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Nearly n-homomorphisms and n-derivations in fuzzy ternary Banach algebras
Journal of Inequalities and Applications volume 2013, Article number: 71 (2013)
Let for some . We investigate the generalized Hyers-Ulam stability of n-homomorphisms and n-derivations on fuzzy ternary Banach algebras related to the generalized Cauchy-Jensen additive functional equation.
MSC:39B52, 46S40, 26E50.
We say a functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to a true solution of (ξ). We say that a functional equation (ξ) is superstable if every approximately solution of (ξ) is an exact solution of it (see ).
Speaking of the stability of a functional equation, we follow the question raised in 1940 by Ulam: When is it true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation? This problem was solved in the next year for the Cauchy functional equation on Banach spaces by Hyers . Let be a mapping between Banach spaces such that
for all and for some . Then there exists a unique additive mapping such that
for all . Moreover, if is continuous in t for each fixed , then T is linear. It gave rise to the Hyers-Ulam type stability of functional equations. Hyers’ theorem was generalized by Rassias  for linear mappings by considering an unbounded Cauchy difference.
Theorem 1.1 (Th.M. Rassias)
Let be a mapping from a normed vector space E into a Banach space subject to the inequality for all , where ϵ and p are constants with and . Then the limit exists for all and is the unique additive mapping which satisfies
for all . Also, if for each , the function is continuous in , then L is linear.
Găvruta  generalized the Rassias result. Beginning around the year 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [5–45]).
Some mathematicians have defined fuzzy norms on a vector space from various points of view (see  and ). Bag and Samanta , following Cheng and Mordeson , gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type . They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces .
In this paper, we consider a mapping satisfying the following functional equation, which is introduced by Azadi Kenary :
for all , where are fixed integers with , . Especially, we observe that in the case , equation (1.1) yields the Cauchy additive equation . We observe that in the case , equation (1.1) yields the Jensen additive equation . Therefore, equation (1.1) is a generalized form of the Cauchy-Jensen additive equation, and thus every solution of equation (1.1) may be analogously called general -Cauchy-Jensen additive. For the case , the authors have established new theorems about the Ulam-Hyers-Rassias stability in quasi β-normed spaces . Let X and Y be linear spaces. For each m with , a mapping satisfies equation (1.1) for all if and only if is Cauchy additive, where if . In particular, we have and for all .
Definition 1.1 Let X be a real vector space. A function is called a fuzzy norm on X if for all and all ,
(N1) for ;
(N2) if and only if for all ;
(N3) if ;
(N5) is a non-decreasing function of ℝ and ;
(N6) for , is continuous on ℝ.
Example 1.1 Let be a normed linear space and . Then
is a fuzzy norm on X.
Definition 1.2 Let be a fuzzy normed vector space. A sequence in X is said to be convergent or converge if there exists an such that for all . In this case, x is called the limit of the sequence in X, and we denote it by .
Definition 1.3 Let be a fuzzy normed vector space. A sequence in X is called Cauchy if for each and each , there exists an such that for all and all , we have .
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping between fuzzy normed vector spaces X and Y is continuous at a point if for each sequence converging to , the sequence converges to . If is continuous at each , then is said to be continuous on X (see ).
Definition 1.4 Let X be a ternary algebra and be a fuzzy normed space.
The fuzzy normed space is called a fuzzy ternary normed algebra if
for all and all positive real numbers s, t and u.
A complete ternary fuzzy normed algebra is called a ternary fuzzy Banach algebra.
Example 1.2 Let be a ternary normed (Banach) algebra. Let
Then is a fuzzy norm on X and is a ternary fuzzy normed (Banach) algebra.
From now on, we suppose that is a fixed positive integer and . Also, we assume that is a fixed positive integer.
Definition 1.5 Let and be two ternary fuzzy normed algebras. Then
a ℂ-linear mapping is called an -homomorphism if
for all ;
a ℂ-linear mapping is called an -derivation if
for all .
We apply the following theorem on weighted spaces.
Theorem 1.2 Let (X,d) be a complete generalized metric space and be a strictly contractive mapping with a Lipschitz constant . Then, for all , either for all nonnegative integers n or there exists a positive integer such that
for all ;
the sequence converges to a fixed point of J;
is the unique fixed point of J in the set ;
for all .
Throughout this paper, we suppose that X is a ternary fuzzy normed algebra and Y is a ternary fuzzy Banach algebra. For convenience, we use the following abbreviations for a given mapping :
There are several recent works on stability of functional equations on Banach algebras (see [10–28]). We investigate the stability of n-homomorphisms and n-derivations on fuzzy ternary Banach algebras.
2 Main results
In this section, by using the idea of Park et al. , we prove the generalized Hyers-Ulam-Rassias stability of 5-homomorphisms and 5-derivations related to the functional equation (1.1) on fuzzy ternary Banach algebras (see also ). We start our main results by the stability of 5-homomorphisms.
Theorem 2.1 Let be a mapping such that there exists an with
for all . Let with be a mapping satisfying
for all , and all . Then there exists a unique -homomorphism such that
for all and all .
Proof Letting and putting , in (2.2), we obtain
for all and . Set and define by
where . By using the same technique as in the proof of Theorem 3.2 of , we can show that is a complete generalized metric space. We define by
for all . It is easy to see that for all . This implies that
By Banach’s fixed point approach, J has a unique fixed point in satisfying
for all . This implies that H is a unique mapping such that (2.5) and that there exists satisfying for all and . Moreover, we have as . This implies the equality
for all .
It follows from (2.2) and (2.6) that
for all , . This means that is additive. By using the same technique as in the proof of Theorem 2.1 of , we can show that H is ℂ-linear. By (2.2), we have
for all and all . Then
for all and all . Hence
for all and all . Hence
for all . This means that H is an -homomorphism. This completes the proof. □
Theorem 2.2 Let be a mapping such that there exists an with
for all . Let be a mapping with satisfying (2.2). Then the limit exists for all and defines an -homomorphism such that
for all and all .
Proof Let be the metric space defined as in the proof of Theorem 2.1. Consider the mapping by for all . One can show that implies that for all positive real numbers ϵ. This means that T is a contraction on . The mapping
is the unique fixed point of T in S. H has the following property:
for all . This implies that H is a unique mapping satisfying (2.8) such that there exists satisfying for all and .
The rest of the proof is similar to the proof of Theorem 2.1. □
Now, we investigate the Hyers-Ulam-Rassias stability of -derivations in ternary fuzzy Banach algebras.
Theorem 2.3 Let be a mapping such that there exists an with (2.1) Let with be a mapping satisfying
for all , and all . Then there exists a unique -derivation such that
for all and all .
Proof By the same reasoning as that in the proof of Theorem 2.1, the mapping defined by
is a unique ℂ-linear mapping which satisfies (2.10). We show that D is an -derivation. By (2.9),
for all and all . Then we have
for all and all . It follows that D is an -derivation. □
Theorem 2.4 Let be a mapping such that there exists an with
for all . Let be a mapping with satisfying (2.9). Then the limit exists for all and defines an -derivation such that
for all and all .
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The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Hassani, F., Ebadian, A., Eshaghi Gordji, M. et al. Nearly n-homomorphisms and n-derivations in fuzzy ternary Banach algebras. J Inequal Appl 2013, 71 (2013). https://doi.org/10.1186/1029-242X-2013-71
- Hyers-Ulam-Rassias stability
- fixed point theorem
- fuzzy ternary Banach algebra