C*-Ternary Biderivations and C*-Ternary Bihomomorphisms

Abstract

In this paper, we investigate $C^{*}$-ternary biderivations and $C^{*}$-ternary bihomomorphism in $C^{*}$-ternary algebras, associated with bi-additive s-functional inequalities.

1. Introduction and Preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms.

The functional equation $f(x+y)=f\left(x\right)+f\left(y\right)$ is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.

Gilányi [6] showed that if f satisfies the functional inequality

$$\begin{array}{c}\hfill \parallel 2f\left(x\right)+2f\left(y\right)-f(x-y)\parallel \le \parallel f(x+y)\parallel \end{array}$$

(1)

then f satisfies the Jordan-von Neumann functional equation

$$2f\left(x\right)+2f\left(y\right)=f(x+y)+f(x-y).$$

See also [7]. Fechner [8] and Gilányi [9] proved the Hyers-Ulam stability of the functional inequality (1).

Park [10,11] defined additive $\rho$-functional inequalities and proved the Hyers-Ulam stability of the additive $\rho$-functional inequalities in Banach spaces and non-Archimedean Banach spaces. The stability problems of various functional equations and functional inequalities have been extensively investigated by a number of authors (see [12,13,14,15,16,17,18,19,20]).

A $C^{*}$-ternary algebra is a complex Banach space A, equipped with a ternary product $(x,y,z)\mapsto [x,y,z]$ of $A^3$ into A, which is $\mathbb{C}$-linear in the outer variables, conjugate $\mathbb{C}$-linear in the middle variable, and associative in the sense that $[x,y,[z,w,v\left]\right]=[x,[w,z,y],v]=\left[\right[x,y,z],w,v]$, and satisfies $\parallel [x,y,z]\parallel \le \parallel x\parallel ·\parallel y\parallel ·\parallel z\parallel$ and $\parallel [x,x,x]\parallel ={\parallel x\parallel }^3$ (see [21]).

If a $C^{*}$-ternary algebra $(A,[·,·,·\left]\right)$ has an identity, i.e., an 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.

Let A and B be $C^{*}$-ternary algebras. A $\mathbb{C}$-linear mapping $H:A\to B$ is called a $C^{*}$-ternary homomorphism if

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

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

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

for all $x,y,z\in A$ (see [22,23]).

Bae and Park [24] defined $C^{*}$-ternary bihomomorphisms and $C^{*}$-ternary biderivations in $C^{*}$-ternary algebras.

Definition 1.

[24] Let A and B be $C^{*}$-ternary algebras. A $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ is called a $C^{*}$-ternary bihomomorphism if

$$\begin{array}{ccc}\hfill H\left(\right[x,y,z],w)& =& \left[H\right(x,w),H(y,w),H(z,w\left)\right],\hfill \\ \hfill H(x,[y,z,w\left]\right)& =& \left[H\right(x,y),H(x,z),H(x,w\left)\right]\hfill \end{array}$$

(2)

for all $x,y,z,w\in A$. A $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ is called a $C^{*}$-ternary biderivation if

$$\begin{array}{ccc}\hfill \delta \left(\right[x,y,z],w)& =& \left[\delta \right(x,w),y,z]+[x,\delta (y,w),z]+[x,y,\delta (z,w\left)\right],\hfill \\ \hfill \delta (x,[y,z,w\left]\right)& =& \left[\delta \right(x,y),z,w]+[y,\delta (x,z),w]+[y,z,\delta (x,w\left)\right]\hfill \end{array}$$

(3)

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

Replacing w by $2w$ in (2), we get

$$\begin{array}{ccc}\hfill 2H\left(\right[x,y,z],w)& =& H\left(\right[x,y,z],2w)=\left[H\right(x,2w),H(y,2w),H(z,2w\left)\right]\hfill \\ \hfill & =& 8\left[H\right(x,w),H(y,w),H(z,w\left)\right]=8H\left(\right[x,y,z],w)\hfill \end{array}$$

and so $H\left(\right[x,y,z],w)=0$ for all $x,y,z,w\in A$.

Replacing w by $iw$ in (3), we get

$$\begin{array}{ccc}\hfill i\delta \left(\right[x,y,z],w)& =& \delta \left(\right[x,y,z],iw)=\left[\delta \right(x,iw),y,z]+[x,\delta (y,iw),z]+[x,y,\delta (z,iw\left)\right]\hfill \\ \hfill & =& i\left[\delta \right(x,w),y,z]-i[x,\delta (y,w),z]+i[x,y,\delta (z,w\left)\right]\ne i\delta \left(\right[x,y,z],w)\hfill \end{array}$$

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

Now we correct the above definition as follows.

Definition 2.

Let A and B be $C^{*}$-ternary algebras. A $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ is called a $C^{*}$-ternary bihomomorphism if

$$\begin{array}{ccc}\hfill H\left(\right[x,y,z],[w,w,w\left]\right)& =& \left[H\right(x,w),H(y,w),H(z,w\left)\right],\hfill \\ \hfill H\left(\right[x,x,x],[y,z,w\left]\right)& =& \left[H\right(x,y),H(x,z),H(x,w\left)\right]\hfill \end{array}$$

for all $x,y,z,w\in A$. A $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ is called a $C^{*}$-ternary biderivation if

$$\begin{array}{ccc}\hfill \delta \left(\right[x,y,z],w)& =& [\delta (x,w),y,z]+[x,\delta (y,w^{*}),z]+[x,y,\delta (z,w)],\hfill \\ \hfill \delta (x,[y,z,w\left]\right)& =& [\delta (x,y),z,w]+[y,\delta (x^{*},z),w]+[y,z,\delta (x,w)]\hfill \end{array}$$

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

In this paper, we prove the Hyers-Ulam stability of $C^{*}$-ternary bihomomorphisms and $C^{*}$-ternary bi-derivations in $C^{*}$-ternary algebras.

