Ternary γ-homomorphisms and ternary γ-derivations on ternary semigroups
© Dehghanian and Modarres; licensee Springer. 2012
Received: 6 July 2011
Accepted: 15 February 2012
Published: 15 February 2012
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 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 : 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  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 [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 ).
Definition 1.1. A nonempty set G with one ternary operation [ ]: G × G × G → G is called a ternary groupoid and denoted by (G, [ ]).
hold for all x, y, z, u, v ∈ G (see ). We shall write x3 instead of [xxx].
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 and called the skew element to x (cf. Dörnte ). As a consequence of results obtained in  we have the following theorem:
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.
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).
for all x, y, z ∈ G.
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
for all x, y, z ∈ G.
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.
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.
for all x, y, z ∈ G, then the mapping T : G → G is a ternary γ-homomorphism.
for all x, y, z ∈ G. Therefore T is a ternary γ-homomorphism.
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.
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
for all x, y, z ∈ G.
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.
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.
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.
for all x, y, z ∈ G and for all positive integer n. Then the mapping D : G → G is a ternary f-derivation.
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).
for all x, y, z ∈ G.
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.
for all x ∈ G.
for all x, y, z ∈ G.
If in Theorem 4.2 replace inequality 12 by equation to obtain the following Theorem.
for all x, y, z ∈ G and for all positive integer n. Then the mapping D : G → G is a ternary (f, h)-derivation.
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