# Ternary γ-homomorphisms and ternary γ-derivations on ternary semigroups

## Abstract

In this paper, we introduce the notions of γ-homomorphism and γ-derivation of a ternary semigroup and investigate γ-homomorphism and γ-derivations on ternary semigroup associated with the following functional in-equality |f([xyz]) - f(x) - f(y) - f(z)| ≤ φ(x, y, z) and |f([xxx]) - 3f(x)| ≤ φ(x, x, x), respectively.

2000 MSC: Primary 39B52, Secondary 39B82; 46B99; 17A40.

## 1 Introduction and preliminaries

Ternary algebraic operations were considered in the 19th century by several mathematicians such as Cayley [1] who introduced the notion of "cubic matrix" which in turn was generalized by Kapranov, Gelfand and Zelevinskii et al. [2]. The simplest example of such non-trivial ternary operation is given by the following composition rule:

${\left\{a,b,c\right\}}_{ijk}=\sum _{1\le l,m,n\le N}{a}_{nil}{b}_{ljm}{c}_{mkn}\phantom{\rule{1em}{0ex}}\left(i,j,k=1,2,\dots \dots ,N\right).$

Ternary structures and their generalization, the so-called n-ary structures, raise certain hopes in view of their possible applications in physics. Some significant physical applications are described in [3, 4].

In 1940, Ulam [5] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homo-morphisms:

We are given a group G and a metric group G' with metric ρ(·, ·). Given ϵ > 0, does there exist a δ > 0 such that if f : GG' satisfies ρ(f(xy), f(x)f(y)) < δ for all x, y G, then a homomorphism h : GG exists with ρ(f(x), h(x)) < ϵ for all x G?

As mentioned above, when this problem has a solution, we say that the homomorphisms from G1 to G2 are stable. In 1941, Hyers [6] gave a partial solution of Ulams problem for the case of approximate additive mappings under the assumption that G1 and G2 are Banach spaces. In 1978, Rassias [7] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. This phenomenon of stability that was introduced by Rassias [7] is called the Hyers-Ulam-Rassias stability. In 1992, a generalization of Rassias theorem was obtained by Găvruta [8].

During the last decades several stability problems of functional equations have been investigated be many mathematicians. A large list of references concerning the stability of functional equations can be found in [915].

In this article, using a sequence of Hyers type, we prove the generalized Hyers-Ulam-Rassias stability of ternary γ-homomorphisms and ternary γ-derivations on commutative ternary semigroups.

In the first section, which have preliminary character, we review some basic definitions and properties related to ternary groups and semigroups (cf. also Rusakov [16]).

Definition 1.1. A nonempty set G with one ternary operation [ ]: G × G × GG is called a ternary groupoid and denoted by (G, [ ]).

We say that (G, [ ]) is a ternary semigroup if the operation [ ] is associative, i.e., if

$\left[\left[xyz\right]uv\right]=\left[x\left[yzu\right]v\right]=\left[xy\left[zuv\right]\right]$

hold for all x, y, z, u, v G (see [17]). We shall write x3 instead of [xxx].

Definition 1.2. A ternary semigroup (G, [ ]) is a ternary group if for all a, b, c G, there are x, y, z G such that

$\left[xab\right]=\left[ayb\right]=\left[abz\right]=c.$

One can prove (post [18]) that elements x, y, z are uniquely determined. Moreover, according to the suggestion of post [18] one can prove (cf, Dudek et al. [19]) that in the above definition, under the assumption of the associativity, it suffices only to postulate the existence of a solution of [ayb] = c, or equivalently, of [xab] = [abz] = c.

In a ternary group, the equation [xxz] = x has a unique solution which is denoted by $z=\stackrel{̄}{x}$ and called the skew element to x (cf. Dörnte [20]). As a consequence of results obtained in [20] we have the following theorem:

Theorem 1.3. In any ternary group (G, [ ]) for all x, y, z G, the following identities take place:

$\begin{array}{c}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left[xx\stackrel{̄}{x}\right]=\left[x\stackrel{̄}{x}x\right]=\left[\stackrel{̄}{x}xx\right]=x,\\ \left[yx\stackrel{̄}{x}\right]=\left[y\stackrel{̄}{x}x\right]=\left[x\stackrel{̄}{x}y\right]=\left[\stackrel{̄}{x}xy\right]=y,\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\overline{\left[xyz\right]}=\left[\stackrel{̄}{z}ȳ\stackrel{̄}{x}\right],\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\overline{\overline{x}}=x.\end{array}$

Other properties of skew elements are described in [21, 22].

