Research Article | Open Access

Jaiok Roh, Ick-Soon Chang, "On the Intuitionistic Fuzzy Stability of Ring Homomorphism and Ring Derivation", *Abstract and Applied Analysis*, vol. 2013, Article ID 192845, 8 pages, 2013. https://doi.org/10.1155/2013/192845

# On the Intuitionistic Fuzzy Stability of Ring Homomorphism and Ring Derivation

**Academic Editor:**Bing Xu

#### Abstract

We take into account the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated with the Jensen functional equation. In addition, we deal with the superstability of functional equation in an intuitionistic fuzzy normed algebra with unit.

#### 1. Introduction and Preliminaries

The study of stability problems has originally been formulated by Ulam [1]: *under what condition does there exists a homomorphism near an approximate homomorphism?* Hyers [2] had answered affirmatively the question of Ulam for Banach spaces. The theorem of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper work of Rassias [4] has had a lot of influence in the development of what is called the generalized Hyers-Ulam stability of functional equations. Thereafter, many interesting results of the generalized Hyers-Ulam stability to a number of functional equations have been investigated. In particular, Badora [5] gave a generalization of Bourgin's result [6], and he also dealt with the stability and the Bourgin-type superstability of derivations in [7]. Recently, fuzzy version is discussed in [8, 9]. Quite recently, the stability results in the setting of intuitionistic fuzzy normed space were studied in [10â€“13]; respectively, while the idea of intuitionistic fuzzy normed space was introduced in [14].

We now demonstrate some notations and basic definitions used in this work.

*Definition 1. *A binary operation is said to be a * continuous t-norm* if it satisfies the following conditions:(1) is associative and commutative;(2) is continuous;(3) for all ;(4) whenever and for each .

*Definition 2. *A binary operation is said to be a * continuous t-conorm* if it satisfies the following conditions:(1) is associative and commutative;(2) is continuous;(3) for all ;(4) whenever and for each .

Using the notions of continuous *t*-norm and *t*-conorm, Saadati and Park [14] have recently introduced the concept of intuitionistic fuzzy normed space as follows.

*Definition 3. *The five-tuple is said to be an * intuitionistic fuzzy normed space* if is a vector space, is a continuous *t*-norm, is a continuous *t*-conorm, and are fuzzy sets on satisfying the following conditions. For every and ,(1);
(2);
(3) if and only if ;(4) for each ;(5);
(6) is continuous;(7) and ;(8);
(9) if and only if ;(10) for each ;(11);
(12) is continuous;(13) and .In this case, is called an *intuitionistic fuzzy norm*.

*Example 4. *Let be a normed space, let , and let for all . For all and every , consider
Then is an intuitionistic fuzzy normed space.

*Definition 5 (see [15]). *The five-tuple is said to be an *intuitionistic fuzzy normed algebra* if is an algebra, is a continuous *t*-norm, is a continuous *t*-conorm, and are fuzzy sets on satisfying the conditions (1)â€“(13) of Definition 3. Furthermore, for every and , (14) , (15) .

For an intuitionistic fuzzy normed algebra , we further assume that (16) and for all .

The concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space are studied in [14]. Let be an intuitionistic fuzzy normed space or intuitionistic fuzzy normed algebra. A sequence is said to be *intuitionistic fuzzy convergent* to if and for all . In this case, we write or as . A sequence in is said to be *intuitionistic fuzzy Cauchy sequence* if and for all and . An intuitionistic fuzzy normed space or intuitionistic fuzzy normed algebra is said to be *complete* if every intuitionistic fuzzy Cauchy sequence in is intuitionistic fuzzy convergent in . A complete intuitionistic fuzzy normed space (resp., intuitionistic fuzzy normed algebra) is also called an *intuitionistic fuzzy Banach space* (resp., *intuitionistic fuzzy Banach algebra*).

In this work, we establish the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen functional equation . Moreover, we consider the superstability of functional equation in intuitionistic fuzzy normed algebra with unit.

#### 2. The Main Results

*Remark 6. *The following theorem introduced in [10] is the generalized Hyers-Ulam theorem in intuitionistic fuzzy normed space for the Jensen functional equation. However, in order to arrive at the conclusion of the theorem, the assumptions
should be added.