This paper is organized as follows: In Section 2 and Section 3, we correct and prove the results on $C^{*}$-ternary bihomomorphisms and $C^{*}$-ternary derivations in $C^{*}$-ternary algebras, given in [24]. In Section 4 and Section 5, we investigate $C^{*}$-ternary biderivations and $C^{*}$-ternary bihomomorphisms in $C^{*}$-ternary algebras associated with the following bi-additive s-functional inequalities

$$\begin{array}{c}\parallel f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\parallel \hfill \\ \le ∥s\left(2f\left(\frac{x+y}{2},z-w\right)+2f\left(\frac{x-y}{2},z+w\right)-2f(x,z)+2f(y,w)\right)∥,\hfill \end{array}$$

(4)

$$\begin{array}{c}∥2f\left(\frac{x+y}{2},z-w\right)+2f\left(\frac{x-y}{2},z+w\right)-2f(x,z)+2f(y,w)∥\hfill \\ \le ∥s\left(f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\right)∥,\hfill \end{array}$$

(5)

where s is a fixed nonzero complex number with $\left|s\right|<1$.

Throughout this paper, let X be a complex normed space and Y a complex Banach space. Assume that s is a fixed nonzero complex number with $\left|s\right|<1$.

2. ${\mathit{C}}^{*}$-Ternary Bihomomorphisms in ${\mathit{C}}^{*}$-Ternary Algebras

In this section, we correct and prove the results on $C^{*}$-ternary bihomomorphisms in $C^{*}$-ternary algebras, given in [24].

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

Lemma 1.

([24], Lemmas 2.1 and 2.2) Let $f:X\times X\to Y$ be a mapping such that

$$f\left(\lambda \right(x+y),\mu (z-w\left)\right)+f\left(\lambda \right(x-y),\mu (z+w\left)\right)=2\lambda \mu f(x,z)-2\lambda \mu f(y,w)$$

for all $\lambda ,\mu \in {\mathbb{T}}^1:=\{\xi \in \mathbb{C}:|\xi |=1\}$ and all $x,y,z,w\in V$. Then $f:X\times X\to Y$ is $\mathbb{C}$-bilinear.

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

$$\begin{array}{c}{D}_{\lambda ,\mu }f(x,y,z,w)\hfill \\ :=f\left(\lambda \right(x+y),\mu (z-w\left)\right)+f\left(\lambda \right(x-y),\mu (z+w\left)\right)-2\lambda \mu f(x,z)+2\lambda \mu f(y,w)\hfill \end{array}$$

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

We prove the Hyers-Ulam stability of $C^{*}$-ternary bihomomorphisms in $C^{*}$-ternary algebras.

Theorem 1.

Let $r<2$ and θ be nonnegative real numbers, and let $f:A\times A\to B$ be a mapping satisfying $f(0,0)=0$ and

$$\begin{array}{c}\hfill \parallel D_{\lambda ,\mu }{f(x,y,z,w)\parallel \le \theta (\parallel x\parallel }^r+{\parallel y\parallel }^r+{\parallel z\parallel }^r+{\parallel w\parallel }^r),\end{array}$$

(6)

$$\begin{array}{c}\parallel f\left(\right[x,y,z],[w,w,w\left]\right)-\left[f\right(x,w),f(y,w),f(z,w\left)\right]\parallel \hfill \\ +\parallel f\left(\right[x,x,x],[y,z,w\left]\right)-\left[f\right(x,y),f(x,z),f(x,w\left)\right]\parallel \hfill \\ \le \theta (\parallel x{\parallel }^r+\parallel y{\parallel }^r+\parallel z{\parallel }^r+\parallel w{\parallel }^r)\hfill \end{array}$$

(7)

for all $\lambda ,\mu \in {\mathbb{T}}^1$ and all $x,y,z,w\in A$. Then there exists a unique $C^{*}$-ternary bi-homomorphism $H:A\times A\to B$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-H(x,z)\parallel \le \frac{6\theta }{4-2^r}{(\parallel x\parallel }^r+{\parallel z\parallel }^r)\end{array}$$

(8)

for all $x,z\in A$.

Proof. 

By the same reasoning as in the proof of ([24] Theorem 2.3), there exists a unique $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ satisfying (8). The $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ is defined by

$$H(x,z)=\underset{n\to \infty }{lim}\frac{1}{4^n}f(2^{n}x,2^{n}z)$$

for all $x,z\in A$.

