\gamma}^{-n} \varphi\bigl( \gamma^{n} x_{11},\ldots,\gamma^{n} x_{1p},\gamma^{n} y_{11},\ldots, \gamma^{n} y_{1p}, \\& \quad \ldots, \gamma^{n} x_{k1},\ldots,\gamma^{n} x_{kp},\ldots,\gamma^{n} y_{k1},\ldots, \gamma^{n} y_{kd}\bigr) = 0, \end{aligned}
(1)
\begin{aligned}& \bigl\Vert \bigl( c_{\mu}f(x_{11},\ldots,x_{1p},y_{11}, \ldots ,y_{1d}),\ldots,c_{\mu}f(x_{k1}, \ldots,x_{kp},y_{k1},\ldots,y_{kd} ) \bigr)\bigr\Vert _{k} \\& \quad \leq \varphi(x_{11},\ldots,x_{1p},y_{11}, \ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1}, \ldots,y_{kd}) , \end{aligned}
(2)
\begin{aligned}& \bigl\Vert \bigl(f\bigl([x_{1}, y_{1}, z_{1}] \bigr) - \bigl[f(x_{1}), f(y_{1}), f(z_{1})\bigr], \\& \qquad \ldots,f\bigl([x_{k}, y_{k}, z_{k}]\bigr) - \bigl[f(x_{k}), f(y_{k}), f(z_{k})\bigr] \bigr) \bigr\Vert _{k} \\& \quad \le\psi(x_{1}, y_{1},z_{1}, \ldots,x_{k},y_{k},z_{k}) , \end{aligned}
(3)
\begin{aligned}& \lim_{n\to\infty} \gamma^{-3n} \psi\bigl( \gamma^{n} x_{1},\gamma^{n} y_{1}, \gamma^{n} z_{1},\ldots,\gamma^{n} x_{k}, \gamma^{n} y_{k},\gamma^{n} z_{k}\bigr) = 0, \end{aligned}
(4)
\begin{aligned}& \lim_{n\to\infty} \gamma^{-2n} \psi\bigl( \gamma^{n} x_{1},\gamma^{n} y_{1}, z_{1},\ldots,\gamma^{n} x_{k},\gamma^{n} y_{k},z_{k}\bigr) = 0 \end{aligned}
(5)

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots, y_{kd}, x_{1},\ldots,x_{k},y_{1}, \ldots,y_{k},z_{1}, \ldots,z_{k}\in A$$, where $$\gamma=\frac{p+2d}{2}$$. If there exists a constant $$L<1$$ such that

\begin{aligned}& \varphi\bigl(\overbrace{\gamma x_{1},\ldots,\gamma x_{1}}^{p+d},\overbrace {\gamma x_{2},\ldots,\gamma x_{2} }^{p+d},\ldots,\overbrace{\gamma x_{k}, \ldots,\gamma x_{k}}^{p+d}\bigr) \\& \quad \le\gamma L \varphi\bigl(\overbrace{x_{1},\ldots ,x_{1}}^{p+d},\overbrace{x_{2}, \ldots,x_{2}}^{p+d},\ldots,\overbrace {x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(6)

for all $$x_{1},x_{2},\ldots,x_{k} \in A$$, then there exists a unique homomorphism $$H : A \rightarrow B$$ such that

\begin{aligned}& \bigl\Vert \bigl(f(x_{1}) - H(x_{1}), \ldots,f(x_{k}) - H(x_{k}) \bigr) \bigr\Vert _{k} \\& \quad \le\frac{1}{(1-L)2\gamma} \varphi\bigl(\overbrace{x_{1}, \ldots,x_{1}}^{p+d},\overbrace{x_{2},\ldots ,x_{2}}^{p+d},\ldots,\overbrace{x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(7)

for all $$x_{1},\ldots,x_{k} \in A$$.

### Proof

Let $$\mu= 1$$ and $$x_{ij} = y_{ij} = x_{i}$$ for $$1\leq i\leq k$$ in (2). Then we get

\begin{aligned}& \bigl\Vert \bigl(f(\gamma x_{1})-\gamma f(x_{1}),\ldots,f(\gamma x_{k})-\gamma f(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad \leq\frac{1}{2} \varphi\bigl(\overbrace{ x_{1}, \ldots,x_{1}}^{p+d},\overbrace{x_{2}, \ldots,x_{2}}^{p+d},\ldots,\overbrace {x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(8)

for all $$x_{1},\ldots,x_{k} \in A$$. Consider the set

$$E: = \{ g : A \rightarrow B\}$$

and introduce the generalized metric on E:

\begin{aligned} d(g, h) = &\inf \bigl\{ C\in{\mathbf{R}}_{+} : \bigl\Vert \bigl(g(x_{1})-h(x_{1}), \ldots,g(x_{k})-h(x_{k}) \bigr)\bigr\Vert _{k} \\ &\le C\varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace{x_{2},\ldots ,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr), \forall x_{1},\ldots,x_{k} \in A \bigr\} . \end{aligned}

It is easy to see that $$(E, d)$$ is complete (see also ).

First we show that d is metric on E. It is obvious $$d(g,g)=0$$ for all $$g\in E$$. If $$d(g,h)=0$$, then, for every fixed $$x_{1},\ldots,x_{k} \in A$$,

$$\bigl\Vert \bigl(g(x_{1}) -h(x_{1}),\ldots,g(x_{k})-h(x_{k}) \bigr)\bigr\Vert _{k}=0$$

and therefore $$g=h$$. If $$d(g,h)=a<\infty$$ and $$d(h,l)=b<\infty$$ for all $$g,h,l\in E$$, then

\begin{aligned}& \bigl\Vert \bigl(g(x_{1})-l(x_{1}),\ldots,g(x_{k})-l(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad = \bigl\Vert \bigl(g(x_{1})-h(x_{1})+h(x_{1})-l(x_{1}), \ldots,g(x_{k})-h(x_{k})+h(x_{k})-l(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad \leq\bigl\Vert \bigl(g(x_{1})-h(x_{1}), \ldots,g(x_{k})-h(x_{k})\bigr)\bigr\Vert _{k} + \bigl\Vert \bigl(h(x_{1})-l(x_{1}),\ldots,h(x_{k})-l(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad \leq a \varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace {x_{2},\ldots,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr) + b \varphi \bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace {x_{2},\ldots,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr) \\& \quad = (a+b) \varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace {x_{2},\ldots,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr). \end{aligned}

So we have $$d(g,l)\leq d(g,h) + d(h,l)$$.

Let $$\{g_{n}\}$$ be a Cauchy sequence in $$(E,d)$$. Then for all $$\epsilon >0$$ there exists N such that $$d(g_{n},g_{i}) < \epsilon$$, if $$n,i \geq N$$, Let $$n,i\geq N$$. Since $$d(g_{n},g_{i}) < \epsilon$$ there exists $$C\in [0,\epsilon)$$ such that

\begin{aligned}& \bigl\Vert \bigl(g_{n}(x_{1}) -g_{i}(x_{1}),\ldots,g_{n}(x_{k})-g_{i}(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad \leq C\varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace {x_{2},\ldots,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr) \\& \quad \leq\epsilon\varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace {x_{2},\ldots,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(9)

for all $$x_{1},\ldots,x_{k} \in A$$, so for each $$x_{1},\ldots,x_{k}\in A$$, $$\{g_{n}(x_{1},\ldots,x_{k})\}$$ is a Cauchy sequence in B. Since B is complete, there exists $$g(x_{1},\ldots,x_{k})\in B$$ such that $$g_{n}(x_{1},\ldots,x_{k})\rightarrow g(x_{1},\ldots,x_{k})$$ as $$n\rightarrow\infty$$. Thus, we have $$g\in E$$. Taking the limit as $$i\rightarrow\infty$$ in (9) we obtain, for $$n\geq N$$,

$$\bigl\Vert \bigl(g_{n}(x_{1}) -g(x_{1}), \ldots,g_{n}(x_{k})-g(x_{k})\bigr)\bigr\Vert _{k} \leq\epsilon\varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace {x_{2},\ldots,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr).$$

Therefore $$d(g_{n},g)\leq\epsilon$$. Hence $$g_{n}\rightarrow g$$ as $$n\rightarrow\infty$$, so $$(E,d)$$ is complete. Now, we consider the linear mapping $$\Lambda: E \rightarrow E$$ such that

$$\Lambda g(x): = \frac{1}{\gamma} g(\gamma x)$$

for all $$x \in A$$. From Theorem 3.1 of  (also see Lemma 3.2 of ),

$$d(\Lambda g, \Lambda h) \le L d(g, h)$$

for all $$g, h \in E$$. Let $$g,h\in E$$ and let $$C\in[0,\infty]$$ be an arbitrary constant with $$d(g,h)\leq C$$. From the definition of d, we have

$$\bigl\Vert \bigl(g(x_{1}) - h(x_{1}) , \ldots,g(x_{k}) - h(x_{k}) \bigr)\bigr\Vert _{k} \leq C \varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \ldots,\overbrace {x_{k},\ldots,x_{k}}^{p+d}\bigr)$$

for all $$x_{1},\ldots,x_{k}\in A$$. From our assumption and the last inequality, we have

\begin{aligned}& \bigl\Vert \bigl(\Lambda g(x_{1}) - \Lambda h(x_{1}), \ldots,\Lambda g(x_{k})- \Lambda h(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad =\frac{1}{\gamma} \bigl\Vert \bigl(g(\gamma x_{1})- h (\gamma x_{1}) ,\ldots,g(\gamma x_{k})- h (\gamma x_{k}) \bigr)\bigr\Vert _{k} \\& \quad \leq\frac{C}{\gamma}\varphi\bigl(\overbrace{\gamma x_{1}, \ldots,\gamma x_{1} }^{p+d},\ldots,\overbrace{\gamma x_{k},\ldots,\gamma x_{k} }^{p+d}\bigr) \\& \quad \leq C L \varphi\bigl(\overbrace {x_{1},\ldots,x_{1}}^{p+d}, \ldots,\overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr) \end{aligned}

