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Homomorphisms and derivations in induced fuzzy -algebras
Journal of Inequalities and Applications volume 2013, Article number: 88 (2013)
Abstract
Using fixed point method, we establish the Hyers-Ulam stability of fuzzy ∗-homomorphisms in fuzzy -algebras and fuzzy ∗-derivations on fuzzy -algebras associated to the following -Cauchy-Jensen additive functional equation:
MSC:47S40, 39B52, 46S40, 47H10, 26E50.
1 Introduction
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Rassias [3] for linear mappings by considering an unbounded Cauchy difference.
Theorem 1.1 (T.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 ℝ-linear.
The functional equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers-Ulam stability of the quadratic functional equation was proved by Skof [4] for mappings , where X is a normed space and Y is a Banach space. Cholewa [5] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [6] proved the Hyers-Ulam stability of the quadratic functional equation.
The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [7–20]).
Katsaras [21] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view (see [22–24]). In particular, Bag and Samanta [25], following Cheng and Mordeson [26], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [27]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [28].
Now, we consider a mapping satisfying the following functional equation, which is introduced by the first author:
for all , where are fixed integers with , . Especially, we observe that in the case , equation (1) yields the Cauchy-type additive equation . We observe that in the case , equation (1) yields the Jensen-type additive equation . Therefore, equation (1) is a generalized form of the Cauchy-Jensen additive equation and thus every solution of equation (1) may be analogously called a general -Cauchy-Jensen additive. For the case , we have established new theorems about the Hyers-Ulam stability in quasi β-normed spaces [29]. Let X and Y be linear spaces. For each m with , a mapping satisfies equation (1) for all if and only if is a Cauchy additive, where if . In particular, we have and for all .
2 Preliminaries
Definition 2.1 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 ;
(N4) ;
(N5) is a non-decreasing function of ℝ and ;
(N6) for , is continuous on ℝ.
Example 2.1 Let be a normed linear space and . Then
is a fuzzy norm on X.
Definition 2.2 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 2.3 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 [28]).
Definition 2.4 Let X be a ∗-algebra and be a fuzzy normed space.
-
(1)
The fuzzy normed space is called a fuzzy normed ∗-algebra if
for all and all positive real numbers s and t.
-
(2)
A complete fuzzy normed ∗-algebra is called a fuzzy Banach ∗-algebra.
Example 2.2 Let be a normed ∗-algebra. Let
Then is a fuzzy norm on X and is a fuzzy normed ∗-algebra.
Definition 2.5 Let be a normed -algebra and N be a fuzzy norm on X.
-
(1)
The fuzzy normed ∗-algebra is called an induced fuzzy normed ∗-algebra.
-
(2)
The fuzzy Banach ∗-algebra is called an induced fuzzy -algebra.
Definition 2.6 Let and be induced fuzzy normed ∗-algebras.
-
(1)
A multiplicative ℂ-linear mapping is called a fuzzy ∗-homomorphism if for all .
-
(2)
A ℂ-linear mapping is called a fuzzy ∗-derivation if and for all .
Definition 2.7 Let X be a nonempty set. A function is called a generalized metric on X if d satisfies the following conditions:
-
(1)
if and only if for all ;
-
(2)
for all ;
-
(3)
for all .
Theorem 2.1 Let 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
-
(1)
for all ;
-
(2)
the sequence converges to a fixed point of J;
-
(3)
is the unique fixed point of J in the set ;
-
(4)
for all .
Throughout this paper, assume that X, Y are unital fuzzy Banach ∗-algebras.
3 Approximate homomorphisms in fuzzy Banach ∗-algebras
In this section, using fixed point method, we prove the Hyers-Ulam stability of homomorphisms in fuzzy Banach ∗-algebras related to functional equation (1).
Theorem 3.1 Let be a function such that there exists an with
for all . Let with be a mapping satisfying
for all and all . Then there exists a fuzzy ∗-homomorphism such that
for all and all .
