- Open Access
Stability and superstability of ternary homomorphisms and ternary derivations on ternary quasi-Banach algebras
© Osbouei et al. ; licensee Springer. 2012
- Received: 2 March 2012
- Accepted: 13 June 2012
- Published: 13 June 2012
In this article, we investigate the generalized Hyers-Ulam-Rassias stability, Isac-Rassias type stability and superstability of ternary homomorphisms and ternary derivations associated to the generalized m- variables Cauchy-Jensen functional equation
for a fixed positive integer m with m ≥ 3 on ternary quasi-Banach algebras.
2010 Mathematics Subject Classification: 39B82; 39B52.
- Hyers-Ulam-Rassias stability
- Isac-Rassias-type stability
- ternary algebra
- quasi-Banach space
A functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to a true solution of (ξ). A functional equation (ξ) is superstable if any function g satisfying the equation (ξ) approximately is a true solution of (ξ).
It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation.
The first stability problem was raised by Ulam  during his talk at the University of Wisconsin in 1940. The stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? If the answer is affirmative, we would say that the equation is stable.
for all x ∈ E. Moreover if f(tx) is continuous in t ∈ ℝ for each fixed x ∈ E, then T is linear. Aoki  and Bourgin  considered the stability problem with unbounded Cauchy differences. In 1978, Rassias  provided a generalization of Hyers' theorem by proving the existence of unique linear mappings near approximate additive mappings. It was shown by Gajda , as well as by Rassias and Šemrl  that one cannot prove a stability theorem of the additive equation for a specific function. Găvruta  obtained generalized result of Rassias' theorem which allows the Cauchy difference to be controlled by a general unbounded function.
Bourgin  is the first mathematician dealing with stability of (ring) ho-momorphism f(xy) = f(x)f(y). The topic of approximate homomorphisms and approximate derivations was studied by a number of mathematicians (see [9–13], and references therein).
ǁx ǁ ≥ 0 for all x ∈ X and ǁx ǁ = 0 if and only if x = 0.
ǁλ.x ǁ = ǀλ ǀ. ǁx ǁ for all λ ∈ ℝ and all x ∈ X.
- (3)There is a constant K ≥ 1 such that ǁx + y ǁ ≤ K(ǁx ǁ + ǁy ǁ) for all x, y ∈ X. The pair (X, ǁ.ǁ) is called a quasi-normed space if ǁ.ǁ is a quasi-norm on X . A quasi-Banach space is a complete quasi-normed space. A quasi-norm ǁ.ǁ is called a p-norm (0 ≤ p ≤ 1) if
for all x, y ∈ X . In this case, a quasi-Banach space is called a p-Banach space.
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 et al. . There are some applications, although still hypothetical, in the fractional quantum Hall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation. The comments on physical applications of ternary structures can be found in [37, 39, 40, 43, 44, 52–59].
Let A be a linear space over a complex field equipped with a mapping : A3 = A × A × A → A with (x, y, z) ↦ [x, y, z] that is linear in variables x, y, z and satisfies the associative identity [[x, y, z], u, v] = [x, [y, z, u], v] = [x, y, [z, u, v]] for all x, y, z, u, v in A. The pair (A, [ ]) is called a ternary algebra.
Assume that A is a ternary algebra. We say A has a unit if there exist an element e ∈ A such that [e, e, a] = [eae] = [a, e, e] = a for all a ∈ A.
Let A be a ternary algebra and let (A, ǁ.ǁ) be a quasi-Banach space (p-Banach space) (with constant K ≥ 1). Then A is called a ternary quasi-Banach algebra (ternary p-Banach algebra) if ǁ[x, y, z] ǁ ≤ K ǁx ǁǁy ǁǁz ǁ for all x, y, z ∈ A.
for a fixed positive integer m with m ≥ 2 in quasi-Banach spaces. In this paper, we establish the generalized Hyers-Ulam-Rassias stability of ternary homomorphisms and ternary derivations on ternary quasi-Banach algebras. Moreover, by using the main theorems, we prove the superstability of ternary homomorphisms and ternary derivations on ternary quasi Banach algebras.
Throughout this article, we assume that A is a ternary quasi-Banach algebra with quasi-norm ǁ.ǁ A and B is a ternary p-Banach algebra with quasi-norm ǁ.ǁ B
for all a, b, c, u, x1, x2, ..., x m ∈ A and all
for all x ∈ A.
for all a, b, c ∈ A. This means that T : A → B is a ternary homomorphism. The uniqueness of T follows from Theorem 2.2 of .
for all x ∈ A
for all a, b, c, u, x1, x1, ..., x m ∈ A.
Now, we investigate the Hyers-Ulam type stability of ternary homomor-phisms on ternary quasi Banach algebras as follows.
for all x ∈ A.
for all u, a, b, c, x1, x2, ..., x m ∈ A.
ψ(ts) ≤ ψ(t)ψ(s); s, t > 0,
ψ(t) < t; t > 1.
These stability results can be applied in stochastic analysis , financial and actuarial mathematics, as well as in psychology and sociology. The following corollary is Isac-Rassias type stability of ternary homomorphisms on ternary quasi-Banach algebras.
for all x ∈ A, where .
for all u, a, b, c, x1, x2,..., x m ∈ A.
Moreover, we have the superstability of ternary homomorphisms on ternary quasi Banach algebras as follows.
for all a, b, c, u, x1, x1, ..., x m ∈ A and all . Then f : A → B is a ternary homomorphism.
for all x ∈ A. This means that f(x) = T(x) for all x ∈ A. Hence f : A → B is a ternary homomorphism.
for all a, b, c, u, x1, x2, ..., x m ∈ A and all .
for all x ∈ A.
for all a, b, c ∈ A. This means that D : A → B is a ternary derivation.
for all x ∈ A.
for all a, b, c, u, x1, x1, ..., x m ∈ A.
We have the Hyers-Ulam type stability of ternary derivations on ternary quasi Banach algebras as follows.
for all x ∈ A.
for all u, a, b, c, x1, x2, ..., x m ∈ A.
By using the same technique of proving Corollary 2.4, we can prove the Isac-Rassias type stability of ternary derivations on ternary quasi-Banach algebras as follows.
for all x ∈ A, where .
Similar to Corollary 2.5, we can prove the superstability of ternary derivations on ternary quasi-Banach algebras as follows.
for all a, b, c, u, x1, x1, ..., x m ∈ A and all . Then f : A → A is a ternary derivation.
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