Definition 1.4. A ternary groupoid (G, [ ]) is called σ-commutative, if

$\left[{x}_{1}{x}_{2}{x}_{3}\right]=\left[{x}_{{\sigma }_{1}}{x}_{{\sigma }_{2}}{x}_{{\sigma }_{3}}\right]$
(1)

holds for all x1, x2, x3 G and all σ S3. If (1) holds for all σ S3, then (G, [ ]) is a commutative groupoid. If (1) holds only for σ = (13), i.e., if [x1x2x3] = [x3x2x1], then (G, [ ]) is called semicommutative.

Definition 1.5. An element e G is called a middle identity or a middle neutral element of (G, [ ]), if for all x G we have

$\left[exe\right]=x.$

An element e G satisfying the identity

$\left[eex\right]=x$

is called a left identity or a left neutral element of (G, [ ]). Similarly, we define a right identity. An element which is a left, middle, and right identity is called a ternary identity (or simply identity).

A mapping f : (G, [ ]) → (G, [ ]) is called a ternary homomorphism if

$f\left(\left[xyz\right]\right)=\left[f\left(x\right)f\left(y\right)f\left(z\right)\right]$

for all x, y, z G.

A mapping f : (G, [ ]) → (G, [ ]) is called a ternary Jordan homomorphism if

$f\left(\left[xxx\right]\right)=\left[f\left(x\right)f\left(x\right)f\left(x\right)\right]$

for all x G.

In Section 2, we define ternary γ-homomorphism on ternary semigroup and investigate their relations.

## 2 Ternary γ-homomorphisms on ternary semigroups

Definition 2.1. Let G be a ternary semigroup. Then the maping H : GG is called a ternary γ-homomorphism if there exists a function γ : G → [0, ∞) such that

$\gamma \left(H\left(\left[xyz\right]\right)\right)=\gamma \left(\left[H\left(x\right)H\left(y\right)H\left(z\right)\right]\right)=\gamma \left(H\left(x\right)\right)+\gamma \left(H\left(y\right)\right)+\gamma \left(H\left(z\right)\right)$

for all x, y, z G.

Theorem 2.2. Let G be a ternary semigroup and φ : G × G × G → [0, ∞) be a function such that

$\stackrel{̃}{\phi }\left(x,y,z\right):=\frac{1}{3}\sum _{n=0}^{\infty }{3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right)<\infty .$

Suppose that H : GG and f : G → [0, ∞) are functions such that

$\left|f\left(\left[xyz\right]\right)-f\left(x\right)-f\left(y\right)-f\left(z\right)\right|\le \phi \left(x,y,z\right)$
(2)
$\left|f\left(H\left(\left[xyz\right]\right)\right)-f\left(\left[H\left(x\right)H\left(y\right)H\left(z\right)\right]\right)\right|\le \phi \left(x,y,z\right)$
(3)

for all x, y, z G. Then there exists a unique function γ : G → [0, ∞) such that

$\left|f\left(x\right)-\gamma \left(x\right)\right|\le \stackrel{̃}{\phi }\left(x,x,x\right)$

and γ(x3) = 3γ(x). If G is commutative and H is a ternary Jordan homomorphism, then mapping H : GG is a ternary γ-homomorphism.

