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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’ [1] (see also [2, 3] and [4]).

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 [4]).

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 [5]).

Ternary structures and their generalization, the so-called n-ary structures, are important in view of their applications in physics (see [6]).

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

([7])

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 [8] and many examples are given in [810].

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

([10])

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 [8]).

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 [12]). 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

([13])

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 [9]).

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 [14] (also see Lemma 3.2 of [9]),

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

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 [9]).

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 [14] (also see Lemma 3.2 of [9]),

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

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 [9]).

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 [14] (also see Lemma 3.2 of [9]),

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

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

([5])

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 [5] 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 [60]).

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

([13])

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. □

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Alsulami, H.H., Kenari, H.M., O’Regan, D. et al. Multi-\(C^{*}\)-ternary algebras and applications. J Inequal Appl 2015, 223 (2015). https://doi.org/10.1186/s13660-015-0746-9

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MSC

  • 39A10
  • 39B72
  • 47H10
  • 46B03

Keywords

  • approximation
  • additive functional equation
  • multi-Banach space
  • multi-\(C^{*}\)-ternary algebra