It follows from (7) that

$$\begin{array}{c}\parallel H\left(\right[x,y,z],[w,w,w\left]\right)-\left[H\right(x,w),H(y,w),H(z,w\left)\right]\parallel \hfill \\ +\parallel H\left(\right[x,x,x],[y,z,w\left]\right)-\left[H\right(x,y),H(x,z),H(x,w\left)\right]\parallel \hfill \\ =\underset{n\to \infty }{lim}\frac{1}{{64}^n}\parallel f(8^n[x,y,z],8^n[w,w,w])-[f(2^{n}x,2^{n}w),f(2^{n}y,2^{n}w),f(2^{n}z,2^{n}w)]\parallel \hfill \\ +\underset{n\to \infty }{lim}\frac{1}{{64}^n}\parallel f(8^n[x,x,x],8^n[y,z,w])-[f(2^{n}x,2^{n}y),f(2^{n}x,2^{n}z),f(2^{n}x,2^{n}w)]\parallel \hfill \\ \le \underset{n\to \infty }{lim}\frac{2^{rn}}{{64}^n}{\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r+{\parallel z\parallel }^r+{\parallel w\parallel }^r)=0\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill H\left(\right[x,y,z],[w,w,w\left]\right)& =& \left[H\right(x,w),H(y,w),H(z,w\left)\right],\hfill \\ \hfill H\left(\right[x,x,x],[y,z,w\left]\right)& =& \left[H\right(x,y),H(x,z),H(x,w\left)\right]\hfill \end{array}$$

for all $x,y,z,w\in A$, as desired. ☐

Similarly, we can obtain the following.

Theorem 2.

Let $r>6$ and θ be nonnegative real numbers, and let $f:A\times A\to B$ be a mapping satisfying $f(0,0)=0$, (6) and (7). Then there exists a unique $C^{*}$-ternary bihomomorphism $H:A\times A\to B$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-H(x,z)\parallel \le \frac{6\theta }{2^r-4}{(\parallel x\parallel }^r+{\parallel z\parallel }^r)\end{array}$$

(9)

for all $x,z\in A$.

Proof. 

By the same reasoning as in the proof of ([24] Theorem 2.5), there exists a unique $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ satisfying (9). The $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ is defined by

$$H(x,z)=\underset{n\to \infty }{lim}{4}^{n}f\left(\frac{x}{2^n},\frac{z}{2^n}\right)$$

for all $x,z\in A$.

It follows from (7) that

$$\begin{array}{c}\parallel H\left(\right[x,y,z],[w,w,w\left]\right)-\left[H\right(x,w),H(y,w),H(z,w\left)\right]\parallel \hfill \\ +\parallel H\left(\right[x,x,x],[y,z,w\left]\right)-\left[H\right(x,y),H(x,z),H(x,w\left)\right]\parallel \hfill \\ =\underset{n\to \infty }{lim}{64}^n∥f\left(\frac{[x,y,z]}{8^n},\frac{[w,w,w]}{8^n}\right)-\left[f\left(\frac{x}{2^n},\frac{w}{2^n}\right),f\left(\frac{y}{2^n},\frac{w}{2^n}\right),f\left(\frac{z}{2^n},\frac{w}{2^n}\right)\right]∥\hfill \\ +\underset{n\to \infty }{lim}{64}^n∥f\left(\frac{[x,x,x]}{8^n},\frac{[y,z,w]}{8^n}\right)-\left[f\left(\frac{x}{2^n},\frac{y}{2^n}\right),f\left(\frac{x}{2^n},\frac{z}{2^n}\right),f\left(\frac{x}{2^n},\frac{w}{2^n}\right)\right]∥\hfill \\ \le \underset{n\to \infty }{lim}\frac{{64}^n}{2^{rn}}{\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r+{\parallel z\parallel }^r+{\parallel w\parallel }^r)=0\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill H\left(\right[x,y,z],[w,w,w\left]\right)& =& \left[H\right(x,w),H(y,w),H(z,w\left)\right],\hfill \\ \hfill H\left(\right[x,x,x],[y,z,w\left]\right)& =& \left[H\right(x,y),H(x,z),H(x,w\left)\right]\hfill \end{array}$$

for all $x,y,z,w\in A$, as desired. ☐

Theorem 3.

Let $r<\frac{1}{2}$ and θ be nonnegative real numbers, and let $f:A\times A\to B$ be a mapping satisfying $f(0,0)=0$ and

$$\begin{array}{c}\hfill \parallel D_{\lambda ,\mu }{f(x,y,z,w)\parallel \le \theta ·\parallel x\parallel }^r·{\parallel y\parallel }^r·{\parallel z\parallel }^r·{\parallel w\parallel }^r,\end{array}$$

(10)

$$\begin{array}{c}\parallel f\left(\right[x,y,z],[w,w,w\left]\right)-\left[f\right(x,w),f(y,w),f(z,w\left)\right]\parallel \hfill \\ +\parallel f\left(\right[x,x,x],[y,z,w\left]\right)-\left[f\right(x,y),f(x,z),f(x,w\left)\right]\parallel \hfill \\ \le \theta ·{\parallel x\parallel }^r·{\parallel y\parallel }^r·{\parallel z\parallel }^r·{\parallel w\parallel }^r\hfill \end{array}$$

(11)

for all $\lambda ,\mu \in {\mathbb{T}}^1$ and all $x,y,z,w\in A$. Then there exists a unique $C^{*}$-ternary bihomomorphism $H:A\times A\to B$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-H(x,z)\parallel \le \frac{2\theta }{4-{16}^r}{(\parallel x\parallel }^r+{\parallel z\parallel }^r)\end{array}$$

(12)

for all $x,z\in A$.