for all $$x_{1},\ldots,x_{k} \in A$$ and so

\begin{aligned}& \bigl\Vert \bigl(\Lambda f(x_{1}) - f(x_{1}),\ldots, \Lambda f(x_{k}) - f(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad = \biggl\Vert \biggl(\frac{1}{\gamma}f(\gamma x_{1}) - f(x_{1}),\ldots,\frac{1}{\gamma}f(\gamma x_{k}) - f(x_{k}) \biggr)\biggr\Vert _{k} \\& \quad = \frac{1}{\gamma}\bigl\Vert \bigl(f(\gamma x_{1}) - \gamma f(x_{1}),\ldots,f(\gamma x_{k}) - \gamma f(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad \leq \frac{1}{2\gamma}\varphi\bigl(\overbrace{x_{1}, \ldots,x_{1}}^{p+d},\ldots ,\overbrace{x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}

for all $$x_{1},\ldots,x_{k} \in A$$. Hence $$d(\Lambda f, f) \le \frac{1}{2\gamma}$$. From Theorem 1.1, the sequence $$\{\Lambda^{n} f\}$$ converges to a fixed point H of Λ, i.e., $$H:A\rightarrow B$$ is a mapping defined by

$$H(x) = \lim_{n\to\infty}\bigl(\Lambda^{n} f\bigr) (x) = \lim_{n\to\infty}\frac{1}{\gamma^{n}}f\bigl(\gamma^{n} x\bigr)$$
(10)

and $$H(\gamma x)=\gamma H(x)$$ for all $$x \in A$$. Also, H is the unique fixed point of Λ in the set $$E'=\{ g \in E : d(f,g)< \infty\}$$ and

$$d(H,f)\leq\frac{1}{1-L}d(\Lambda f , f)\leq\frac{1}{(1-L)2\gamma},$$

i.e., the inequality (7) hold for all $$x_{1},\ldots ,x_{k} \in A$$. Thus it follows from the definition of H, (1), and (2) that

\begin{aligned}& \Biggl\Vert \Biggl(2H \Biggl(\frac{\sum_{j=1}^{p} \mu x_{1j}}{2}+\sum _{j=1}^{d} \mu y_{1j} \Biggr) - \sum _{j=1}^{p} \mu H(x_{1j})-2 \sum _{j=1}^{d} \mu H(y_{1j}), \\& \qquad \ldots, 2H \Biggl(\frac{\sum_{j=1}^{p} \mu x_{kj}}{2}+\sum_{j=1}^{d} \mu y_{kj} \Biggr) - \sum_{j=1}^{p} \mu H(x_{kj})-2 \sum_{j=1}^{d} \mu H(y_{kj}) \Biggr) \Biggr\Vert _{k} \\& \quad = \lim_{n\to\infty} \frac{1}{\gamma^{n}} \Biggl\Vert \Biggl(2 f \Biggl(\gamma^{n}\frac{\sum_{j=1}^{p} \mu x_{1j}}{2}+\gamma^{n}\sum _{j=1}^{d} \mu y_{1j} \Biggr) - \sum _{j=1}^{p} \mu f\bigl(\gamma^{n} x_{1j}\bigr)-2 \sum_{j=1}^{d} \mu f\bigl(\gamma^{n}y_{1j}\bigr), \\& \qquad \ldots, 2 f \Biggl(\gamma^{n}\frac{\sum_{j=1}^{p} \mu x_{kj}}{2}+ \gamma^{n}\sum_{j=1}^{d} \mu y_{kj} \Biggr) - \sum_{j=1}^{p} \mu f\bigl(\gamma^{n} x_{kj}\bigr)-2 \sum _{j=1}^{d} \mu f\bigl(\gamma ^{n}y_{kj} \bigr) \Biggr) \Biggr\Vert _{k} \\& \quad \leq\lim_{n\to\infty} \frac{1}{\gamma^{n}}\bigl\Vert \bigl(C_{\mu}f\bigl(\gamma^{n} x_{11},\ldots, \gamma^{n} x_{1p},\gamma^{n} y_{11}, \ldots,\gamma^{n}y_{1d}\bigr), \\& \qquad \ldots, C_{\mu}f\bigl(\gamma^{n} x_{k1}, \ldots,\gamma^{n} x_{kp},\gamma^{n} y_{k1},\ldots,\gamma^{n}y_{kd}\bigr) \bigr)\bigr\Vert _{k} \\& \quad \leq\lim_{n\to\infty}\frac{1}{\gamma^{n}}\varphi\bigl( \gamma^{n} x_{11},\ldots,\gamma^{n} x_{1p},\gamma^{n} y_{11},\ldots, \gamma^{n} y_{1d}, \\& \qquad \ldots, \gamma^{n} x_{k1},\ldots,\gamma^{n} x_{kp},\gamma^{n} y_{k1},\ldots, \gamma^{n} y_{kd}\bigr)=0 \end{aligned}

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots,y_{kd} \in A$$. Hence we have

$$2H \Biggl(\frac{\sum_{j=1}^{p} \mu x_{ij}}{2}+\sum_{j=1}^{d} \mu y_{ij} \Biggr) = \sum_{j=1}^{p} \mu H(x_{ij})+2 \sum_{j=1}^{d} \mu H(y_{ij})$$

for all $$\mu\in\mathbf{T}^{1}$$, $$x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A$$ and $$1\leq i \leq k$$ and so $$H(\lambda x + \mu y)=\lambda H(x) + \mu H(y)$$ for all $$\lambda, \mu\in\mathbf{T}^{1}$$ and $$x,y \in A$$. Therefore, by Lemma 3.1, the mapping $$H : A \rightarrow B$$ is C-linear.

Also it follows from (3) and (4) that

\begin{aligned}& \bigl\Vert \bigl(H\bigl([x_{1}, y_{1}, z_{1}] \bigr)- \bigl[H(x_{1}), H(y_{1}), H(z_{1})\bigr], \ldots ,H\bigl([x_{k}, y_{k}, z_{k}]\bigr)- \bigl[H(x_{k}), H(y_{k}), H(z_{k})\bigr] \bigr)\bigr\Vert _{k} \\& \quad =\lim_{n\to\infty}\frac{1}{\gamma^{3n}} \bigl\Vert \bigl(f \bigl(\bigl[\gamma^{n} x_{1}, \gamma^{n} y_{1}, \gamma^{n} z_{1}\bigr] \bigr) - \bigl[f \bigl(\gamma^{n} x_{1}\bigr), f\bigl(\gamma^{n} y_{1}\bigr), f\bigl(\gamma^{n} z_{1}\bigr) \bigr], \\& \qquad \ldots, f \bigl(\bigl[\gamma^{n} x_{k}, \gamma^{n} y_{k}, \gamma^{n} z_{k}\bigr] \bigr) - \bigl[f\bigl(\gamma^{n} x_{k}\bigr), f\bigl( \gamma^{n} y_{k}\bigr), f\bigl(\gamma^{n} z_{k}\bigr) \bigr] \bigr) \bigr\Vert _{k} \\& \quad \leq\lim_{n\to\infty} \frac{1}{\gamma^{3n}}\psi\bigl( \gamma^{n}x_{1},\gamma^{n}y_{1},\gamma ^{n}z_{1},\ldots,\gamma^{n}x_{k}, \gamma^{n}y_{k},\gamma^{n}z_{k}\bigr)=0 \end{aligned}

for all $$x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} \in A$$. Thus we have

$$H\bigl([x, y,z]\bigr) = \bigl[H(x), H(y), H(z)\bigr]$$

for all $$x, y, z \in A$$. Thus $$H : A \rightarrow B$$ is a homomorphism satisfying (7).

Now, let $$T : A \rightarrow B$$ be another $$C^{*}$$-ternary-algebras homomorphism satisfying (7). Since $$d(f,T)\leq \frac{1}{(1-L)2\gamma}$$ and T is C-linear, we get $$T\in E'$$ and $$(\Lambda T)(x)=\frac{1}{\gamma}(T\gamma x)=T(x)$$ for all $$x\in A$$, i.e., T is a fixed point of Λ. Since H is the unique fixed point of $$\Lambda\in E'$$, we get $$H=T$$. This completes the proof. □

### Theorem 3.4

Let $$(( {B}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ be a multi-$$C^{*}$$-ternary algebra. Let $$f: A \rightarrow B$$ be a mapping for which there are the functions $$\varphi: A^{(p+d)k} \rightarrow[0, \infty)$$ and $$\psi: A^{3k} \rightarrow[0, \infty)$$ satisfying the inequalities (2) and (3) such that

\begin{aligned}& \lim_{n\to\infty} {\gamma}^{n} \varphi \biggl( \frac{x_{11}}{\gamma^{n}} ,\ldots,\frac{x_{1p}}{\gamma^{n}} ,\frac{y_{11}}{\gamma^{n}} ,\ldots, \frac{y_{1p}}{\gamma^{n}}, \ldots,\frac{x_{k1}}{\gamma^{n}} ,\ldots,\frac{x_{kp}}{\gamma ^{n}} ,\ldots, \frac{y_{k1}}{\gamma^{n}} ,\ldots,\frac{y_{kd}}{\gamma^{n}} \biggr) = 0, \end{aligned}
(11)
\begin{aligned}& \lim_{n\to\infty} \gamma^{3n} \psi \biggl( \frac{x_{1}}{\gamma^{n}} ,\frac{y_{1}}{\gamma^{n}} ,\frac{z_{1}}{\gamma^{n}} ,\ldots, \frac{x_{k}}{\gamma^{n}} ,\frac{y_{k}}{\gamma^{n}} ,\frac{z_{k}}{\gamma^{n}} \biggr) = 0, \end{aligned}
(12)
\begin{aligned}& \lim_{n\to\infty} \gamma^{2n} \psi \biggl( \frac{x_{1}}{\gamma^{n}} ,\frac{y_{1}}{\gamma^{n}} ,z_{1} ,\ldots,\frac{x_{k}}{\gamma^{n}} ,\frac{y_{k}}{\gamma^{n}} ,z_{k} \biggr) = 0 \end{aligned}
(13)

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots, y_{kd}, x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k} \in A$$, where $$\gamma=\frac{p+2d}{2}$$. If the constant $$L<1$$ exists such that

\begin{aligned}& \varphi\biggl(\overbrace{\frac{x_{1}}{\gamma} ,\ldots, \frac{x_{1}}{\gamma} }^{p+d},\overbrace{\frac{x_{2}}{\gamma} ,\ldots, \frac{x_{2}}{\gamma} }^{p+d},\ldots,\overbrace{\frac{x_{k}}{\gamma} ,\ldots, \frac{x_{k}}{\gamma } }^{p+d}\biggr) \\& \quad \le\frac{L}{\gamma} \varphi\bigl(\overbrace{x_{1},\ldots ,x_{1}}^{p+d},\overbrace{x_{2}, \ldots,x_{2}}^{p+d},\ldots,\overbrace {x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(14)

for all $$x_{1},x_{2},\ldots,x_{k} \in A$$, then there exists a unique homomorphism $$H : A \rightarrow B$$ such that

\begin{aligned}& \bigl\Vert \bigl(f(x_{1}) - H(x_{1}), \ldots,f(x_{k}) - H(x_{k}) \bigr) \bigr\Vert _{k} \\& \quad \le\frac{1}{(1-L)2\gamma} \varphi\bigl(\overbrace{x_{1}, \ldots,x_{1}}^{p+d},\overbrace{x_{2},\ldots ,x_{2}}^{p+d},\ldots,\overbrace{x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(15)

for all $$x_{1},\ldots,x_{k} \in A$$.

