# 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  who introduced the notion of "cubic matrix" which in turn was generalized by Kapranov, Gelfand and Zelevinskii et al. . 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  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  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  generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. This phenomenon of stability that was introduced by Rassias  is called the Hyers-Ulam-Rassias stability. In 1992, a generalization of Rassias theorem was obtained by Găvruta .

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 .

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 ).

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 ). 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 ) that elements x, y, z are uniquely determined. Moreover, according to the suggestion of post  one can prove (cf, Dudek et al. ) 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 ). As a consequence of results obtained in  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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Correspondence to Mehdi Dehghanian.

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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

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• DOI: https://doi.org/10.1186/1029-242X-2012-34

### Keywords

• ternary semigroup
• ternary γ-homomorphism
• ternary γ-derivation
• ternary (γ, h)-derivation. 