Proof. 

By the same reasoning as in the proof of ([24] Theorem 2.6), there exists a unique $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ satisfying (12). The $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ is defined by

$$H(x,z)=\underset{n\to \infty }{lim}\frac{1}{4^n}f\left(2^{n}x,2^{n}z\right)$$

for all $x,z\in A$.

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

Theorem 4.

Let $r>\frac{3}{2}$ and θ be nonnegative real numbers, and let $f:A\times A\to B$ be a mapping satisfying $f(0,0)=0$, (10) and (11). Then there exists a unique $C^{*}$-ternary bihomomorphism $H:A\times A\to B$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-H(x,z)\parallel \le \frac{2\theta }{{16}^r-4}{(\parallel x\parallel }^r+{\parallel z\parallel }^r)\end{array}$$

(13)

for all $x,z\in A$.

Proof. 

By the same reasoning as in the proof of ([24] Theorem 2.7), there exists a unique $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ satisfying (13). The $\mathbb{C}$-bilinear mapping $H:A\times A\to B$ is defined by

$$H(x,z)=\underset{n\to \infty }{lim}{4}^{n}f\left(\frac{x}{2^n},\frac{z}{2^n}\right)$$

for all $x,z\in A$.

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

3. ${\mathit{C}}^{*}$-Ternary Biderivations on ${\mathit{C}}^{*}$-Ternary Algebras

In this section, we correct and prove the results on $C^{*}$-ternary biderivations on $C^{*}$-ternary algebras, given in [24].

Throughout this paper, assume that A is a $C^{*}$-ternary algebra.

Theorem 5.

Let $r<2$ and θ be nonnegative real numbers, and let $f:A\times A\to A$ be a mapping satisfying $f(0,0)=0$ and

$$\begin{array}{c}\hfill \parallel D_{\lambda ,\mu }{f(x,y,z,w)\parallel \le \theta (\parallel x\parallel }^r+{\parallel y\parallel }^r+{\parallel z\parallel }^r+{\parallel w\parallel }^r),\end{array}$$

(14)

$$\begin{array}{c}\parallel f([x,y,z],w)-[f(x,w),y,z]-[x,f(y,w^{*}),z]-[x,y,f(z,w)]\parallel \hfill \\ +\parallel f(x,[y,z,w])-[f(x,y),z,w]-[y,f(x^{*},z),w]-[y,z,f(x,w)]\parallel \hfill \\ \le \theta (\parallel x{\parallel }^r+\parallel y{\parallel }^r+\parallel z{\parallel }^r+\parallel w{\parallel }^r)\hfill \end{array}$$

(15)

for all $\lambda ,\mu \in {\mathbb{T}}^1$ and all $x,y,z,w\in A$. Then there exists a unique $C^{*}$-ternary biderivation $\delta :A\times A\to A$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-\delta (x,z)\parallel \le \frac{6\theta }{4-2^r}{(\parallel x\parallel }^r+{\parallel z\parallel }^r)\end{array}$$

(16)

for all $x,z\in A$.

Proof. 

By the same reasoning as in the proof of ([24] Theorems 2.3 and 3.1), there exists a unique $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ satisfying (16). The $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ is defined by

$$\delta (x,z)=\underset{n\to \infty }{lim}\frac{1}{4^n}f(2^{n}x,2^{n}z)$$

for all $x,z\in A$.

It follows from (15) that

$$\begin{array}{c}\parallel \delta ([x,y,z],w)-[\delta (x,w),y,z]-[x,\delta (y,w^{*}),z]-[x,y,\delta (z,w)]\parallel \hfill \\ +\parallel \delta (x,[y,z,w])-[\delta (x,y),z,w]-[y,\delta (x^{*},z),w]-[y,z,\delta (x,w)]\parallel \hfill \\ =\underset{n\to \infty }{lim}\frac{1}{{16}^n}(\parallel f(8^n[x,y,z],2^{n}w)-[f(2^{n}x,2^{n}w),2^{n}y,2^{n}w]\hfill \\ -[2^{n}x,f(2^{n}y,2^n{w}^{*}),2^{n}z]-[2^{n}x,2^{n}y,f(2^{n}z,2^{n}w)]\parallel )\hfill \\ +\underset{n\to \infty }{lim}\frac{1}{{16}^n}(\parallel f(2^{n}x,8^n[y,z,w])-[f(2^{n}x,2^{n}y),2^{n}z,2^{n}w]\hfill \\ -[2^{n}y,f(2^n{x}^{*},2^{n}z),2^{n}w]-[2^{n}y,2^{n}z,f(2^{n}x,2^{n}w)]\parallel )\hfill \\ \le \underset{n\to \infty }{lim}\frac{2^{rn}}{{16}^n}{\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r+{\parallel z\parallel }^r+{\parallel w\parallel }^r)=0\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill \delta \left(\right[x,y,z],w)& =& [\delta (x,w),y,z]+[x,\delta (y,w^{*}),z]+[x,y,\delta (z,w)],\hfill \\ \hfill \delta (x,[y,z,w\left]\right)& =& [\delta (x,y),z,w]+[y,\delta (x^{*},z),w]+[y,z,\delta (x,w)]\hfill \end{array}$$

for all $x,y,z,w\in A$, as desired. ☐

Similarly, we can obtain the following.

Theorem 6.