### Proof

If we replace $$x_{i}$$ in (8) by $$\frac{x_{i}}{\gamma}$$ for $$1\leq i\leq k$$, then we get

\begin{aligned}& \biggl\Vert \biggl(f(x_{1})-\gamma f\biggl( \frac{1}{x_{1}}\biggr),\ldots,f(x_{k})-\gamma f\biggl( \frac{1}{x_{k}}\biggr) \biggr)\biggr\Vert _{k} \\& \quad \leq\frac{1}{2} \varphi\biggl(\overbrace{ \frac{1}{x_{1}},\ldots, \frac{1}{x_{1}}}^{p+d},\overbrace{\frac {1}{x_{2}},\ldots, \frac{1}{x_{2}}}^{p+d},\ldots,\overbrace{\frac {1}{x_{k}},\ldots, \frac{1}{x_{k}}}^{p+d}\biggr) \end{aligned}
(16)

for all $$x_{1},\ldots,x_{k} \in A$$. Consider the set

$$E: = \{ g : A \rightarrow B\}$$

and introduce the generalized metric on E:

\begin{aligned} d(g, h) =& \inf \bigl\{ C\in{\mathbf{R}}_{+} : \bigl\Vert \bigl(g(x_{1})-h(x_{1}), \ldots,g(x_{k})-h(x_{k}) \bigr)\bigr\Vert _{k} \\ &\le C\varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace{x_{2},\ldots ,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr), \forall x_{1},\ldots,x_{k} \in A \bigr\} . \end{aligned}

It is easy to see that $$(E, d)$$ is complete (see ).

Now, we consider the linear mapping $$\Lambda: E \rightarrow E$$ such that

$$\Lambda g(x): = \gamma g\biggl(\frac{x}{\gamma} \biggr)$$

for all $$x \in A$$. From Theorem 3.1 of  (also see Lemma 3.2 of ),

$$d(\Lambda g, \Lambda h) \le L d(g, h)$$

for all $$g, h \in E$$. Let $$g,h\in E$$ and let $$C\in[0,\infty]$$ be an arbitrary constant with $$d(g,h)\leq C$$. From the definition of d, we have

$$\bigl\Vert \bigl(g(x_{1}) - h(x_{1}) , \ldots,g(x_{k}) - h(x_{k}) \bigr)\bigr\Vert _{k} \leq C \varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \ldots,\overbrace {x_{k},\ldots,x_{k}}^{p+d}\bigr)$$

for all $$x_{1},\ldots,x_{k}\in A$$. From our assumption and the last inequality, we have

\begin{aligned}& \bigl\Vert \bigl(\Lambda g(x_{1}) - \Lambda h(x_{1}), \ldots,\Lambda g(x_{k})- \Lambda h(x_{k}) \bigr)\bigr\Vert _{k} \\& \quad =\gamma\biggl\Vert \biggl(g\biggl(\frac{x_{1}}{\gamma} \biggr)- h \biggl( \frac{x_{1}}{\gamma} \biggr) ,\ldots,g\biggl(\frac{x_{k}}{\gamma}\biggr)- h \biggl( \frac{x_{k}}{\gamma}\biggr) \biggr)\biggr\Vert _{k} \\& \quad \leq{C} {\gamma}\varphi\biggl(\overbrace{\frac{x_{1}}{\gamma} ,\ldots, \frac{x_{1}}{\gamma} }^{p+d},\ldots,\overbrace{\frac{x_{k}}{\gamma} ,\ldots, \frac{x_{k} }{\gamma} }^{p+d}\biggr) \\& \quad \leq C L \varphi\bigl(\overbrace {x_{1},\ldots,x_{1}}^{p+d}, \ldots,\overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr) \end{aligned}

for all $$x_{1},\ldots,x_{k} \in A$$ and so $$d(\Lambda g ,\Lambda h)\leq Ld(g,h)$$ for any $$g,h \in E$$. It follows from (16) that $$d(\Lambda f,f)\leq\frac{1}{2\gamma}$$. Therefore, according to Theorem 1.1, the sequence $$\{\Lambda^{n} f\}$$ converges to a fixed point H of Λ, i.e., $$H:A\rightarrow B$$ is a mapping defined by

$$H(x) = \lim_{n\to\infty}\bigl(\Lambda^{n} f\bigr) (x) = \lim_{n\to\infty}{\gamma^{n}}f\biggl(\frac{ x}{\gamma^{n}} \biggr)$$
(17)

for all $$x \in A$$.

The rest of the proof is similar to the proof of Theorem 3.3 and so we omit it. This completes the proof. □

### Theorem 3.5

Let r and θ be non-negative real numbers such that $$r\notin[1,3]$$ and let $$(( {B}^{k},{\|\cdot\|_{k}}): k\in\mathbf{N})$$ be a multi-$$C^{*}$$-ternary algebra. Let $$f : A \rightarrow B$$ be a mapping such that

\begin{aligned}& \bigl\Vert \bigl(C_{\mu}f(x_{11}, \ldots,x_{1p},y_{11},\ldots ,y_{1d}),\ldots, C_{\mu}f(x_{k1},\ldots,x_{kp},y_{k1}, \ldots,y_{kd}) \bigr)\bigr\Vert _{k} \\& \quad \le\theta \Biggl( \sum_{j=1}^{p} \|x_{1j}\|^{r}_{A} + \sum _{j=1}^{d}\|y_{1j}\| ^{r}_{A} + \cdots+\sum_{j=1}^{p}\|x_{kj} \|^{r}_{A} + \sum_{j=1}^{d} \|y_{kj}\| ^{r}_{A} \Biggr) \end{aligned}
(18)

and

\begin{aligned}& \bigl\Vert \bigl(f\bigl([x_{1},y_{1},z_{1}] \bigr)-\bigl[f(x_{1}),f(y_{1}),f(z_{1})\bigr],\ldots ,f\bigl([x_{k},y_{k},z_{k}]\bigr)- \bigl[f(x_{k}),f(y_{k}),f(z_{k})\bigr] \bigr)\bigr\Vert _{k} \\& \quad \le\theta\bigl(\|x_{1}\|^{r}_{A}\cdot \|y_{1}\|^{r}_{A} . \|z_{1} \|^{r}_{A} + \cdots+ \| x_{k}\|^{r}_{A} \cdot\|y_{k}\|^{r}_{A} \cdot\|z_{k} \|^{r}_{A}\bigr) \end{aligned}
(19)

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd},x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k} \in A$$. Then there exists a unique $$C^{*}$$-ternary algebra homomorphism $$H : A \rightarrow B$$ such that

\begin{aligned} \begin{aligned}[b] &\bigl\Vert \bigl(f(x_{1}) - H(x_{1}), \ldots,f(x_{k}) - H(x_{k}) \bigr)\bigr\Vert _{B} \\ &\quad \le\frac{2^{r}(p+d)\theta}{|2(p+2d)^{r} -(p+2d)2^{r}|}\bigl(\|x_{1}\|^{r}_{A} + \cdots + \|x_{k}\|^{r}_{A}\bigr) \end{aligned} \end{aligned}
(20)

for all $$x_{1},\ldots,x_{k} \in A$$.

### Proof

The proof follows from Theorem 3.3 by taking

\begin{aligned}& \varphi(x_{11},\ldots,x_{1p},y_{11}, \ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1}, \ldots,y_{kd}) \\& \quad := \theta \Biggl( \sum_{j=1}^{p} \|x_{ij}\|^{r}_{A} + \sum _{j=1}^{d}\|y_{ij}\|^{r}_{A} + \cdots+ \sum_{j=1}^{p}\|x_{kj} \|^{r}_{A} + \sum_{j=1}^{d} \|y_{kj}\|^{r}_{A} \Biggr), \\& \psi(x_{1},y_{1},z_{1},\ldots,x_{k},y_{k},z_{k}) \\& \quad :=\theta\bigl(\|x_{1}\|^{r}_{A} \cdot \|y_{1}\|^{r}_{A} \cdot\|z_{1} \|^{r}_{A} + \cdots+\| x_{k}\|^{r}_{A} \cdot\|y_{k}\|^{r}_{A} \cdot\|z_{k} \|^{r}_{A} \bigr) \end{aligned}

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots, y_{kd},x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k}\in A$$. Then we can choose $$L=2^{1-r}(p+2d)^{r-1}$$, when $$0< r<1$$, and $$L=2-2^{1-r}(p+2d)^{r-1}$$, when $$r>3$$, and so we get the desired result. This completes the proof. □