Proof Letting and replacing by in (2), we have
for all and . Consider the set and the generalized metric d in S defined by
where . It is easy to show that is complete (see [30]). Now, we consider a linear mapping such that for all . Let be such that . Then for all and . Hence,
for all and . Thus, implies that . This means that for all . It follows from (6) that
for all and all . So,
This implies that . By Theorem 2.1, there exists a mapping satisfying the following:
-
(1)
A is a fixed point of J, that is,
(7)
for all . The mapping H is a unique fixed point of J in the set . This implies that H is a unique mapping satisfying (7) such that there exists satisfying for all and .
-
(2)
as . This implies the equality
(8)
for all .
-
(3)
with , which implies the inequality
This implies that the inequality (5) holds. Furthermore, it follows from (2) and (8) that
for all , all and all . Hence,
for all . So, the mapping is additive and ℂ-linear. By (3)
for all and all . Then
for all and all . So,
for all and all . Thus, .
On the other hand, by (4)
for all and all . So,
for all and all . Since for all and , we get
for all and all . Thus, for all . This completes the proof. □
Theorem 3.2 Let be a function such that there exists an with
for all . Let be a mapping satisfying , (2)-(4). Then the limit exists for each and defines a fuzzy ∗-homomorphism such that
for all and all .
Proof Let be a generalized metric space defined as in the proof of Theorem 3.1. Consider the linear mapping such that for all . Let be such that . Then for all and . Hence,
for all and . Thus, implies that . This means that for all . It follows from (6) that
for all and . So, . By Theorem 2.1, there exists a mapping satisfying the following:
-
(1)
A is a fixed point of J, that is,
(12)
for all . The mapping H is a unique fixed point of J in the set . This implies that H is a unique mapping satisfying (12) such that there exists satisfying for all and .
-
(2)
as . This implies the equality
for all .
-
(3)
with , which implies the inequality
This implies that the inequality (10) holds.
The rest of the proof is similar to the proof of Theorem 3.1. □
From now on, we assume that X has a unit e and a unitary group .
Theorem 3.3 Let be a function such that there exists an with
for all . Let be a mapping satisfying , (2) and
for all and all . Then there exists a fuzzy ∗-homomorphism satisfying (5).
Proof By the same reasoning as in the proof of Theorem 3.1, there is a ℂ-linear mapping satisfying (5). The mapping is given by
for all . By (13)
for all and all . Then
as for all and all . So,
for all and all . Thus,
Since H is ℂ-linear and each is a finite linear combination of unitary elements, i.e.,
it follows from (15) that
for all . So, . Similarly, one can obtain that for all . Thus by induction, one can easily show that . By (4)
for all and all . So,
for all and all . Since for all and all , we get
for all and all . Thus,
for all . Since H is ℂ-linear and each is a finite linear combination of unitary elements, i.e., (, ), it follows from (16) that
for all . So, for all . Therefore, the mapping is a ∗-homomorphism. □
Similarly, we have the following. We will omit the proof.
Theorem 3.4 Let be a function such that there exists an with
for all . Let be a mapping satisfying , (2), (13) and (14). Then the limit exists for each and defines a fuzzy ∗-homomorphism such that
for all and all .
4 Approximate derivations on fuzzy Banach ∗-algebras
In this section, we assume that is a fuzzy Banach ∗-algebra. Using fixed point method, we prove the Hyers-Ulam stability of derivations on fuzzy Banach ∗-algebras related to functional equation (1).
Theorem 4.1 Let be a function such that there exists an with
for all . Let be a mapping satisfying ,
for all and all . Then exists for all and defines a fuzzy ∗-derivation such that
for all and all .
Proof The proof is similar to the proof of Theorem 3.1. □
Theorem 4.2 Let be a function such that there exists an with
for all . Let be a mapping satisfying , (17), (18) and (19). Then the limit exists for all and defines a fuzzy ∗-derivation such that
for all and all .
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Azadi Kenary, H., Ghirati, M., Park, C. et al. Homomorphisms and derivations in induced fuzzy -algebras. J Inequal Appl 2013, 88 (2013). https://doi.org/10.1186/1029-242X-2013-88
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DOI: https://doi.org/10.1186/1029-242X-2013-88