Proof. Putting y = z = x in inequality (2), we get

$\left|f\left({x}^{3}\right)-3f\left(x\right)\right|\le \phi \left(x,x,x\right).$

By induction, one can show that

$\left|{3}^{-n}f\left({x}^{{3}^{n}}\right)-f\left(x\right)\right|\le \frac{1}{3}\sum _{k=0}^{n-1}{3}^{-k}\phi \left({x}^{{3}^{k}},{x}^{{3}^{k}},{x}^{{3}^{k}}\right),$
(4)

for all x G and for all positive integer n, and

$\left|{3}^{-n}f\left({3}^{{3}^{n}}\right)-{3}^{-m}f\left({x}^{{3}^{m}}\right)\right|\le \frac{1}{3}\sum _{k=m}^{n-1}{3}^{-k}\phi \left({x}^{{3}^{k}},{x}^{{3}^{k}},{x}^{{3}^{k}}\right)$

for all x G and for all nonnegative integers m, n with m < n. Hence, $\left\{{3}^{-n}f\left({x}^{{3}^{n}}\right)\right\}$ is a Cauchy sequence in [0, ∞). Due to the completeness of [0, ∞) we conclude that this sequence is convergent. Now, let

$\gamma \left(x\right)={\text{lim}}_{n\to \infty }{3}^{-n}f\left({x}^{{3}^{n}}\right),\phantom{\rule{1em}{0ex}}x\in G.$

Hence

$\gamma \left({x}^{3}\right)={\text{lim}}_{n\to \infty }{3}^{-n}f\left({x}^{{3}^{n+1}}\right)=3{\text{lim}}_{n\to \infty }{3}^{-\left(n+1\right)}f\left({x}^{{3}^{n+1}}\right)=3\gamma \left(x\right)$

for all x G. If n → ∞ in inequality (4), we obtain

$\left|f\left(x\right)-\gamma \left(x\right)\right|\le \stackrel{̃}{\phi }\left(x,x,x\right).$

Next, assume that G is commutative and H : GG is a ternary Jordan homomorphism. Replace x by ${x}^{{3}^{n}}$, y by ${y}^{{3}^{n}}$ and z by ${z}^{{3}^{n}}$ in inequalities (2) and (3) and divide both sides by 3nto obtain the following:

$\left|{3}^{-n}f\left({\left[xyz\right]}^{{3}^{n}}\right)-{3}^{-n}f\left({x}^{{3}^{n}}\right)-{3}^{-n}f\left({y}^{{3}^{n}}\right)-{3}^{-n}f\left({z}^{{3}^{n}}\right)\right|\le {3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right),$

and

$\left|{3}^{-n}f\left({\left(H\left[xyz\right]\right)}^{{3}^{n}}\right)-{3}^{-n}f\left({\left[H\left(x\right)H\left(y\right)H\left(z\right)\right]}^{{3}^{n}}\right)\right|\le {3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right).$

If n tends to infinity. Then

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

for all x, y, z G. If γ' is another mapping with the required properties, then

$\begin{array}{ll}\hfill \left|\gamma \left(x\right)-{\gamma }^{\prime }\left(x\right)\right|& =\frac{1}{{3}^{n}}\left|{3}^{n}\gamma \left(x\right)-{3}^{n}{\gamma }^{\prime }\left(x\right)\right|\phantom{\rule{2em}{0ex}}\\ =\frac{1}{{3}^{n}}\left|\gamma \left({x}^{{3}^{n}}\right)-{\gamma }^{\prime }\left({x}^{{3}^{n}}\right)\right|\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{{3}^{n}}\left(\left|\gamma \left({x}^{{3}^{n}}\right)-f\left({x}^{{3}^{n}}\right)\right|+\left|f\left({x}^{{3}^{n}}\right)-{\gamma }^{\prime }\left({x}^{{3}^{n}}\right)\right|\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{2}{{3}^{n}}\stackrel{̃}{\phi }\left({x}^{{3}^{n}},{x}^{{3}^{n}},{x}^{{3}^{n}}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Passing to the limit as n → ∞ we get γ(x) = γ'(x), x G. So γ is unique. Therefore, the mapping H : GG is a unique ternary γ-homomorphism.

Theorem 2.3. Let G be a commutative ternary semigroup and φ : G × G × G → [0, ∞) be a function such that

$\stackrel{̃}{\phi }\left(x,y,z\right):=\frac{1}{3}\sum _{n=0}^{\infty }{3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right)<\infty .$

Suppose that H : GG and f : G → [0, ∞) are functions satisfying (2) and (3). If there exists a mapping T : GG such that T is a ternary Jordan homomorphism and

$\left|f\left(H\left(\left[xyz\right]\right)\right)-f\left(\left[H\left(x\right)H\left(y\right)T\left(z\right)\right]\right)\right|\le \phi \left(x,y,z\right)$
(5)

for all x, y, z G, then the mapping T : GG is a ternary γ-homomorphism.