### Theorem 3.6

Let $$(( {B}^{k},\|\cdot\|_{k}): k\mathbf{N})$$ be a multi-$$C^{*}$$-ternary algebra. Let $$f: A \rightarrow B$$ be a mapping for which there are functions $$\varphi: A^{(p+d)k} \rightarrow[0, \infty)$$ and $$\psi: A^{3k} \rightarrow[0, \infty)$$ such that

\begin{aligned}& \lim_{n\to\infty} {d}^{-n} \varphi \bigl(d^{n} x_{11},\ldots,d^{n} x_{1p},d^{n} y_{11},\ldots,d^{n} y_{1p}, \\& \quad \ldots, d^{n} x_{k1},\ldots,d^{n} x_{kp},\ldots,d^{n} y_{k1},\ldots,d^{n} y_{kd}\bigr) = 0, \end{aligned}
(21)
\begin{aligned}& \bigl\Vert \bigl( c_{\mu}f(x_{11},\ldots,x_{1p},y_{11}, \ldots,y_{1d}),\ldots,c_{\mu }f(x_{k1}, \ldots,x_{kp},y_{k1},\ldots,y_{kd} ) \bigr)\bigr\Vert _{k} \\& \quad \leq \varphi(x_{11},\ldots,x_{1p},y_{11}, \ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1}, \ldots,y_{kd}) , \end{aligned}
(22)
\begin{aligned}& \bigl\Vert \bigl(f\bigl([x_{1}, y_{1}, z_{1}] \bigr) - \bigl[f(x_{1}), f(y_{1}), f(z_{1})\bigr], \\& \qquad \ldots,f\bigl([x_{k}, y_{k}, z_{k}]\bigr) - \bigl[f(x_{k}), f(y_{k}), f(z_{k})\bigr]\bigr) \bigr\Vert _{k} \\& \quad \le\psi(x_{1}, y_{1},z_{1}, \ldots,x_{k},y_{k},z_{k}) , \end{aligned}
(23)
\begin{aligned}& \lim_{n\to\infty} d^{-3n} \psi \bigl(d^{n} x_{1},d^{n} y_{1},d^{n} z_{1},\ldots,d^{n} x_{k},d^{n} y_{k},d^{n} z_{k}\bigr)= 0, \end{aligned}
(24)
\begin{aligned}& \lim_{n\to\infty} d^{-2n} \psi \bigl(d^{n} x_{1},d^{n} y_{1},z_{1}, \ldots,d^{n} x_{k},d^{n} y_{k}, z_{k}\bigr) = 0 \end{aligned}
(25)

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots,y_{kd}, x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k}\in A$$, where $$\gamma=\frac{p+2d}{2}$$. If there exists the constant $$L<1$$ such that

\begin{aligned}& \varphi\bigl(\overbrace{d x_{1},\ldots,d x_{1}}^{p+d},\overbrace{d x_{2},\ldots,d x_{2} }^{p+d},\ldots,\overbrace{d x_{k},\ldots,d x_{k}}^{p+d}\bigr) \\& \quad \le d L \varphi\bigl( \overbrace{0,\ldots,0}^{p},\overbrace {x_{1},\ldots,x_{1}}^{d},\overbrace{0, \ldots,0}^{p}, \overbrace{x_{2},\ldots,x_{2}}^{d}, \ldots,\overbrace{0,\ldots ,0}^{p},\overbrace{x_{k}, \ldots,x_{k}}^{d}\bigr) \end{aligned}
(26)

for all $$x_{1},x_{2},\ldots,x_{k} \in A$$, then there exists a unique homomorphism $$H : A \rightarrow B$$ such that

\begin{aligned}& \bigl\Vert \bigl(f(x_{1}) - H(x_{1}), \ldots,f(x_{k}) - H(x_{k})\bigr) \bigr\Vert _{k} \\& \quad \le\frac{1}{(1-L)2d} \varphi\bigl(\overbrace{0,\ldots,0}^{p}, \overbrace{x_{1},\ldots ,x_{1}}^{d},\overbrace{0, \ldots,0}^{p},\overbrace{x_{2},\ldots ,x_{2}}^{d}, \ldots, \overbrace{0,\ldots,0}^{p},\overbrace{x_{k}, \ldots,x_{k}}^{d}\bigr) \end{aligned}
(27)

for all $$x_{1},\ldots,x_{k} \in A$$.

### Proof

Let $$\mu= 1$$ and $$x_{ij} = 0$$, $$y_{ij} = x_{i}$$ for $$1\leq i\leq k$$ in (22). Then we get

\begin{aligned}& \bigl\Vert \bigl(f(d x_{1})- d f(x_{1}), \ldots,f(d x_{k})-d f(x_{k})\bigr)\bigr\Vert _{k} \\ & \quad \leq\frac{1}{2} \varphi\bigl(\overbrace{0,\ldots,0}^{p}, \overbrace{x_{1},\ldots ,x_{1}}^{d},\overbrace{0, \ldots,0}^{p},\overbrace{x_{2},\ldots ,x_{2}}^{d}, \ldots, \overbrace{0,\ldots,0}^{p},\overbrace{x_{k}, \ldots,x_{k}}^{d}\bigr) \end{aligned}
(28)

for all $$x_{1},\ldots,x_{k} \in A$$. Consider the set

$$E: = \{ g : A \rightarrow B\}$$

and introduce the generalized metric on E:

\begin{aligned} d(g, h) =& \inf \bigl\{ C\in{\mathbf{R}}_{+} : \bigl\Vert \bigl(g(x_{1})-h(x_{1}), \ldots,g(x_{k})-h(x_{k})\bigr)\bigr\Vert _{k} \\ & \le C\varphi\bigl(\overbrace{0,\ldots,0}^{p},\overbrace{x_{1}, \ldots ,x_{1}}^{d},\overbrace{0,\ldots,0}^{p}, \overbrace{x_{2},\ldots ,x_{2}}^{d},\ldots, \overbrace{0,\ldots,0}^{p},\overbrace{x_{k}, \ldots,x_{k}}^{d}\bigr), \\ &\forall x_{1}, \ldots,x_{k} \in A \bigr\} . \end{aligned}

It is easy to see that $$(E, d)$$ is complete (see ).

Now, we consider the linear mapping $$\Lambda: E \rightarrow E$$ such that

$$\Lambda g(x): = \frac{1}{d} g(d x)$$

for all $$x \in A$$. From Theorem 3.1 of  (also see Lemma 3.2 of ),

$$d(\Lambda g, \Lambda h) \le L d(g, h)$$

for all $$g, h \in E$$. Let $$g,h\in E$$ and let $$C\in[0,\infty]$$ be an arbitrary constant with $$d(g,h)\leq C$$. From the definition of d, we have

$$\bigl\Vert \bigl(g(x_{1}) - h(x_{1}) , \ldots,g(x_{k}) - h(x_{k}) \bigr)\bigr\Vert _{k} \leq C \varphi\bigl(\overbrace{0,\ldots,0}^{p},\overbrace {x_{1},\ldots,x_{1}}^{d},\ldots,\overbrace{0, \ldots,0}^{p},\overbrace {x_{k},\ldots,x_{k}}^{d} \bigr)$$

for all $$x_{1},\ldots,x_{k}\in A$$. From our assumption and the last inequality, we have

\begin{aligned}& \bigl\Vert \bigl(\Lambda g(x_{1}) - \Lambda h(x_{1}), \ldots,\Lambda g(x_{k})- \Lambda h(x_{k})\bigr)\bigr\Vert _{k} \\& \quad =\frac{1}{d} \bigl\Vert \bigl(g(d x_{1})- h (d x_{1}) ,\ldots,g(d x_{k})- h (d x_{k})\bigr)\bigr\Vert _{k} \\& \quad \leq\frac{C}{d}\varphi \bigl(\overbrace{0,\ldots,0}^{p}, \overbrace{d x_{1},\ldots,d x_{1}}^{d},\ldots, \overbrace{0,\ldots,0}^{p},\overbrace{d x_{k},\ldots,d x_{k} }^{d}\bigr) \\& \quad \leq C L \varphi\bigl(\overbrace{0,\ldots,0}^{p},\overbrace {x_{1},\ldots,x_{1}}^{d},\ldots,\overbrace{0, \ldots,0}^{p},\overbrace {x_{k},\ldots,x_{k}}^{d} \bigr) \end{aligned}

for all $$x_{1},\ldots,x_{k} \in A$$. Thus we have

\begin{aligned}& \bigl\Vert \bigl(\Lambda f(x_{1}) - f(x_{1}),\ldots, \Lambda f(x_{k}) - f(x_{k}) \bigr) \bigr\Vert _{k} \\& \quad = \biggl\Vert \biggl(\frac{1}{d}f(d x_{1}) - f(x_{1}),\ldots,\frac{1}{d}f(d x_{k}) - f(x_{k})\biggr)\biggr\Vert _{k} \\& \quad = \frac{1}{d }\bigl\Vert \bigl(f(d x_{1}) - d f(x_{1}),\ldots,f(d x_{k}) - d f(x_{k})\bigr)\bigr\Vert _{k} \\& \quad \leq \frac{1}{2d}\varphi\bigl(\overbrace{0,\ldots,0}^{p}, \overbrace{x_{1},\ldots ,x_{1}}^{d},\ldots, \overbrace{0,\ldots,0}^{p},\overbrace{x_{k}, \ldots,x_{k}}^{d}\bigr) \end{aligned}

for all $$x_{1},\ldots,x_{k} \in A$$. Hence $$d(\Lambda f, f) \le \frac{1}{2d}$$. From Theorem 1.1, the sequence $$\{\Lambda^{n} f\}$$ converges to a fixed point H of Λ, i.e., $$H:A\rightarrow B$$ is a mapping defined by

$$H(x) = \lim_{n\to\infty}\bigl(\Lambda^{n} f\bigr) (x) = \lim_{n\to\infty}\frac{1}{d^{n}}f\bigl(d^{n} x\bigr)$$
(29)

and $$H(d x)=d H(x)$$ for all $$x \in A$$. Also, H is the unique fixed point of Λ in the set $$E'=\{ g \in E : d(f,g)< \infty\}$$ and

$$d(H,f)\leq\frac{1}{1-L}d(\Lambda f , f)\leq\frac{1}{(1-L)2d},$$

i.e., the inequality (27) hold for all $$x_{1},\ldots ,x_{k} \in A$$. It follows from the definition of H, (21), and (22) that

