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:

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

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 : G → G' satisfies ρ(f(xy), f(x)f(y)) < δ for all x, y ∈ G, then a homomorphism h : G → G 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 [9–15].

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 × G → G 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 $z=\bar{x}$ 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:

$$\begin{array}{c}\left[xx\bar{x}\right]=\left[x\bar{x}x\right]=\left[\bar{x}xx\right]=x,\\ \left[yx\bar{x}\right]=\left[y\bar{x}x\right]=\left[x\bar{x}y\right]=\left[\bar{x}xy\right]=y,\\ \overline{\left[xyz\right]}=\left[\bar{z}ȳ\bar{x}\right],\\ \overline{\overline{x}}=x.\end{array}$$

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 : G → G 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

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

Suppose that H : G → G 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

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

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

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

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

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

Next, assume that G is commutative and H : G → G 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

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

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

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

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

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

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

and H : G → G 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,

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

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

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

Suppose that H : G → G 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

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

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

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

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 : G → G. Then there exists a unique function γ : G → [0, ∞) such that

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

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

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

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

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

Next, assume that G is commutative and D : G → G 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

$$\begin{array}{c}\left|3^{-n}f\left(D{\left(\left[xyz\right]\right)}^{3^n}\right)-3^{-n}f\left({\left[D\left(x\right)yz\right]}^{3^n}\right)\right.\\ \left.-3^{-n}f\left({\left[xD\left(y\right)z\right]}^{3^n}\right)-3^{-n}f\left({\left[xyD\left(z\right)\right]}^{3^n}\right)\right|\\ \le 3^{-n}\phi \left(x^{3^n},y^{3^n},z^{3^n}\right).\end{array}$$

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

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

Passing to the limit as n → ∞ we get γ(x) = γ'(x), x ∈ G. This proves the uniqueness of γ. Thus, the mapping D : G → G 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 : G → G. 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 : G → G 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

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

Suppose that D : G → G 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 : G → G is a ternary f-derivation.

Proof. Since G is commutative and D : G → G 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:

$$\begin{array}{c}\left|3^{-n}f\left(D{\left(\left[xyz\right]\right)}^{3^n}\right)-3^{-n}f\left({\left[D\left(x\right)yz\right]}^{3^n}\right)\right.\\ \left.-3^{-n}f\left({\left[xD\left(y\right)z\right]}^{3^n}\right)-3^{-n}f\left({\left[xyD\left(z\right)\right]}^{3^n}\right)\right|\\ \le 3^{-n}\phi \left(x^{3^n},y^{3^n},z^{3^n}\right).\end{array}$$

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 : G → G 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 : G → G is called ternary (γ, h)-derivation if there exists mappings h : G → G 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

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

Suppose that D, h : G → G 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)

$$\left.-f\left(\left[h\left(x\right)h\left(y\right)D\left(z\right)\right]\right)\right|\le \phi \left(x,y,z\right)$$

(14)

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

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

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

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

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

Such that

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

and

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

for all x ∈ G.

Now suppose that G is commutative and D, h : G → G 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:

$$\begin{array}{c}\left|3^{-n}f\left(D{\left(\left[xyz\right]\right)}^{3^n}\right)-3^{-n}f\left({\left[D\left(x\right)h\left(y\right)h\left(z\right)\right]}^{3^n}\right)\right.\\ \left.-3^{-n}f\left({\left[h\left(x\right)D\left(y\right)h\left(z\right)\right]}^{3^n}\right)-3^{-n}f\left({\left[h\left(x\right)h\left(y\right)D\left(z\right)\right]}^{3^n}\right)\right|\\ \le 3^{-n}\phi \left(x^{3^n},y^{3^n},z^{3^n}\right).\end{array}$$

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

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

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

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

$$\begin{array}{c}\left|f\left(D\left(\left[xyz\right]\right)\right)-f\left(\left[D\left(x\right)h\left(y\right)h\left(z\right)\right]\right)-f\left(\left[h\left(x\right)D\left(y\right)h\left(z\right)\right]\right)\right.\\ \left.-f\left(\left[h\left(x\right)h\left(y\right)D\left(z\right)\right]\right)\right|\le \phi \left(x,y,z\right)\end{array}$$

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