Proof. By Theorem 2.2, there exists a unique mapping γ : G → [0, ∞) such that

$\gamma \left(x\right)={\text{lim}}_{n\to \infty }{3}^{-n}f\left({x}^{{3}^{n}}\right),\phantom{\rule{1em}{0ex}}x\in G,$

and H : GG is a ternary γ-homomorphism. It follows from (5) that

$\begin{array}{c}\left|\gamma \left(\left[H\left(x\right)H\left(y\right)H\left(z\right)\right]\right)-\gamma \left(\left[H\left(x\right)H\left(y\right)T\left(z\right)\right]\right)\right|\\ =\left|\gamma \left(H\left[xyz\right]\right)-\gamma \left(\left[H\left(x\right)H\left(y\right)T\left(z\right)\right]\right)\right|\\ ={\text{lim}}_{n\to \infty }\frac{1}{{3}^{n}}\left|f\left({\left(H\left[xyz\right]\right)}^{{3}^{n}}\right)-f\left({\left[H\left(x\right)H\left(y\right)T\left(z\right)\right]}^{{3}^{n}}\right)\right|\\ \le {\text{lim}}_{n\to \infty }\frac{1}{{3}^{n}}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right)=0\end{array}$

for all x, y, z G. So, γ([H(x)H(y)H(z)]) = γ([H(x)H(y)T(z)]) for all x, y, z G. By (2), γ is ternary additive. Hence, γ(H(x)) = γ(T(x)) for all x G. Thus,

$\begin{array}{ll}\hfill \gamma \left(T\left[xyz\right]\right)& =\gamma \left(H\left[xyz\right]\right)=\gamma \left(H\left(x\right)\right)+\gamma \left(H\left(y\right)\right)+\gamma \left(H\left(z\right)\right)\phantom{\rule{2em}{0ex}}\\ =\gamma \left(T\left(x\right)\right)+\gamma \left(T\left(y\right)\right)+\gamma \left(T\left(z\right)\right)=\gamma \left(\left[T\left(x\right)T\left(y\right)T\left(z\right)\right]\right)\phantom{\rule{2em}{0ex}}\end{array}$

for all x, y, z G. Therefore T is a ternary γ-homomorphism.

Corollary 2.4. Let G be a ternary group with identity element e and φ : G5 → [0, ∞) be a function such that

$\stackrel{̃}{\phi }\left(x,y,u.v.w\right):=\frac{1}{3}\sum _{n=0}^{\infty }{3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{u}^{{3}^{n}},{v}^{{3}^{n}},{w}^{{3}^{n}}\right)<\infty .$

Suppose that H : GG and f : G → [0, ∞) are functions such that f(e) = 0, H(e) = e and

$\left|f\left(\left[xyH\left(\left[uvw\right]\right)\right]\right)-f\left(x\right)-f\left(y\right)-f\left(\left[H\left(u\right)H\left(v\right)H\left(w\right)\right]\right)\right|$
(6)
$\le \phi \left(x,y,H\left(u\right),v,w\right)$
(7)

for all x, y, u, v, w G. Then there exists a unique function γ : G → [0, ∞) such that

$\left|f\left(x\right)-\gamma \left(x\right)\right|\le \stackrel{̃}{\phi }\left(x,x,x,e,e\right)$

and γ(x3) = 3γ(x). If G is commutative and H is a ternary Jordan homomorphism, then the mapping H : GG is a ternary γ-homomorphism.

Proof. Letting v = w = e in (6), we get

$\left|f\left(\left[xyH\left(u\right)\right]\right)-f\left(x\right)-f\left(y\right)-f\left(H\left(u\right)\right)\right|\le \phi \left(x,y,H\left(u\right),e,e\right)$

and by putting x = y = e in (6) we get

$\left|f\left(\left[H\left(\left[uvw\right]\right)\right]\right)-f\left(\left[H\left(u\right)H\left(v\right)H\left(w\right)\right]\right)\right|\le \phi \left(e,e,H\left(u\right),v,w\right).$

The rest of the proof are similar to the proof of Theorem 2.2.