\begin{aligned}& \Biggl\Vert 2H \Biggl(\frac{\sum_{j=1}^{p} \mu x_{1j}}{2}+\sum_{j=1}^{d} \mu y_{1j} \Biggr) - \sum_{j=1}^{p} \mu H(x_{1j})-2 \sum_{j=1}^{d} \mu H(y_{1j}), \\& \qquad \ldots,2H \Biggl(\frac{\sum_{j=1}^{p} \mu x_{kj}}{2}+\sum_{j=1}^{d} \mu y_{kj} \Biggr) - \sum_{j=1}^{p} \mu H(x_{kj})-2 \sum_{j=1}^{d} \mu H(y_{kj}) \Biggr\Vert _{k} \\& \quad = \lim_{n\to\infty} \frac{1}{d^{n}} \Biggl\Vert 2 f \Biggl(d^{n}\frac{\sum_{j=1}^{p} \mu x_{1j}}{2}+d^{n}\sum _{j=1}^{d} \mu y_{1j} \Biggr) - \sum _{j=1}^{p} \mu f\bigl(d^{n} x_{1j}\bigr)-2 \sum_{j=1}^{d} \mu f\bigl(d^{n}y_{1j}\bigr), \\& \qquad \ldots,2 f \Biggl(d^{n}\frac{\sum_{j=1}^{p} \mu x_{kj}}{2}+d^{n}\sum _{j=1}^{d} \mu y_{kj} \Biggr) - \sum_{j=1}^{p} \mu f\bigl(d^{n} x_{kj}\bigr)-2 \sum_{j=1}^{d} \mu f\bigl(d^{n}y_{kj}\bigr) \Biggr\Vert _{k} \\& \quad \leq\lim_{n\to\infty} \frac{1}{d^{n}}\bigl\Vert \bigl(C_{\mu}f\bigl(d^{n} x_{11}, \ldots,d^{n} x_{1p},d^{n} y_{11}, \ldots,d^{n}y_{1d}\bigr), \\& \qquad \ldots, C_{\mu}f\bigl(d^{n} x_{k1}, \ldots,d^{n} x_{kp},d^{n} y_{k1}, \ldots,d^{n}y_{kd}\bigr)\bigr)\bigr\Vert _{k} \\& \qquad {}+ \lim_{n\to\infty}\frac{1}{d^{n}}\varphi \bigl(d^{n} x_{11},\ldots,d^{n} x_{1p},d^{n} y_{11},\ldots,d^{n} y_{1d}, \\& \qquad \ldots,d^{n} x_{k1},\ldots,d^{n} x_{kp},d^{n} y_{k1},\ldots,d^{n} y_{kd}\bigr)=0 \end{aligned}

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A$$. Hence we have

$$2H \Biggl(\frac{\sum_{j=1}^{p} \mu x_{ij}}{2}+\sum_{j=1}^{d} \mu y_{ij} \Biggr) = \sum_{j=1}^{p} \mu H(x_{ij})+2 \sum_{j=1}^{d} \mu H(y_{ij})$$

for all $$\mu\in\mathbf{T}^{1}$$, $$x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A$$ and $$1\leq i \leq k$$ and so $$H(\lambda x + \mu y)=\lambda H(x) + \mu H(y)$$ for all $$\lambda, \mu\in\mathbf{T}^{1}$$ and all $$x,y \in A$$. Therefore, by Lemma 3.1, the mapping $$H : A \rightarrow B$$ is C-linear.

Also it follows from (23) and (24) that

\begin{aligned}& \bigl\Vert H\bigl([x_{1}, y_{1}, z_{1}]\bigr)- \bigl[H(x_{1}), H(y_{1}), H(z_{1})\bigr],\ldots,H \bigl([x_{k}, y_{k}, z_{k}]\bigr)- \bigl[H(x_{k}), H(y_{k}), H(z_{k})\bigr]\bigr\Vert _{k} \\& \quad =\lim_{n\to\infty}\frac{1}{d^{3n}} \bigl\Vert f \bigl( \bigl[d^{n} x_{1}, d^{n} y_{1}, d^{n} z_{1}\bigr] \bigr) - \bigl[f\bigl(d^{n} x_{1}\bigr), f\bigl(d^{n} y_{1}\bigr), f \bigl(d^{n} z_{1}\bigr) \bigr], \\& \qquad \ldots, f \bigl(\bigl[d^{n} x_{k}, d^{n} y_{k}, d^{n} z_{k}\bigr] \bigr) - \bigl[f \bigl(d^{n} x_{k}\bigr), f\bigl(d^{n} y_{k}\bigr), f\bigl(d^{n} z_{k}\bigr) \bigr]\bigr\Vert _{k} \\& \quad \leq\lim_{n\to\infty} \frac{1}{d^{3n}}\psi \bigl(d^{n}x_{1},d^{n}y_{1},d^{n}z_{1}, \ldots ,d^{n}x_{k},d^{n}y_{k},d^{n}z_{k} \bigr)=0 \end{aligned}

for all $$x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} \in A$$. Thus

$$H\bigl([x, y,z]\bigr) = \bigl[H(x), H(y), H(z)\bigr]$$

for all $$x, y, z \in A$$.Thus $$H : A \rightarrow B$$ is a homomorphism satisfying (26).

Now, let $$T : A \rightarrow B$$ be another $$C^{*}$$-ternary algebras homomorphism satisfying (27). Since $$d(f,T)\leq \frac{1}{(1-L)2d}$$ and T is C-linear, we get $$T\in E'$$ and $$(\Lambda T)(x)=\frac{1}{d}(T\gamma x)=T(x)$$ for all $$x\in A$$, i.e., T is a fixed point of Λ. Since H is the unique fixed point of $$\Lambda\in E'$$, we get $$H=T$$. This completes the proof. □

### Theorem 3.7

Let r, s, and θ be non-negative real numbers such that $$0< r\neq1$$, $$0< s\neq3$$, and let $$d\ge2$$. Suppose that $$f : A \rightarrow B$$ is a mapping with $$f(0)=0$$ satisfying (18) and

\begin{aligned}& \bigl\Vert \bigl( f\bigl([x_{1},y_{1},z_{1}] \bigr)-\bigl[f(x_{1}),f(y_{1}),f(z_{1})\bigr],\ldots ,f\bigl([x_{k},y_{k},z_{k}]\bigr)- \bigl[f(x_{k}),f(y_{k}),f(z_{k})\bigr] \bigr)\bigr\Vert \\& \quad \leq\theta \bigl( \|x_{1}\|^{s}_{A}\cdot \|y_{1}\|^{s}_{A}\cdot\|z_{1} \|^{s}_{A} + \cdots+ \|x_{k}\|^{s}_{A} \cdot\|y_{k}\|^{s}_{A}\cdot\|z_{k} \|^{s}_{A} \bigr) \end{aligned}
(30)

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k}, z_{1},\ldots,z_{k} \in A$$. Then there exists a unique $$C^{*}$$-ternary algebra homomorphism $$H : A \rightarrow B$$ such that

\begin{aligned}& \bigl\Vert \bigl(f(x_{1}) - H(x_{1}), \ldots,f(x_{k}) - H(x_{k}) \bigr)\bigr\Vert _{K} \\& \quad \le\frac{d\theta}{2|d-d^{r}|} \bigl(\|x_{1}\|^{r}_{A} + \cdots+\|x_{k}\| ^{r}_{A} \bigr) \end{aligned}
(31)

for all $$x_{1},\ldots,x_{k} \in A$$.

### Proof

We only prove the theorem when $$0< r<1$$ and $$0< s<3$$. One can prove the theorem for the other cases in a similar way. The proof follows from Theorem 3.6 by taking

\begin{aligned}& \varphi (x_{11},\ldots,x_{1p},y_{11}, \ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1}, \ldots,y_{kd}) \\& \quad :=\theta \Biggl(\sum_{j=1}^{p} \|x_{1j}\|^{r}_{A} + \sum _{j=1}^{d}\|y_{1j}\|^{r}_{A} + \cdots+ \sum_{j=1}^{p}\|x_{kj} \|^{r}_{A} + \sum_{j=1}^{d} \|y_{kj}\|^{r}_{A} \Biggr), \\& \psi(x_{1},y_{1},z_{1},\ldots,x_{k},y_{k},z_{k}):= \theta\bigl(\|x_{1}\|^{s}_{A} \cdot \|y_{1}\|^{s}_{A} \cdot \|z_{1} \|^{s}_{A} + \cdots+\|x_{k}\|^{s}_{A} \cdot \|y_{k}\|^{s}_{A} \cdot \|z_{k} \|^{s}_{A}\bigr) \end{aligned}

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd},x_{1},\ldots,x_{k},y_{1},\ldots,y_{k}, z_{1}, \ldots,z_{k} \in A$$. Then we can choose $$L=d^{r-1}$$, when $$0< r<1$$ and $$0< s<3$$, and $$L=2-d^{r-1}$$, when $$r>1$$ and $$s>3$$, and so we get the desired result. □

Now, assume that A is a unital $$C^{*}$$-ternary algebra with norm $$\| \cdot\|$$ and unit e and B is a unital $$C^{*}$$-ternary algebra with norm $$\| \cdot\|$$ and unit $$e'$$.

We investigate homomorphisms in $$C^{*}$$-ternary algebras associated with the functional equation $$C_{\mu}f(x_{1}, \ldots, x_{p}, y_{1}, \ldots, y_{d})=0$$.

### Theorem 3.8

()

Let $$r > 1$$ (resp., $$r<1$$) and θ be non-negative real numbers and let $$f : A \rightarrow B$$ be a bijective mapping satisfying (18) and

$$f\bigl([x, y, z]\bigr) = \bigl[f(x), f(y), f(z)\bigr]$$

for all $$x, y, z \in A$$. If $$\lim_{n\rightarrow\infty} \frac{(p+2d)^{n}}{2^{n}} f(\frac{2^{n}e}{(p+2d)^{n}}) = e'$$ (resp., $$\lim_{n\rightarrow\infty} \frac{2^{n}}{(p+2d)^{n}} f(\frac{(p+2d)^{n}}{2^{n}} e) = e'$$), then the mapping $$f : A \rightarrow B$$ is a $$C^{*}$$-ternary algebra isomorphism.

