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.
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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:
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
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
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 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:
Other properties of skew elements are described in [21, 22].
Definition 1.4. A ternary groupoid (G, [ ]) is called σ-commutative, if
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
An element e ∈ G satisfying the identity
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
for all x, y, z ∈ G.
A mapping f : (G, [ ]) → (G, [ ]) is called a ternary Jordan homomorphism if
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
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
Suppose that H : G → G and f : G → [0, ∞) are functions such that
for all x, y, z ∈ G. Then there exists a unique function γ : G → [0, ∞) such that
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
By induction, one can show that
for all x ∈ G and for all positive integer n, and
for all x ∈ G and for all nonnegative integers m, n with m < n. Hence, is a Cauchy sequence in [0, ∞). Due to the completeness of [0, ∞) we conclude that this sequence is convergent. Now, let
Hence
for all x ∈ G. If n → ∞ in inequality (4), we obtain
Next, assume that G is commutative and H : G → G is a ternary Jordan homomorphism. Replace x by , y by and z by in inequalities (2) and (3) and divide both sides by 3nto obtain the following:
and
If n tends to infinity. Then
for all x, y, z ∈ G. If γ' is another mapping with the required properties, then
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
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
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
and H : G → G is a ternary γ-homomorphism. It follows from (5) that
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,
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
Suppose that H : G → G and f : G → [0, ∞) are functions such that f(e) = 0, H(e) = e and
for all x, y, u, v, w ∈ G. Then there exists a unique function γ : G → [0, ∞) such that
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
and by putting x = y = e in (6) we get
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
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
Suppose that f : G → [0, ∞) is a function such that
for all x, y, z ∈ G and mapping D : G → G. Then there exists a unique function γ : G → [0, ∞) such that
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
for all x ∈ G and for all positive integer n, and
for all x ∈ G and for all nonnegative integers m, n with m < n. Hence, is a Cauchy sequence in [0, ∞). Due to the completeness of [0, ∞) we conclude that this sequence is convergent. Set now
Hence
for all x ∈ G. If n → ∞ in inequality (10), we obtain
Next, assume that G is commutative and D : G → G is a ternary Jordan homomorphism. Replace x by , y by and z by in inequality (9) and divide both sides by 3n, we have
If n tends to infinity. Then
for all x, y, z ∈ G. If γ' is another mapping with the required properties, then
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
for all x, y, z ∈ G and mapping D : G → G. Then there exists a unique function γ : G → [0, ∞) such that
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
Suppose that D : G → G is a ternary Jordan homomorphism and f : G → [0, ∞) is a function such that
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 , y by and z by in inequality (11) and divide both sides by 3nto obtain the following:
If n tends to infinity. Then
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
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
Suppose that D, h : G → G and f : G → [0, ∞) are functions such that
for all x, y, z ∈ G. Then there exist a unique function γ : G → [0, ∞) such that
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
Such that
and
for all x ∈ G.
Now suppose that G is commutative and D, h : G → G are ternary homomorphism. Replace x by , y by and z by in inequality (13) and divide both sides by 3nto obtain the following:
Let n tend to infinity. Then
for all x, y, z ∈ G.
If in Theorem 4.2 replace inequality 12 by equation 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
Suppose that D, h : G → G are ternary Jordan homomorphism and f : G → [0, ∞) is a function such that
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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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