Theorem 7. *Let be a vector space and a mapping from to an intuitionistic fuzzy Banach space with . Suppose that is a function from to an intuitionistic fuzzy normed space such that
**
for all , , and . If for some real number with , then there exists a unique additive mapping such that ,
**
where
*

We begin with a generalized Hyers-Ulam theorem in intuitionistic fuzzy Banach algebra for the ring homomorphism.

Theorem 8. *Let be an algebra and a mapping from to an intuitionistic fuzzy Banach algebra with . Suppose that is a function from to an intuitionistic fuzzy normed algebra satisfying (3) and that is a function from to an intuitionistic fuzzy normed space such that
**
for all , , and . If for some real number with and for some real number with , then there exists a unique ring homomorphism satisfying (4).*

*Proof. *It follows by Theorem 7 that there exists a unique additive mapping satisfying (4), where .

Without loss of generality, we suppose that . We prove that is a ring homomorphism. Note that
for all and . On the other hand, we see that
for all and and
for all and . Letting in (7), (8), and (9), we get
This implies that
for all .

Using additivity of and (11), we find that
So we obtain ; that is,
for all and . This relation yields that
for all and . On the other hand, we see that
for all and . Sending in (14) and (15), we have
Thus, we conclude that
for all .

Therefore, by combining (11) and (17), we get , which completes the proof.

Now we recall that an additive mapping on an algebra is said to be a *ring derivation* if the functional equation holds for all .

Theorem 9. *Let be an intuitionistic fuzzy Banach algebra and a mapping with . Assume that is a function from to an intuitionistic fuzzy normed space satisfying (3) and that is a function from to an intuitionistic fuzzy normed space such that
**
for all , , and . If for some real number with and for some real number with , then there exists a unique ring derivation satisfying (4). Moreover,
**
is fulfilled for all .*

*Proof. *By Theorem 7, there exists a unique additive mapping satisfying (4), where .

As in the proof of Theorem 8, we consider . We show that is a ring derivation. Observe that
for all and . On the other hand, we yield that
for all and and
for all and . Letting in (20), (21), and (22), we have
So we get
for all .

Due to additivity of and (24), we see that
So we obtain ; that is,
for all and . From this, it follows that
for all and . On the other hand, we find that
for all and . Taking in (27) and (28), we obtain that
Therefore,
for all , which implies that condition (19) holds.

Comparing (24) and (30), we have . This completes the proof.

By considering the unit, the following result can be obtained easily from Theorem 9.

Corollary 10. *Let be an intuitionistic fuzzy Banach algebra with unit, and let be a mapping with . Assume that is a function from to an intuitionistic fuzzy normed space satisfying (3), and that is a function from to an intuitionistic fuzzy normed space satisfying (18). If for some real number with and for some real number with , then is a ring derivation.*

We can also prove the preceding results for the case when and . In this case, the mapping . In the case of intuitionistic fuzzy normed algebras with unit, we can also prove the superstability of functional equation as follows.

Theorem 11. *Let be an intuitionistic fuzzy normed algebra with unit. Suppose that is a mapping for which there exists a function from to an intuitionistic fuzzy normed space satisfying (18). If is a fixed integer and for some real number with , then satisfies the functional equation for all .*

*Proof. *Without loss of generality, we assume that . Here, we will denote the unit by . We first note that
for all and , which mean that
for all . Secondly, we find that
for all and . This implies that
for all .

We also know that
for all and . On the other hand, we see that
for all and and
for all and . In particular, we obtain the following:
for all and . Also, we arrive at

Letting in (35)â€“(39) with (32) and (34), we get the desired result. This completes the proof.

We remark that we can also verify Theorem 11 for the case when .

#### Acknowledgments

The authors would like to thank the referees for giving useful suggestions and for the improvement of this paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2010-002338 and no. 2013R1A1A2A10004419). This research was also supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (no. 2012R1A1A2021721).

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#### Copyright

Copyright © 2013 Jaiok Roh and Ick-Soon Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.