### Theorem 3.9

Let $$r<1$$ and θ be non-negative real numbers and let $$f : A \rightarrow B$$ be a mapping satisfying (18) and (19). If there exist a real number $$\lambda>1$$ (resp., $$0<\lambda<1$$) and an element $$x_{0}\in A$$ such that $$\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f(\lambda^{n} x_{0}) = e'$$ (resp., $$\lim_{n\rightarrow\infty} \lambda^{n} f(\frac{x_{0}}{\lambda^{n}}) = e'$$), then the mapping $$f : A \rightarrow B$$ is a multi-$$C^{*}$$-ternary algebra homomorphism.

### Proof

By using the proof of Theorem 3.5, there exists a unique multi-$$C^{*}$$-ternary algebra homomorphism $$H : A \rightarrow B$$ satisfying (20). It follows from (20) that

$$H(x)=\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f\bigl( \lambda^{n} x\bigr)\qquad \biggl(\mbox{resp.}, H(x)=\lim _{n\rightarrow\infty} \lambda^{n} f\biggl(\frac{x}{\lambda^{n}}\biggr) \biggr)$$

for all $$x\in A$$ and $$\lambda>1$$ ($$0<\lambda<1$$). Therefore, from our assumption, we get $$H(x_{0})=e'$$.

Let $$\lambda>1$$ and $$\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f(\lambda^{n} x_{0}) = e'$$. It follows from (19) that

\begin{aligned}& \bigl\Vert \bigl( \bigl[H(x_{1}),H(y_{1}),H(z_{1}) \bigr] - \bigl[H(x_{1}),H(y_{1}),f(z_{1})\bigr], \\& \qquad \ldots,\bigl[H(x_{k}),H(y_{k}),H(z_{k}) \bigr] - \bigl[H(x_{k}),H(y_{k}),f(z_{k})\bigr] \bigr)\bigr\Vert \\& \quad = \bigl\Vert \bigl( H[x_{1},y_{1},z_{1}] - \bigl[H(x_{1}),H(y_{1}),f(z_{1})\bigr], \\& \qquad \ldots,H[x_{k},y_{k},z_{k}] - \bigl[H(x_{k}),H(y_{k}),f(z_{k})\bigr] \bigr)\bigr\Vert \\& \quad =\lim_{n\to\infty}\frac{1}{\lambda ^{2n}}\bigl\Vert \bigl(f\bigl( \bigl[\lambda^{n}x_{1},\lambda^{n}y_{1},z_{1} \bigr]\bigr) - \bigl[f\bigl(\lambda^{n}x_{1}\bigr),f\bigl( \lambda^{n}y_{1}\bigr),f(z_{1})\bigr], \\& \qquad \ldots,f\bigl(\bigl[\lambda^{n}x_{k},\lambda ^{n}y_{k},z_{k}\bigr]\bigr) - \bigl[f\bigl( \lambda^{n}x_{k}\bigr),f\bigl(\lambda^{n}y_{k} \bigr),f(z_{k})\bigr] \bigr)\bigr\Vert \\& \quad \leq\lim_{n\to\infty}\frac{\lambda^{rn}}{\lambda^{3n}}\theta\bigl( \|x_{1}\|^{r}_{A} \cdot\|y_{1} \|^{r}_{A} \cdot\|z_{1}\|^{r}_{A} + \cdots+ \|x_{k}\|^{r}_{A} \cdot\|y_{k} \| ^{r}_{A} \cdot\|z_{k}\|^{r}_{A} \bigr)= 0 \end{aligned}

for all $$x_{1},\ldots,x_{k}\in A$$. Thus $$[H(x),H(y),H(z)]=[H(x), H(y), f(z)]$$ for all $$x,y,z\in A$$. Letting $$x=y=x_{0}$$ in the last equality, we get $$f(z)=H(z)$$ for all $$z\in A$$. Similarly, one can show that $$H(x)=f(x)$$ for all $$x\in A$$ when $$0<\lambda<1$$ and $$\lim_{n\rightarrow\infty} \lambda^{n} f(\frac{x_{0}}{\lambda^{n}})=e'$$.

Similarly, one can show the theorem for the case $$\lambda>1$$. Therefore, the mapping $$f : A \rightarrow B$$ is a multi-$$C^{*}$$-ternary algebra homomorphism. This completes the proof. □

### Theorem 3.10

Let $$r>1$$ and θ be non-negative real numbers and let $$f : A \rightarrow B$$ be a mapping satisfying (18) and (19). If there exist a real number $$\lambda>1$$ (resp., $$0<\lambda<1$$) and an element $$x_{0}\in A$$ such that $$\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f(\lambda^{n} x_{0}) = e'$$ (resp., $$\lim_{n\rightarrow\infty} \lambda^{n} f(\frac{x_{0}}{\lambda^{n}}) = e'$$), then the mapping $$f : A \rightarrow B$$ is a multi-$$C^{*}$$-ternary algebra homomorphism.

### Proof

The proof is similar to the proof of Theorem 3.9 and we omit it. □

## Approximation of derivations on multi-$$C^{*}$$-ternary algebras

Throughout this section, assume that A is a $$C^{*}$$-ternary algebra with norm $$\| \cdot\|$$.

Park  studied approximation of derivations on $$C^{*}$$-ternary algebras for the functional equation $$C_{\mu}f(x_{1}, \ldots, x_{p}, y_{1}, \ldots, y_{d}) =0$$ (see also [5, 13, 1559] and ).

For any mapping $$f : A \rightarrow A$$, let

$${\mathbf{D}} f(x,y,z)=f\bigl([x, y, z]\bigr)-\bigl[f(x), y, z\bigr]-\bigl[x, f(y), z\bigr]-\bigl[x, y, f(z)\bigr]$$

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

### Theorem 4.1

()

Let r and θ be non-negative real numbers such that $$r\notin[1,3]$$ and let $$f:A \rightarrow A$$ be a mapping satisfying (19) and

$$\bigl\Vert {\mathbf{D}}f(x,y,z)\bigr\Vert \le\theta\bigl(\|x\|^{r} + \|y\|^{r} + \|z\|^{r}\bigr)$$

for all $$x, y, z \in A$$. Then there exists a unique $$C^{*}$$-ternary derivation $$\delta:A\rightarrow A$$ such that

$$\bigl\Vert f(x)-\delta(x) \bigr\Vert \le\frac{2^{r}(p+d)}{|2(p+2d)^{r} -(p+2d)2^{r}|} \theta \|x\|^{r}$$

for all $$x \in A$$.

In the following theorem, we generalize and improve the result in Theorem 4.1.

### Theorem 4.2

Let $$(( {A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ be a multi-$$C^{*}$$-ternary algebra. Let $$f:A\rightarrow A$$ be a mapping for which there are the functions $$\varphi: A^{(p+d)k} \rightarrow[0, \infty)$$ and $$\psi : A^{3k} \rightarrow[0, \infty)$$ satisfying the inequalities (1), (2), and (4) such that

$$\bigl\Vert \bigl({\mathbf{D}} f(x_{1}, y_{1}, z_{1}),\ldots,{\mathbf{D}} f(x_{k}, y_{k}, z_{k}) \bigr)\bigr\Vert \le\psi(x_{1}, y_{1},z_{1},\ldots,x_{k},y_{k},z_{k})$$
(32)

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots ,y_{1d},\ldots,x_{k1},\ldots,x_{kp},y_{k1},\ldots,y_{kd}, x_{1},\ldots,x_{k}, y_{1},\ldots,y_{k},z_{1}, \ldots,z_{k}\in A$$, where $$\gamma=\frac{p+2d}{2}$$. If the constant $$L<1$$ exists such that

\begin{aligned}& \varphi\bigl(\overbrace{\gamma x_{1},\ldots,\gamma x_{1}}^{p+d},\overbrace {\gamma x_{2},\ldots,\gamma x_{2} }^{p+d},\ldots,\overbrace{\gamma x_{k}, \ldots,\gamma x_{k}}^{p+d}\bigr) \\& \quad \le\gamma L \varphi\bigl(\overbrace{x_{1},\ldots,x_{1}}^{p+d}, \overbrace {x_{2},\ldots,x_{2}}^{p+d},\ldots, \overbrace{x_{k},\ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(33)

for all $$x_{1},x_{2},\ldots,x_{k} \in A$$, then there exists a unique $$C^{*}$$-ternary derivation $$\delta: A \rightarrow B$$ such that

\begin{aligned}& \bigl\Vert \bigl(f(x_{1}) - \delta(x_{1}), \ldots,f(x_{k}) - \delta(x_{k}) \bigr) \bigr\Vert _{k} \\& \quad \le\frac{1}{(1-L)2\gamma} \varphi\bigl(\overbrace{x_{1}, \ldots,x_{1}}^{p+d},\overbrace{x_{2},\ldots ,x_{2}}^{p+d},\ldots,\overbrace{x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(34)

for all $$x_{1},\ldots,x_{k} \in A$$.

