Abstract and Applied Analysis

Volume 2012 (2012), Article ID 503671, 16 pages

doi:10.1155/2012/503671

Research Article

Higher Ring Derivation and Intuitionistic Fuzzy Stability

Ick-Soon Chang

Department of Mathematics, Mokwon University, Daejeon 302-729, Republic of Korea

Received 3 May 2012; Accepted 12 June 2012

Academic Editor: Bing Xu

Copyright © 2012 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 account of the stability of higher ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen type functional equation. In addition, we deal with the superstability of higher ring derivation in intuitionistic fuzzy Banach algebra with unit.

1. Introduction and Preliminaries

The stability problem of functional equations has originally been formulated by Ulam [1]: under what condition does there exist a homomorphism near an approximate homomorphism? Hyers [2] answered the problem of Ulam under the assumption that the groups are Banach spaces. A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [3] and for approximately linear mappings was presented by Rassias [4] 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. Since then, more generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings have been investigated (e.g., [5–7]). In particular, Badora [8] gave a generalization of the Bourgin's result [9], and he also dealt with the stability and the Bourgin-type superstability of derivations in [10]. Recently, fuzzy version is discussed in [11, 12]. Quite recently, the intuitionistic fuzzy stability problem for Jensen functional equation and cubic functional equation is considered in [13–15], respectively, while the idea of intuitionistic fuzzy normed space was introduced in [16], and there are some recent and important results which are directly related to the central theme of this paper, that is, intuitionistic fuzziness (see e.g., [17–20]).

In this paper, we establish the stability of higher ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen type functional equation . Moreover, we consider the superstability of higher ring derivation in intuitionistic fuzzy Banach algebra with unit.

We now recall some notations and basic definitions used in this paper.

Definition 1.1 (see [5]). Let and be algebras over the real or complex field . Let be the set of the natural numbers. From , a sequence (resp., ) of additive operators from into is called a higher ring derivation of rank (resp., infinite rank) if the functional equation holds for each (resp., ) and for all . A higher ring derivation of additive operators on , particularly, is called strong if is an identity operator.

Of course, a higher ring derivation of rank 0 from into (resp., a strong higher ring derivation of rank 1 on ) is a ring homomorphism (resp., a ring derivation). Note that a higher ring derivation is a generalization of both a ring homomorphism and a ring derivation.

Definition 1.2. 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 whenever and for each .

Definition 1.3. 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 whenever and for each .

Using the notions of continuous t-norm and t-conorm, Saadati and Park [16] have recently introduced the concept of intuitionistic fuzzy normed space as follows.

Definition 1.4. 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 1.5. Let be a normed space, , and for all . For all and every , consider

Then is an intuitionistic fuzzy normed space.

Example 1.6. Let be a normed space, , and for all . For all and every and , consider

Then is an intuitionistic fuzzy normed space.

Definition 1.7 (see [21]). 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 the Definition 1.4. 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 [16]. 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 (resp., 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).

2. Stability of Higher Ring Derivation in Intuitionistic Fuzzy Banach Algebra

As a matter of convenience in this paper, we use the following abbreviation:

In addition,

We begin with a generalized Hyers-Ulam theorem in intuitionistic fuzzy Banach space for the Jensen type functional equation. The following result is also the generalization of the theorem introduced in [13].

Theorem 2.1. Let be a vector space, and let be 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 is a fixed integer, and for some real number with , then there exists a unique additive mapping such that ,

for all and , where

Proof. Without loss of generality, we assume that . From (2.3) and (2.4), we get

for all and . Again, by (2.3) and (2.4), we obtain

for all and . Combining (2.7) and (2.8), we arrive at

for all and . This implies that

for all and . Now we define

for all and . Then we have by assumption

for all and . Using (2.10) and (2.12), we get

for all and . Therefore, for all , we have

for all and . Let and be given. Since and , there exists some such that . Since , there exists a positive integer such that for all .

Then

This shows that is a Cauchy sequence in . Since is complete, we can define a mapping by for all . Moreover, if we let in (2.14), then we get

for all and . Therefore, we find that

Next, we will show that is additive mapping. Note that

On the other hand, (2.3) and (2.4) give the following:

Letting in (2.18) and (2.19), we yield

So we see that is additive mapping.

Now, we approximate the difference between and in an intuitionistic fuzzy sense. By (2.17), we get

for all and and sufficiently large .

In order to prove the uniqueness of , we assume that is another additive mapping from to , which satisfies the inequality (2.5). Then

for all and . Therefore, due to the additivity of and , we obtain that

Since , and we get

that is, and for all . So , which completes the proof.

In particular, we can prove the preceding result for the case when . In this case, the mapping . We now establish a generalized Hyers-Ulam stability in intuitionistic fuzzy Banach algebra for the higher ring derivation.

Theorem 2.2. Let be an algebra, and let be a sequence of mappings from to an intuitionistic fuzzy Banach algebra with for each . Suppose that is a function from to an intuitionistic fuzzy normed algebra such that for each ,

for all and , and that is a function from to an intuitionistic fuzzy normed space such that for each ,

for all , and . If is a fixed integer, , and for some real numbers and with and , then there exists a unique higher ring derivation of any rank such that for each ,

for all and . In this case,

Moreover, the identity

holds for each and all .

Proof. It follows by Theorem 2.1 that for each and all , there exists a unique additive mapping given by

satisfying (2.27) since is an intuitionistic fuzzy normed algebra.

Without loss of generality, we suppose that . Now, we need to prove that the sequence satisfies the identity for each and all . It is observed that for each ,

for all and . On account of (2.26), we see that for each ,

for all and . Due to additivity of , for each ,

for all and . In addition, we feel that

Letting in (2.31), (2.32), (2.33), and (2.34), we get and . This implies that

for each and all .

Using additivity of and (2.35), we find that

So we obtain . Hence for each ,

for all and . This relation yields that for each ,

for all and . On the other hand, we see that

Sending in (2.38) and (2.40), we have that for each ,

for all and . Thus, we conclude that

for each and all .

Therefore, by combining (2.35) and (2.42), we get the required result, which completes the proof.

As a consequence of Theorem 2.2, we get the following superstability.

Corollary 2.3. Let be an intuitionistic fuzzy Banach algebra with unit, and let a sequence of operators on satisfy for each , where is an identity operator. Suppose that is a function from to an intuitionistic fuzzy normed algebra satisfying (2.25) and (2.14) and that is a function from to an intuitionistic fuzzy normed space satisfying (2.26). If is a fixed integer, , and for some real numbers and with and , then is a strong higher ring derivation on .

Proof. According to (2.30), we have for all , and so (=) is an identity operator on . By induction, we get the conclusion. If , then it follows from (2.29) that holds for all since contains the unit element. Let us assume that is valid for all and . Then (2.29) implies that for all . Since has the unit element, for all . Hence we conclude that for each and all . So this tells us that is a higher ring derivation of any rank from and . The proof of the corollary is complete.

We remark that we can prove the preceding result for the case when and .

Acknowledgments

The authors would like to thank the referees for giving useful suggestions and for the improvement of this paper. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2012-0002410).

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