In next section, firstly we define ternary γ-derivation on ternary semigroup and investigate ternary γ-derivations on ternary semigroups with the following functional inequality |f([xxx]) - 3f(x)| ≤ φ(x, x, x).

## 3 Ternary γ-derivations on ternary semigroups

Definition 3.1. Let G be a ternary semigroup. Then the map D : GG is called a ternary γ-derivation if there exists a function γ : G → [0, ∞) such that

$\gamma \left(D\left(\left[xyz\right]\right)\right)=\gamma \left(\left[D\left(x\right)yz\right]\right)+\gamma \left(\left[xD\left(y\right)z\right]\right)+\gamma \left(\left[xyD\left(z\right)\right]\right)$

for all x, y, z G.

Theorem 3.2. Let G be a ternary semigroup and φ : G × G × G → [0, ∞) be a function such that

$\stackrel{̃}{\phi }\left(x,y,z\right):=\frac{1}{3}\sum _{n=0}^{\infty }{3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right)<\infty .$

Suppose that f : G → [0, ∞) is a function such that

$\left|f\left({x}^{3}\right)-3f\left(x\right)\right|\le \phi \left(x,x,x\right)$
(8)
$\left|f\left(D\left(\left[xyz\right]\right)\right)-f\left(\left[D\left(x\right)yz\right]\right)-f\left(\left[xD\left(y\right)z\right]\right)-f\left(\left[xyD\left(z\right)\right]\right)\right|\le \phi \left(x,y,z\right)$
(9)

for all x, y, z G and mapping D : GG. Then there exists a unique function γ : G → [0, ∞) such that

$\left|f\left(x\right)-\gamma \left(x\right)\right|\le \stackrel{̃}{\phi }\left(x,x,x\right)$

and γ (x3) = 3γ(x). If G is commutative and D is a ternary Jordan homomorphism, then mapping D : GG is a ternary γ-derivation.

Proof. By induction in (8), one can show that

$\left|{3}^{-n}f\left({x}^{{3}^{n}}\right)-f\left(x\right)\right|\le \frac{1}{3}\sum _{k=0}^{n-1}{3}^{-k}\phi \left({x}^{{3}^{k}},{x}^{{3}^{k}},{x}^{{3}^{k}}\right),$
(10)

for all x G and for all positive integer n, and

$\left|{3}^{-n}f\left({3}^{{3}^{n}}\right)-{3}^{-m}f\left({x}^{{3}^{m}}\right)\right|\le \frac{1}{3}\sum _{k=m}^{n-1}{3}^{-k}\phi \left({x}^{{3}^{k}},{x}^{{3}^{k}},{x}^{{3}^{k}}\right)$

for all x G and for all nonnegative integers m, n with m < n. Hence, $\left\{{3}^{-n}f\left({x}^{{3}^{n}}\right)\right\}$ is a Cauchy sequence in [0, ∞). Due to the completeness of [0, ∞) we conclude that this sequence is convergent. Set now

$\gamma \left(x\right)={\text{lim}}_{n\to \infty }{3}^{-n}f\left({x}^{{3}^{n}}\right),\phantom{\rule{1em}{0ex}}x\in G.$

Hence

$\gamma \left({x}^{3}\right)={\text{lim}}_{n\to \infty }{3}^{-n}f\left({x}^{{3}^{n+1}}\right)=3{\text{lim}}_{n\to \infty }{3}^{-\left(n+1\right)}f\left({x}^{{3}^{n+1}}\right)=3\gamma \left(x\right)$

for all x G. If n → ∞ in inequality (10), we obtain

$\left|f\left(x\right)-\gamma \left(x\right)\right|\le \stackrel{̃}{\phi }\left(x,x,x\right).$

Next, assume that G is commutative and D : GG is a ternary Jordan homomorphism. Replace x by ${x}^{{3}^{n}}$, y by ${y}^{{3}^{n}}$ and z by ${z}^{{3}^{n}}$ in inequality (9) and divide both sides by 3n, we have

