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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 192845, 8 pages
http://dx.doi.org/10.1155/2013/192845
Research Article

On the Intuitionistic Fuzzy Stability of Ring Homomorphism and Ring Derivation

1Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea
2Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Republic of Korea

Received 7 June 2013; Accepted 25 July 2013

Academic Editor: Bing Xu

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.

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 [1013]; 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)–(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)–(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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