Let $r>4$ and θ be nonnegative real numbers, and let $f:A\times A\to A$ be a mapping satisfying $f(0,0)=0$, (14) and (15). Then there exists a unique $C^{*}$-ternary biderivation $\delta :A\times A\to A$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-\delta (x,z)\parallel \le \frac{6\theta }{2^r-4}{(\parallel x\parallel }^r+{\parallel z\parallel }^r)\end{array}$$

(17)

for all $x,z\in A$.

Proof. 

By the same reasoning as in the proof of ([24] Theorem 2.5), there exists a unique $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ satisfying (17). The $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ is defined by

$$\delta (x,z)=\underset{n\to \infty }{lim}{4}^{n}f\left(\frac{x}{2^n},\frac{z}{2^n}\right)$$

for all $x,z\in A$.

It follows from (15) that

$$\begin{array}{c}\parallel \delta ([x,y,z],w)-[\delta (x,w),y,z]-[x,\delta (y,w^{*}),z]-[x,y,\delta (z,w)]\parallel \hfill \\ +\parallel \delta (x,[y,z,w])-[\delta (x,y),z,w]-[y,\delta (x^{*},z),w]-[y,z,\delta (x,w)]\parallel \hfill \\ =\underset{n\to \infty }{lim}{16}^n\left(∥f\left(\frac{[x,y,z]}{8^n},\frac{w}{2^n}\right)-\left[f\left(\frac{x}{2^n},\frac{w}{2^n}\right),\frac{y}{2^n},\frac{w}{2^n}\right]\right.\hfill \\ \left.-\left[\frac{x}{2^n},f\left(\frac{y}{2^n},\frac{w^{*}}{2^n}\right),\frac{z}{2^n}\right]-\left[\frac{x}{2^n},\frac{y}{2^n},f\left(\frac{z}{2^n},\frac{w}{2^n}\right)\right]∥\right)\hfill \\ +\underset{n\to \infty }{lim}{16}^n\left(∥f\left(\frac{x}{2^n},\frac{[y,z,w]}{8^n}\right)-\left[f\left(\frac{x}{2^n},\frac{y}{2^n}\right),\frac{z}{2^n},\frac{w}{2^n}\right]\right.\hfill \\ \left.-\left[\frac{y}{2^n},f\left(\frac{x^{*}}{2^n},\frac{z}{2^n}\right),\frac{w}{2^n}\right]-\left[\frac{y}{2^n},\frac{z}{2^n},f\left(\frac{x}{2^n},\frac{w}{2^n}\right)\right]∥\right)\hfill \\ \le \underset{n\to \infty }{lim}\frac{{16}^n}{2^{rn}}{\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r+{\parallel z\parallel }^r+{\parallel w\parallel }^r)=0\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill \delta \left(\right[x,y,z],w)& =& [\delta (x,w),y,z]+[x,\delta (y,w^{*}),z]+[x,y,\delta (z,w)],\hfill \\ \hfill \delta (x,[y,z,w\left]\right)& =& [\delta (x,y),z,w]+[y,\delta (x^{*},z),w]+[y,z,\delta (x,w)]\hfill \end{array}$$

for all $x,y,z,w\in A$, as desired. ☐

Theorem 7.

Let $r<\frac{1}{2}$ and θ be nonnegative real numbers, and let $f:A\times A\to A$ be a mapping satisfying $f(0,0)=0$ and

$$\begin{array}{c}\hfill \parallel D_{\lambda ,\mu }{f(x,y,z,w)\parallel \le \theta ·\parallel x\parallel }^r·{\parallel y\parallel }^r·{\parallel z\parallel }^r·{\parallel w\parallel }^r,\end{array}$$

(18)

$$\begin{array}{c}\parallel f([x,y,z],w)-[f(x,w),y,z]-[x,f(y,w^{*}),z]-[x,y,f(z,w)]\parallel \hfill \\ +\parallel f(x,[y,z,w])-[f(x,y),z,w]-[y,f(x^{*},z),w]-[y,z,f(x,w)]\parallel \hfill \\ \le \theta ·{\parallel x\parallel }^r·{\parallel y\parallel }^r·{\parallel z\parallel }^r·{\parallel w\parallel }^r\hfill \end{array}$$

(19)

for all $\lambda ,\mu \in {\mathbb{T}}^1$ and all $x,y,z,w\in A$. Then there exists a unique $C^{*}$-ternary biderivation $\delta :A\times A\to A$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-\delta (x,z)\parallel \le \frac{2\theta }{4-{16}^r}{(\parallel x\parallel }^r+{\parallel z\parallel }^r)\end{array}$$

(20)

for all $x,z\in A$.

Proof. 

By the same reasoning as in the proof of ([24] Theorem 2.6), there exists a unique $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ satisfying (20). The $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ is defined by

$$\delta (x,z)=\underset{n\to \infty }{lim}\frac{1}{4^n}f\left(2^{n}x,2^{n}z\right)$$

for all $x,z\in A$.

The rest of the proof is similar to the proof of Theorem 5. ☐

Theorem 8.

Let $r>\frac{3}{2}$ and θ be nonnegative real numbers, and let $f:A\times A\to A$ be a mapping satisfying $f(0,0)=0$, (18) and (19) . Then there exists a unique $C^{*}$-ternary biderivation $\delta :A\times A\to A$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-\delta (x,z)\parallel \le \frac{2\theta }{{16}^r-4}{(\parallel x\parallel }^r+{\parallel z\parallel }^r)\end{array}$$

(21)

for all $x,z\in A$.