If n tends to infinity. Then

$\gamma \left(D\left(\left[xyz\right]\right)\right)=\gamma \left(\left[D\left(x\right)yz\right]\right)+\gamma \left(\left[xD\left(y\right)z\right]\right)+\gamma \left(\left[xyD\left(z\right)\right]\right)$

for all x, y, z G. If γ' is another mapping with the required properties, then

$\begin{array}{ll}\hfill \left|\gamma \left(x\right)-{\gamma }^{\prime }\left(x\right)\right|& =\frac{1}{{3}^{n}}\left|{3}^{n}\gamma \left(x\right)-{3}^{n}{\gamma }^{\prime }\left(x\right)\right|\phantom{\rule{2em}{0ex}}\\ =\frac{1}{{3}^{n}}\left|\gamma \left({x}^{{3}^{n}}\right)-{\gamma }^{\prime }\left({x}^{{3}^{n}}\right)\right|\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{{3}^{n}}\left(\left|\gamma \left({x}^{{3}^{n}}\right)-f\left({x}^{{3}^{n}}\right)\right|+\left|f\left({x}^{{3}^{n}}\right)-{\gamma }^{\prime }\left({x}^{{3}^{n}}\right)\right|\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{2}{{3}^{n}}\stackrel{̃}{\phi }\left({x}^{{3}^{n}},{x}^{{3}^{n}},{x}^{{3}^{n}}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Passing to the limit as n → ∞ we get γ(x) = γ'(x), x G. This proves the uniqueness of γ. Thus, the mapping D : GG is a unique ternary γ-derivation.

Corollary 3.3. Let G be a ternary semigroup, and ϵ > 0. Suppose that f : G → [0, ∞) is a function such that

$\left|f\left({x}^{3}\right)-3f\left(x\right)\right|\le \epsilon ,$
$\left|f\left(D\left(\left[xyz\right]\right)\right)-f\left(\left[D\left(x\right)yz\right]\right)-f\left(\left[xD\left(y\right)z\right]\right)-f\left(\left[xyD\left(z\right)\right]\right)\right|\le \epsilon$

for all x, y, z G and mapping D : GG. Then there exists a unique function γ : G → [0, ∞) such that

$\left|f\left(x\right)-\gamma \left(x\right)\right|\le \frac{1}{2}\epsilon$

and γ(x3) = 3γ(x). If G is commutative and D is a ternary Jordan homomorphism, then mapping D : GG is a ternary γ-derivation.

Theorem 3.4. Let G be a commutative ternary semigroup and φ : G × G × G → [0, ∞) be a function such that

$\stackrel{̃}{\phi }\left(x,y,z\right):=\frac{1}{3}\sum _{n=0}^{\infty }{3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right)<\infty .$

Suppose that D : GG is a ternary Jordan homomorphism and f : G → [0, ∞) is a function such that

$f\left({x}^{{3}^{n}}\right)={3}^{n}f\left(x\right)$
$\left|f\left(D\left(\left[xyz\right]\right)\right)-f\left(\left[D\left(x\right)yz\right]\right)-f\left(\left[xD\left(y\right)z\right]\right)-f\left(\left[xyD\left(z\right)\right]\right)\right|\le \phi \left(x,y,z\right)$
(11)

for all x, y, z G and for all positive integer n. Then the mapping D : GG is a ternary f-derivation.

Proof. Since G is commutative and D : GG is ternary Jordan homomorphism. Replace x by ${x}^{{3}^{n}}$, y by ${y}^{{3}^{n}}$ and z by ${z}^{{3}^{n}}$ in inequality (11) and divide both sides by 3nto obtain the following:

If n tends to infinity. Then

$f\left(D\left(\left[xyz\right]\right)\right)=f\left(\left[D\left(x\right)yz\right]\right)+f\left(\left[xD\left(y\right)z\right]\right)+f\left(\left[xyD\left(z\right)\right]\right)$

for all x, y, z G. Thus, the mapping D : GG is a ternary f-derivation.

## 4 Ternary (γ, h)-derivations on ternary semigroups

In this section, we introduce concept ternary (γ, h)-derivations on ternary semigroups and investigate ternary (γ, h)-derivations on ternary semigroups with the following functional inequality |f([xxx]) - 3f(x)| < φ(x, x, x).

