Research Article | Open Access

# Higher Ring Derivation and Intuitionistic Fuzzy Stability

**Academic Editor:**Bing Xu

#### 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., [5â€“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 [13â€“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., [17â€“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 ,â€‰â€‰(1)â€‰â€‰,â€‰â€‰(2)â€‰â€‰,â€‰â€‰(3)â€‰â€‰ if and only if ,â€‰â€‰(4)â€‰â€‰â€‰ for each ,â€‰â€‰(5)â€‰â€‰,â€‰â€‰(6)â€‰â€‰ is continuous, (7) and ,â€‰â€‰(8)â€‰â€‰,â€‰â€‰(9)â€‰â€‰ if and only if ,â€‰â€‰(10)â€‰â€‰ for each ,â€‰â€‰(11)â€‰â€‰,â€‰â€‰(12)â€‰â€‰ 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)â€“(13) of the Definition 1.4. Furthermore, for every and ,â€‰â€‰(14)â€‰â€‰,â€‰â€‰(15)â€‰â€‰.

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 *