Proof. 

By the same reasoning as in the proof of ([24] Theorem 2.7), there exists a unique $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ satisfying (21). The $\mathbb{C}$-bilinear mapping $\delta :A\times A\to A$ is defined by

$$\delta (x,z)=\underset{n\to \infty }{lim}{4}^{n}f\left(\frac{x}{2^n},\frac{z}{2^n}\right)$$

for all $x,z\in A$.

The rest of the proof is similar to the proof of Theorem 5. ☐

4. ${\mathit{C}}^{*}$-Ternary Biderivations on ${\mathit{C}}^{*}$-Ternary Algebras Associated with the Bi-Additive Functional Inequalities (4) and (5)

In [25], Park introduced and investigated the bi-additive s-functional inequalities (4) and (5) in complex Banach spaces.

Theorem 9.

([25] Theorem 2.2) Let $r>1$ and θ be nonnegative real numbers and let $f:X^2\to Y$ be a mapping satisfying $f(x,0)=f(0,z)=0$ and

$$\begin{array}{c}\parallel f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\parallel \hfill \\ \le ∥s\left(2f\left(\frac{x+y}{2},z-w\right)+2f\left(\frac{x-y}{2},z+w\right)-2f(x,z)+2f(y,w)\right)∥\hfill \\ +{\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r{\left)\right(\parallel z\parallel }^r+{\parallel w\parallel }^r)\hfill \end{array}$$

(22)

for all $x,y,z,w\in X$. Then there exists a unique bi-additive mapping $A:X^2\to Y$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-A(x,z)\parallel \le \frac{2\theta }{2^r-2}{\parallel x\parallel }^r{\parallel z\parallel }^r\end{array}$$

(23)

for all $x,z\in X$.

Theorem 10.

([25] Theorem 2.3) Let $r<1$ and θ be nonnegative real numbers and let $f:X^2\to Y$ be a mapping satisfying (22) and $f(x,0)=f(0,z)=0$ for all $x,z\in X$. Then there exists a unique bi-additive mapping $A:X^2\to Y$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-A(x,z)\parallel \le \frac{2\theta }{2-2^r}{\parallel x\parallel }^r{\parallel z\parallel }^r\end{array}$$

(24)

for all $x,z\in X$.

Theorem 11.

([25] Theorem 3.2) Let $r>1$ and θ be nonnegative real numbers and let $f:X^2\to Y$ be a mapping satisfying $f(x,0)=f(0,z)=0$ and

$$\begin{array}{c}∥2f\left(\frac{x+y}{2},z-w\right)+2f\left(\frac{x-y}{2},z+w\right)-2f(x,z)+2f(y,w)∥\hfill \\ \le ∥s\left(f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\right)∥\hfill \\ +{\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r{\left)\right(\parallel z\parallel }^r+{\parallel w\parallel }^r)\hfill \end{array}$$

(25)

for all $x,y,z,w\in X$. Then there exists a unique bi-additive mapping $A:X^2\to Y$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-A(x,z)\parallel \le \frac{2^{r-1}\theta }{2^r-2}{\parallel x\parallel }^r{\parallel z\parallel }^r\end{array}$$

(26)

for all $x,z\in X$.

Theorem 12.

([25] Theorem 3.3) Let $r<1$ and θ be nonnegative real numbers and let $f:X^2\to Y$ be a mapping satisfying (25) and $f(x,0)=f(0,z)=0$ for all $x,z\in X$. Then there exists a unique bi-additive mapping $A:X^2\to Y$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-A(x,z)\parallel \le \frac{\theta }{2(2-2^r)}{\parallel x\parallel }^r{\parallel z\parallel }^r\end{array}$$

(27)

for all $x,z\in X$.

Now, we investigate $C^{*}$-ternary biderivations on $C^{*}$-ternary algebras associated with the bi-additive s-functional inequalities (4) and (5).

From now on, assume that A is a $C^{*}$-ternary algebra.

Theorem 13.

Let $r>2$ and θ be nonnegative real numbers, and let $f:A^2\to A$ be a mapping satisfying $f(x,0)=f(0,z)=0$ and

$$\begin{array}{c}\parallel f\left(\lambda \right(x+y),\mu (z-w\left)\right)+f\left(\lambda \right(x-y),\mu (z+w\left)\right)-2\lambda \mu f(x,z)+2\lambda \mu f(y,w)\parallel \hfill \\ \le ∥s\left(2f\left(\frac{x+y}{2},z-w\right)+2f\left(\frac{x-y}{2},z+w\right)-2f(x,z)+2f(y,w)\right)∥\hfill \\ +{\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r{\left)\right(\parallel z\parallel }^r+{\parallel w\parallel }^r)\hfill \end{array}$$

(28)

for all $\lambda ,\mu \in {\mathbb{T}}^1$ and all $x,y,z,w\in A$. Then there exists a unique $\mathbb{C}$-bilinear mapping $D:A^2\to A$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-D(x,z)\parallel \le \frac{2\theta }{2^r-2}{\parallel x\parallel }^r{\parallel z\parallel }^r\end{array}$$

(29)

for all $x,z\in A$.