### Proof

The same reasoning as in the proof of Theorem 3.3, guarantees there exists a unique C-linear mapping $$\delta :A\rightarrow A$$ satisfying (32). The mapping $$\delta:A\rightarrow A$$ is given by

$$\delta(x) = \lim_{n\to\infty}\bigl( \Lambda^{n} f\bigr) (x) = \lim_{n\to\infty}\frac{1}{\gamma^{n}}f \bigl(\gamma^{n} x\bigr)$$
(35)

and $$\delta(\gamma x)=\gamma\delta(x)$$ for all $$x \in A$$. Also, H is the unique fixed point of Λ in the set $$E'=\{ g \in E : d(f,g)< \infty\}$$ and

$$d(\delta,f)\leq\frac{1}{1-L}d(\Lambda f , f)\leq\frac{1}{(1-L)2\gamma },$$

i.e., the inequality (6) holds for all $$x_{1},\ldots ,x_{k} \in A$$. It follows from the definition of δ, (1) and (2), and (35) that

\begin{aligned}& \bigl\Vert \bigl(C_{\mu}\delta(x_{11},\ldots,x_{1p}y_{11}, \ldots, y_{1d}),\ldots,C_{\mu}\delta(x_{k1}, \ldots,x_{kp}y_{k1},\ldots, y_{kd}) \bigr) \bigr\Vert _{k} \\& \quad = \lim_{n\to\infty} \frac{1}{\gamma^{n}} \bigl\Vert \bigl(C_{\mu}f\bigl(\gamma^{n} x_{11},\ldots, \gamma^{n} x_{1p},\gamma^{n} y_{11}, \ldots,\gamma^{n} y_{1d}\bigr), \\& \qquad \ldots,C_{\mu}f \bigl(\gamma^{n} x_{k1}, \ldots,\gamma^{n} x_{kp},\gamma^{n} y_{k1},\ldots,\gamma^{n} y_{kd}\bigr) \bigr) \bigr\Vert _{k} \\& \quad \leq\lim_{n\to\infty}\frac{1}{\gamma^{n}}\varphi\bigl( \gamma^{n} x_{11},\ldots,\gamma^{n} x_{1p}.\gamma^{n} y_{11},\ldots, \gamma^{n} y_{1d}, \\& \qquad \ldots,\gamma^{n} x_{k1},\ldots,\gamma^{n} x_{kp},\gamma^{n} y_{k1},\ldots, \gamma^{n} y_{kd}\bigr)=0 \end{aligned}

for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A$$. Hence we have

$$2\delta \Biggl(\frac{\sum_{j=1}^{p} \mu x_{ij}}{2}+\sum_{j=1}^{d} \mu y_{ij} \Biggr) = \sum_{j=1}^{p} \mu\delta(x_{ij})+2 \sum_{j=1}^{d} \mu\delta(y_{ij})$$

for all $$\mu\in\mathbf{T}^{1}$$, $$x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd} \in A$$ and $$1\leq i \leq k$$ and so $$\delta(\lambda x + \mu y)=\lambda \delta(x) + \mu \delta(y)$$ for all $$\lambda, \mu\in\mathbf{T}^{1}$$ and $$x,y \in A$$. Therefore, by Lemma 3.1, the mapping $$\delta: A \rightarrow B$$ is C-linear.

Also it follows from (4) and (32) that

\begin{aligned}& \bigl\Vert \bigl({\mathbf{D}}\delta(x_{1},y_{1},z_{1}), \ldots,{\mathbf{D}}\delta (x_{k},y_{k},z_{k}) \bigr) \bigr\Vert _{k} \\& \quad =\lim_{n\to\infty}\frac{1}{\gamma^{3n}} \bigl\Vert f \bigl({ \mathbf{D}}f\bigl(\gamma^{n} x_{1}, \gamma^{n} y_{1}, \gamma^{n} z_{1}\bigr),\ldots,f\bigl( \gamma^{n} x_{k}, \gamma^{n} y_{k}, \gamma^{n} z_{k}\bigr) \bigr)\bigr\Vert \\& \quad \leq\lim_{n\to\infty} \frac{1}{\gamma^{3n}}\psi\bigl( \gamma^{n}x_{1},\gamma^{n}y_{1},\gamma ^{n}z_{1},\ldots,\gamma^{n}x_{k}, \gamma^{n}y_{k},\gamma^{n}z_{k}\bigr)=0 \end{aligned}

for all $$x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} \in A$$ and hence

\begin{aligned}& \bigl(\delta\bigl([x_{1}, y_{1},z_{1}] \bigr),\ldots,\delta\bigl([x_{k}, y_{k},z_{k}]\bigr) \bigr) \\& \quad {} + \bigl(\bigl[\delta(x_{1}),(y_{1}), (z_{1})\bigr] + \bigl[x_{1},\delta(y_{1}),z_{1} \bigr] + \bigl[x_{1},y_{1},\delta(z_{1})\bigr], \\& \quad \ldots,\bigl[\delta(x_{k}),(y_{k}), (z_{k}) \bigr] + \bigl[x_{k},\delta(y_{k}),z_{k}\bigr] + \bigl[x_{k},y_{k},\delta(z_{k})\bigr] \bigr) \end{aligned}
(36)

for all $$x, y, z \in A$$ and so the mapping $$\delta: A \rightarrow A$$ is a $$C^{*}$$-ternary derivation. It follows from (32) and (4) that

\begin{aligned}& \bigl\Vert \bigl( \delta[x_{1},y_{1},z_{1}]- \bigl[\delta(x_{1}),y_{1},z_{1}\bigr]- \bigl[x_{1},\delta (y_{1}),z_{1}\bigr]- \bigl[x,y,f(z_{1})\bigr], \\& \qquad \ldots,\delta[x_{k},y_{k},z_{k}]-\bigl[ \delta(x_{k}),y_{k},z_{k}\bigr]- \bigl[x_{k},\delta (y_{k}),z_{k}\bigr]- \bigl[x,y,f(z_{k})\bigr] \bigr)\bigr\Vert \\& \quad =\lim_{n\to\infty}\frac{1}{\gamma^{2n}} \bigl\Vert \bigl(f\bigl[ \gamma^{n} x_{1},\gamma^{n} y_{1},z_{1} \bigr]-\bigl[f\bigl(\gamma^{n} x_{1}\bigr), \gamma^{n} y_{1},z_{1}\bigr] \\& \qquad {}-\bigl[\gamma^{n} x_{1},f\bigl(\gamma^{n} y_{1}\bigr),z_{1}\bigr]-\bigl[\gamma^{n} x_{1},\gamma^{n} y_{1},f(z_{1})\bigr], \\& \qquad \ldots,f\bigl[\gamma^{n} x_{k},\gamma^{n} y_{k},z_{k}\bigr]-\bigl[f\bigl(\gamma^{n} x_{k}\bigr),\gamma^{n} y_{k},z_{k}\bigr] \\& \qquad {}-\bigl[\gamma^{n} x_{k},f\bigl(\gamma^{n} y_{k}\bigr),z_{k}\bigr]-\bigl[\gamma^{n} x_{k},\gamma^{n} y_{k},f(z_{k})\bigr] \bigr)\bigr\Vert \\& \quad \le\lim_{n\rightarrow\infty}\frac{1}{\gamma^{2n}}\psi\bigl(\gamma ^{n}x_{1},\gamma^{n}y_{1},z_{1}, \ldots,\gamma^{n}x_{k},\gamma^{n}y_{k},z_{k} \bigr)=0 \end{aligned}

for all $$x_{1},y_{1},z_{1},\ldots,x_{k},y_{k},z_{k}\in A$$ and so we have

$$\bigl(\delta[x,y,z]\bigr) =\bigl[\delta(x),y,z\bigr]+\bigl[x, \delta(y),z\bigr]+\bigl[x,y,f(z)\bigr]$$
(37)

for all $$x,y,z\in A$$. Hence it follows from (36) and (37) that

$$\bigl[x,y,\delta(z)\bigr]=\bigl[x,y,f(z)\bigr]$$
(38)

for all $$x,y,z\in A$$. Letting $$x=y=f(z)-\delta(z)$$ in (38), we get

$$\bigl\Vert f(z)-\delta(z)\bigr\Vert ^{3}= \bigl\Vert \bigl[f(z)-\delta(z),f(z)-\delta(z),f(z)-\delta(z) \bigr]\bigr\Vert =0$$
(39)

for all $$z_{1},\ldots,z_{k} \in A$$ and hence $$f(z)=\delta(z)$$ for all $$z\in A$$. Therefore, the mapping $$f:A\rightarrow A$$ is a $$C^{*}$$-ternary derivation. This completes the proof. □

### Corollary 4.3

Let $$r<1$$, $$s<2$$, and θ be non-negative real numbers and let $$f:A \rightarrow A$$ be a mapping satisfying (18) and

\begin{aligned}& \bigl\Vert \bigl({\mathbf{D}} f(x_{1},y_{1},z_{1}), \ldots,{\mathbf {D}}f(x_{k},y_{k},z_{k}) \bigr)\bigr\Vert \\& \quad \le\theta\bigl(\|x_{1}\|^{s}_{A} \cdot \|y_{1}\|^{s}_{A} \cdot\|z_{1} \|^{s}_{A} + \cdots+\|x_{k}\| ^{s}_{A} \cdot \|y_{k}\|^{s}_{A} \cdot\|z_{k} \|^{s}_{A}\bigr) \end{aligned}

for all $$x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} \in A$$. Then the mapping $$f:A \rightarrow A$$ is a $$C^{*}$$-ternary derivation.

### Proof

Define

\begin{aligned}& \varphi(x_{11},\ldots,x_{1p},y_{11}, \ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1}, \ldots,y_{kd}) \\& \quad =\theta \Biggl( \sum_{j=1}^{p} \|x_{1j}\|^{r}_{A} + \sum _{j=1}^{d}\|y_{1j}\|^{r}_{A}, \ldots,\sum_{j=1}^{p}\|x_{kj} \|^{r}_{A} + \sum_{j=1}^{d} \|y_{kj}\|^{r}_{A} \Biggr) \end{aligned}

and

\begin{aligned} \begin{aligned} &\psi(x_{1},y_{1},z_{1},\ldots,x_{k},y_{k},z_{k}) \\ &\quad =\theta \bigl(\|x_{1}\|^{s}_{A}\cdot \|y_{1}\|^{s}_{A}\cdot\|z_{1} \|^{s}_{A} + \cdots+ \|x_{k}\|^{s}_{A} \cdot\|y_{k}\|^{s}_{A}\cdot\|z_{k} \|^{s}_{A} \bigr) \end{aligned} \end{aligned}

for all $$x_{1}, y_{1}, z_{1},\ldots,x_{k},y_{k},z_{k} , x_{11},\ldots,x_{1p},y_{11},\ldots,y_{1d},\ldots,x_{k1},\ldots ,x_{kp},y_{k1},\ldots,y_{kd}\in A$$ and applying Theorem 4.2, we get the desired result. □