Definition 4.1. Let G be a ternary semigroup. Then the maping D : GG is called ternary (γ, h)-derivation if there exists mappings h : GG and γ : G → [0, ∞) such that

$\gamma \left(D\left(\left[xyz\right]\right)\right)=\gamma \left(\left[D\left(x\right)h\left(y\right)h\left(z\right)\right]\right)+\gamma \left(\left[h\left(x\right)D\left(y\right)h\left(z\right)\right]\right)+\gamma \left(\left[h\left(x\right)h\left(y\right)D\left(z\right)\right]\right)$

for all x, y, z G.

Theorem 4.2. Let G be a ternary semigroup, and let φ : G × G × G → [0, ∞) be a function such that

$\stackrel{̃}{\phi }\left(x,y,z\right):=\frac{1}{3}\sum _{n=0}^{\infty }{3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right)<\infty .$

Suppose that D, h : GG and f : G → [0, ∞) are functions such that

$\left|f\left({x}^{3}\right)-3f\left(x\right)\right|\le \phi \left(x,x,x\right)$
(12)
$\left|f\left(D\left(\left[xyz\right]\right)\right)-f\left(\left[D\left(x\right)h\left(y\right)h\left(z\right)\right]\right)-f\left(\left[h\left(x\right)D\left(y\right)h\left(z\right)\right]\right)\right$
(13)
(14)

for all x, y, z G. Then there exist a unique function γ : G → [0, ∞) such that

$\left|f\left(x\right)-\gamma \left(x\right)\right|\le \stackrel{̃}{\phi }\left(x,x,x\right)$

and γ(x3) = 3γ(x). If G is commutative and D, h are ternary homomorphisms, then mapping D : GG is a ternary (γ, h)-derivation.

Proof. By a similar method to the proof of Theorem 3.2 we obtain

$\gamma \left(x\right)={\text{lim}}_{n\to \infty }{3}^{-n}f\left({x}^{{3}^{n}}\right),\phantom{\rule{1em}{0ex}}x\in G.$

Such that

$\left|f\left(x\right)-\gamma \left(x\right)\right|\le \stackrel{̃}{\phi }\left(x,x,x\right)$

and

$\gamma \left({x}^{3}\right)=3\gamma \left(x\right)$

for all x G.

Now suppose that G is commutative and D, h : GG are ternary homomorphism. Replace x by ${x}^{{3}^{n}}$, y by ${y}^{{3}^{n}}$ and z by ${z}^{{3}^{n}}$ in inequality (13) and divide both sides by 3nto obtain the following:

Let n tend to infinity. Then

$\gamma \left(D\left(\left[xyz\right]\right)\right)=\gamma \left(\left[D\left(x\right)h\left(y\right)h\left(z\right)\right]\right)+\gamma \left(\left[h\left(x\right)D\left(y\right)h\left(z\right)\right]\right)+\gamma \left(\left[h\left(x\right)h\left(y\right)D\left(z\right)\right]\right)$

for all x, y, z G.

If in Theorem 4.2 replace inequality 12 by equation $f\left({x}^{{3}^{n}}\right)={3}^{n}f\left(x\right)$ to obtain the following Theorem.

Theorem 4.3. Let G be a commutative ternary semigroup and φ : G × G × G → [0, ∞) be a function such that

$\stackrel{̃}{\phi }\left(x,y,z\right):=\frac{1}{3}\sum _{n=0}^{\infty }{3}^{-n}\phi \left({x}^{{3}^{n}},{y}^{{3}^{n}},{z}^{{3}^{n}}\right)<\infty .$

Suppose that D, h : GG are ternary Jordan homomorphism and f : G → [0, ∞) is a function such that

$f\left({x}^{{3}^{n}}\right)={3}^{n}f\left(x\right)$

for all x, y, z G and for all positive integer n. Then the mapping D : GG is a ternary (f, h)-derivation.

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Dehghanian, M., Modarres, M.S. Ternary γ-homomorphisms and ternary γ-derivations on ternary semigroups. J Inequal Appl 2012, 34 (2012). https://doi.org/10.1186/1029-242X-2012-34