If, in addition, the mapping $f:A^2\to A$ satisfies $f(2x,z)=2f(x,z)$ and

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

(30)

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

(31)

for all $x,y,z,w\in A$, then the mapping $f:A^2\to A$ is a $C^{*}$-ternary biderivation.

Proof. 

Let $\lambda =\mu =1$ in (28). By Theorem 9, there is a unique bi-additive mapping $D:A^2\to A$ satisfying (29) defined by

$$D(x,z):=\underset{n\to \infty }{lim}{2}^{n}f\left(\frac{x}{2^n},z\right)$$

for all $x,z\in A$.

Letting $y=w=0$ in (28), we get $f(\lambda x,\mu z)=\lambda \mu f(x,z)$ for all $x,z\in A$ and all $\lambda ,\mu \in {\mathbb{T}}^1$. By Lemma 1, the bi-additive mapping $D:A^2\to A$ is $\mathbb{C}$-bilinear.

If $f(2x,z)=2f(x,z)$ for all $x,z\in A$, then we can easily show that $D(x,z)=f(x,z)$ for all $x,z\in A$.

It follows from (30) that

$$\begin{array}{c}\parallel D([x,y,z],w)-[D(x,w),y,z]-[x,D(y,w^{*}),z]-[x,y,D(z,w)]\parallel \hfill \\ =\underset{n\to \infty }{lim}{16}^n\left(∥f\left(\frac{[x,y,z]}{8^n},\frac{w}{2^n}\right)-\left[f\left(\frac{x}{2^n},\frac{w}{2^n}\right),\frac{y}{2^n},\frac{z}{2^n}\right]\right.\hfill \\ \left.-\left[\frac{x}{2^n},f\left(\frac{y}{2^n},\frac{w^{*}}{2^n}\right),\frac{z}{2^n}\right]-\left[\frac{x}{2^n},\frac{y}{2^n},f\left(\frac{z}{2^n},\frac{w}{2^n}\right)\right]∥\right)\hfill \\ \le \underset{n\to \infty }{lim}\frac{{16}^n\theta }{4^{rn}}{(\parallel x\parallel }^r+{\parallel y\parallel }^r{\left)\right(\parallel z\parallel }^r+{\parallel w\parallel }^r)=0\hfill \end{array}$$

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

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

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

Similarly, one can show that

$$D(x,[y,z,w])=[D(x,y),z,w]-[y,D(x^{*},z,w]-[y,z,D(x,w)]$$

for all $x,y,z,w\in A$. Hence the mapping $f:A^2\to A$ is a $C^{*}$-ternary biderivation. ☐

Theorem 14.

Let $r<1$ and θ be nonnegative real numbers, and let $f:A^2\to A$ be a mapping satisfying (28) and $f(x,0)=f(0,z)=0$ for all $x,z\in A$. Then there exists a unique $\mathbb{C}$-bilinear mapping $D:A^2\to A$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-D(x,z)\parallel \le \frac{2\theta }{2-2^r}{\parallel x\parallel }^r{\parallel z\parallel }^r\end{array}$$

(32)

for all $x,z\in A$.

If, in addition, the mapping $f:A^2\to A$ satisfies (30), (31) and $f(2x,z)=2f(x,z)$ for all $x,z\in A$, then the mapping $f:A^2\to A$ is a $C^{*}$-ternary biderivation.

Proof. 

The proof is similar to the proof of Theorem 13. ☐

Similarly, we can obtain the following results.

Theorem 15.

Let $r>2$ and θ be nonnegative real numbers, and let $f:A^2\to A$ be a mapping satisfying $f(x,0)=f(0,z)=0$ and

$$\begin{array}{c}∥2f\left(\lambda \frac{x+y}{2},\mu (z-w)\right)+2f\left(\lambda \frac{x-y}{2},\mu (z+w)\right)-2\lambda \mu f(x,z)+2\lambda \mu f(y,w)∥\hfill \\ \le ∥s\left(f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\right)∥\hfill \\ +{\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r{\left)\right(\parallel z\parallel }^r+{\parallel w\parallel }^r)\hfill \end{array}$$

(33)

for all $\lambda ,\mu \in {\mathbb{T}}^1$ and all $x,y,z,w\in A$. Then there exists a unique $\mathbb{C}$-bilinear mapping $D:A^2\to A$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-D(x,z)\parallel \le \frac{2^{r-1}\theta }{2^r-2}{\parallel x\parallel }^r{\parallel z\parallel }^r\end{array}$$

(34)

for all $x,z\in A$.

If, in addition, the mapping $f:A^2\to A$ satisfies (30), (31) and $f(2x,z)=2f(x,z)$ for all $x,z\in A$, then the mapping $f:A^2\to A$ is a $C^{*}$-ternary biderivation.

Theorem 16.

Let $r<1$ and θ be nonnegative real numbers, and let $f:A^2\to A$ be a mapping satisfying (33) and $f(x,0)=f(0,z)=0$ for all $x,z\in A$. Then there exists a unique $\mathbb{C}$-bilinear mapping $D:A^2\to A$ such that

$$\begin{array}{c}\hfill \parallel f(x,z)-D(x,z)\parallel \le \frac{\theta }{2(2-2^r)}{\parallel x\parallel }^r{\parallel z\parallel }^r\end{array}$$

(35)

for all $x,z\in A$.