### Theorem 4.4

Let $$(( {A}^{k},\|\cdot\|_{k}): k\in\mathbf{N})$$ be a multi-$$C^{*}$$-ternary algebra. Let $$f: A \rightarrow A$$ be a mapping for which there are the functions $$\varphi: A^{(p+d)k} \rightarrow[0, \infty)$$ and $$\psi: A^{3k} \rightarrow[0, \infty)$$ satisfying the inequalities (2), (11), (12), and (32) for all $$\mu\in\mathbf{T}^{1}$$ and $$x_{11},\ldots ,x_{1p},y_{11},\ldots, y_{1d},\ldots,x_{k1}, \ldots,x_{kp},y_{k1}, \ldots,y_{kd}, x_{1},\ldots,x_{k},y_{1},\ldots,y_{k},z_{1},\ldots,z_{k}\in A$$, where $$\gamma=\frac{p+2d}{2}$$. If there exists the constant $$L<1$$ such that

\begin{aligned}& \varphi\biggl(\overbrace{\frac{x_{1}}{\gamma} ,\ldots, \frac{x_{1}}{\gamma} }^{p+d},\overbrace{\frac{x_{2}}{\gamma} ,\ldots, \frac{x_{2}}{\gamma} }^{p+d},\ldots,\overbrace{\frac{x_{k}}{\gamma} ,\ldots, \frac{x_{k}}{\gamma } }^{p+d}\biggr) \\& \quad \le\frac{L}{\gamma} \varphi\bigl(\overbrace{x_{1},\ldots ,x_{1}}^{p+d},\overbrace{x_{2}, \ldots,x_{2}}^{p+d},\ldots,\overbrace {x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(40)

for all $$x_{1},x_{2},\ldots,x_{k} \in A$$, then there exists a unique homomorphism $$\delta: A \rightarrow A$$ such that

\begin{aligned}& \bigl\Vert \bigl(f(x_{1}) - \delta(x_{1}), \ldots,f(x_{k}) - \delta(x_{k})\bigr) \bigr\Vert _{k} \\& \quad \le\frac{1}{(1-L)2\gamma} \varphi\bigl(\overbrace{x_{1}, \ldots,x_{1}}^{p+d},\overbrace{x_{2},\ldots ,x_{2}}^{p+d},\ldots,\overbrace{x_{k}, \ldots,x_{k}}^{p+d}\bigr) \end{aligned}
(41)

for all $$x_{1},\ldots,x_{k} \in A$$.

### Proof

The same reasoning as in the proof of Theorem 3.4 guarantees there exists a unique C-linear mapping $$\delta :A\rightarrow A$$ satisfying (32). The rest of the proof is similar to the proof of Theorem 4.2 and so we omit it. □

## References

1. Takhtajan, L: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160, 295-315 (1994)

2. Abramov, V, Kerner, R, Le Roy, B: Hypersymmetry: a $$Z_{3}$$ graded generalization of supersymmetry. J. Math. Phys. 38, 1650-1669 (1997)

3. Vainerman, L, Kerner, R: On special classes of n-algebras. J. Math. Phys. 37, 2553-2565 (1996)

4. Zettl, H: A characterization of ternary rings of operators. Adv. Math. 48, 117-143 (1983)

5. Park, C: Isomorphisms between $$C^{*}$$-ternary algebras. J. Math. Anal. Appl. 327, 101-115 (2007)

6. Kerner, R: The cubic chessboard: geometry and physics. Class. Quantum Gravity 14, A203-A225 (1997)

7. Diaz, J, Margolis, B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305-309 (1968)

8. Dales, HG, Polyakov, ME: Multi-normed spaces and multi-Banach algebras. Preprint

9. Dales, HG, Moslehian, MS: Stability of mappings on multi-normed spaces. Glasg. Math. J. 49, 321-332 (2007)

10. Moslehian, MS, Nikodem, K, Popa, D: Asymptotic aspect of the quadratic functional equation in multi-normed spaces. J. Math. Anal. Appl. 355, 717-724 (2009)

11. Moslehian, MS: Superstability of higher derivations in multi-Banach algebras. Tamsui Oxf. J. Math. Sci. 24, 417-427 (2008)

12. Dales, HG: Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, New Series, vol. 24. Oxford University Press, Oxford (2000)

13. Park, C: Homomorphisms between Poisson $$JC^{*}$$-algebras. Bull. Braz. Math. Soc. 36, 79-97 (2005)

14. Cădariu, L, Radu, V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4, Article ID 4 (2003)

15. Ulam, SM: A Collection of the Mathematical Problems. Interscience, New York (1960)

16. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222-224 (1941)

17. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297-300 (1978)

18. Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64-66 (1950)

19. Bourgin, DG: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 57, 223-237 (1951)

20. Găvruţa, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431-436 (1994)

21. Agarwal, RP, Cho, YJ, Saadati, R, Wang, S: Nonlinear L-fuzzy stability of cubic functional equations. J. Inequal. Appl. 2012, 77 (2012)

22. Baktash, E, Cho, YJ, Jalili, M, Saadati, R, Vaezpour, SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. J. Inequal. Appl. 2008, Article ID 902187 (2008)

23. Chahbi, A, Bounader, N: On the generalized stability of d’Alembert functional equation. J. Nonlinear Sci. Appl. 6(3), 198-204 (2013)

24. Yao, Z: Uniqueness and global exponential stability of almost periodic solution for hematopoiesis model on time scales. J. Nonlinear Sci. Appl. 8(2), 142-152 (2015)

25. Cho, YJ, Kang, JI, Saadati, R: Fixed points and stability of additive functional equations on the Banach algebras. J. Comput. Anal. Appl. 14, 1103-1111 (2012)

26. Cho, YJ, Kang, SM, Sadaati, R: Nonlinear random stability via fixed-point method. J. Appl. Math. 2012, Article ID 902931 (2012)

27. Wang, Z, Sahoo, PK: Stability of an ACQ-functional equation in various matrix normed spaces. J. Nonlinear Sci. Appl. 8(1), 64-85 (2015)

28. Cho, YJ, Park, C, Yang, Y-O: Stability of derivations in fuzzy normed algebras. J. Nonlinear Sci. Appl. 8(1), 1-7 (2015)

29. Raffoul, Y, Unal, M: Stability in nonlinear delay Volterra integro-differential systems. J. Nonlinear Sci. Appl. 7(6), 422-428 (2014)

30. Shen, Y, Lan, Y: On the general solution of a quadratic functional equation and its Ulam stability in various abstract spaces. J. Nonlinear Sci. Appl. 7(6), 368-378 (2014)

31. Wang, F, Shen, Y: On the Ulam stability of a quadratic set-valued functional equation. J. Nonlinear Sci. Appl. 7(5), 359-367 (2014)

32. Chahbi, A, Bounader, N: On the generalized stability of d’Alembert functional equation. J. Nonlinear Sci. Appl. 6(3), 198-204 (2013)

33. Urs, C: Ulam-Hyers stability for coupled fixed points of contractive type operators. J. Nonlinear Sci. Appl. 6(2), 124-136 (2013)

34. Mlesnite, O: Existence and Ulam-Hyers stability results for coincidence problems. J. Nonlinear Sci. Appl. 6(2), 108-116 (2013)

35. Zaharia, C: On the probabilistic stability of the monomial functional equation. J. Nonlinear Sci. Appl. 6(1), 51-59 (2013)

36. Cădariu, L, Găvruţa, L, Găvruţa, P: On the stability of an affine functional equation. J. Nonlinear Sci. Appl. 6(2), 60-67 (2013)

37. Park, C: On the stability of the linear mapping in Banach modules. J. Math. Anal. Appl. 275, 711-720 (2002)

38. Park, C: Modified Trif’s functional equations in Banach modules over a $$C^{*}$$-algebra and approximate algebra homomorphisms. J. Math. Anal. Appl. 278, 93-108 (2003)

39. Park, C: Orthogonal stability of a cubic-quartic functional equation. J. Nonlinear Sci. Appl. 5(1), 28-36 (2012) (special issue)

40. Park, C: On an approximate automorphism on a $$C^{*}$$-algebra. Proc. Am. Math. Soc. 132, 1739-1745 (2004)

41. Park, C: Lie -homomorphisms between Lie $$C^{*}$$-algebras and Lie -derivations on Lie $$C^{*}$$-algebras. J. Math. Anal. Appl. 293, 419-434 (2004)

42. Park, C: Homomorphisms between Lie $$JC^{*}$$-algebras and Cauchy-Rassias stability of Lie $$JC^{*}$$-algebra derivations. J. Lie Theory 15, 393-414 (2005)

43. Park, C: Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between $$C^{*}$$-algebras. Bull. Belg. Math. Soc. Simon Stevin 13, 619-631 (2006)

44. Park, C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007, Article ID 50175 (2007)

45. Park, C: Additive ρ-functional inequalities. J. Nonlinear Sci. Appl. 7(5), 296-310 (2014)

46. Park, C, Cui, J: Generalized stability of $$C^{*}$$-ternary quadratic mappings. Abstr. Appl. Anal. 2007, Article ID 23282 (2007)

47. Park, C, Hou, J: Homomorphisms between $$C^{*}$$-algebras associated with the Trif functional equation and linear derivations on $$C^{*}$$-algebras. J. Korean Math. Soc. 41, 461-477 (2004)

48. Park, C, Najati, A: Homomorphisms and derivations in $$C^{*}$$-algebras. Abstr. Appl. Anal. 2007, Article ID 80630 (2007)

49. Park, C: Proper $$CQ^{*}$$-ternary algebras. J. Nonlinear Sci. Appl. 7(4), 278-287 (2014)

50. Rassias, TM: Problem 16; 2, Report of the 27th International Symp. on Functional Equations. Aequ. Math. 39, 292-293, 309 (1990)

51. Rassias, TM: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246, 352-378 (2000)

52. Rassias, TM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264-284 (2000)

53. Rassias, JM: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126-130 (1982)

54. Rassias, JM: On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math. 108, 445-446 (1984)

55. Rassias, JM: Solution of a problem of Ulam. J. Approx. Theory 57, 268-273 (1989)

56. Ravi, K, Thandapani, E, Senthil Kumar, BV: Solution and stability of a reciprocal type functional equation in several variables. J. Nonlinear Sci. Appl. 7(1), 18-27 (2014)

57. Saadati, R, Cho, YJ, Vahidi, J: The stability of the quartic functional equation in various spaces. Comput. Math. Appl. 60, 1994-2002 (2010)

58. Radu, V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91-96 (2003)

59. Cădariu, L, Radu, V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43-52 (2004)

60. Jung, S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, Berlin (2011)

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### Authors’ contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

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