If, in addition, the mapping $f:A^2\to A$ satisfies (30), (31) and $f(2x,z)=2f(x,z)$ for all $x,z\in A$, then the mapping $f:A^2\to A$ is a $C^{*}$-ternary biderivation.

5. ${\mathit{C}}^{*}$-Ternary Bihomomorphisms in ${\mathit{C}}^{*}$-Ternary Algebras Associated with the Bi-Additive Functional Inequalities (4) and (5)

In this section, we investigate $C^{*}$-ternary bihomomorphisms in $C^{*}$-ternary algebras associated with the bi-additive s-functional inequalities (4) and (5).

Theorem 17.

Let $r>3$ and θ be nonnegative real numbers, and let $f:A^2\to B$ be a mapping satisfying $f(x,0)=f(0,z)=0$ and (28). Then there exists a unique $\mathbb{C}$-bilinear mapping $H:A^2\to B$ satisfying (29), where D is replaced by H in (29).

If, in addition, the mapping $f:A^2\to B$ satisfies $f(2x,z)=2f(x,z)$ and

$$\begin{array}{c}\hfill \parallel f([x,y,z],[w,w,w])-[f(x,w),f(y,w),f(z,w)]\parallel \le {\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r{\left)\right(\parallel z\parallel }^r+{\parallel w\parallel }^r),\end{array}$$

(36)

$$\begin{array}{c}\hfill \parallel f([x,x,x],[y,z,w])-[f(x,y),f(x,z),f(x,w)]\parallel \le {\theta (\parallel x\parallel }^r+{\parallel y\parallel }^r{\left)\right(\parallel z\parallel }^r+{\parallel w\parallel }^r)\end{array}$$

(37)

for all $x,y,z,w\in A$, then the mapping $f:A^2\to B$ is a $C^{*}$-ternary bihomomorphism.

Proof. 

By the same reasoning as in the proof of Theorem 13, there is a unique $\mathbb{C}$-bilinear mapping $H:A^2\to B$, which is defined by

$$H(x,z)=\underset{n\to \infty }{lim}{2}^{n}f\left(\frac{x}{2^n},z\right)$$

for all $x,z\in A$.

If $f(2x,z)=2f(x,z)$ for all $x,z\in A$, then we can easily show that $H(x,z)=f(x,z)$ for all $x,z\in A$.

It follows from (36) that

$$\begin{array}{c}\parallel H\left(\right[x,y,z],[w,w,w\left]\right)-\left[H\right(x,w),H(y,w),H(z,w\left)\right]\parallel \hfill \\ =\underset{n\to \infty }{lim}{4}^{3n}∥f\left(\frac{[x,y,z]}{8^n},\frac{[w,w,w]}{8^n}\right)-\left[f\left(\frac{x}{2^n},\frac{w}{2^n}\right),f\left(\frac{y}{2^n},\frac{w}{2^n}\right),f\left(\frac{z}{2^n},\frac{w}{2^n}\right)\right]∥\hfill \\ \le \underset{n\to \infty }{lim}\frac{4^{3n}\theta }{4^{rn}}{(\parallel x\parallel }^r+{\parallel y\parallel }^r{\left)\right(\parallel z\parallel }^r+{\parallel w\parallel }^r)=0\hfill \end{array}$$

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

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

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

Similarly, one can show that

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

for all $x,y,z,w\in A$. Hence the mapping $f:A^2\to B$ is a $C^{*}$-ternary bihomomorphism. ☐

Theorem 18.

Let $r<1$ and θ be nonnegative real numbers, and let $f:A^2\to B$ be a mapping satisfying (28) and $f(x,0)=f(0,z)=0$ for all $x,z\in A$. Then there exists a unique $\mathbb{C}$-bilinear mapping $H:A^2\to B$ satisfying (32), where D is replaced by H in (32).

If, in addition, the mapping $f:A^2\to B$ satisfies (36), (37) and $f(2x,z)=2f(x,z)$ for all $x,z\in A$, then the mapping $f:A^2\to B$ is a $C^{*}$-ternary bihomomorphism.

Proof. 

The proof is similar to the proof of Theorem 17. ☐

Similarly, we can obtain the following results.

Theorem 19.

Let $r>3$ and θ be nonnegative real numbers, and let $f:A^2\to B$ be a mapping satisfying $f(x,0)=f(0,z)=0$ and (33). Then there exists a unique $\mathbb{C}$-bilinear mapping $H:A^2\to B$ satisfying (34), where D is replaced by H in (34).

If, in addition, the mapping $f:A^2\to B$ satisfies (36), (37) and $f(2x,z)=2f(x,z)$ for all $x,z\in A$, then the mapping $f:A^2\to B$ is a $C^{*}$-ternary bihomomorphism.

Theorem 20.

Let $r<1$ and θ be nonnegative real numbers, and let $f:A^2\to B$ be a mapping satisfying (33) and $f(x,0)=f(0,z)=0$ for all $x,z\in A$. Then there exists a unique $\mathbb{C}$-bilinear mapping $H:A^2\to B$ satisfying (35), where D is replaced by H in (35).

If, in addition, the mapping $f:A^2\to B$ satisfies (36), (37) and $f(2x,z)=2f(x,z)$ for all $x,z\in A$, then the mapping $f:A^2\to B$ is a $C^{*}$-ternary bihomomorphism.