Research Article | Open Access

# Stability of Various Functional Equations in Non-Archimedean Intuitionistic Fuzzy Normed Spaces

**Academic Editor:**Seenith Sivasundaram

#### Abstract

We define and study the concept of non-Archimedean intuitionistic fuzzy normed space by using the idea of *t*-norm and *t*-conorm. Furthermore, by using the non-Archimedean intuitionistic fuzzy normed space, we investigate the stability of various functional equations. That is, we determine some stability results concerning the Cauchy, Jensen and its Pexiderized functional equations in the framework of non-Archimedean IFN spaces.

#### 1. Introduction

The study of stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms.

Let be a group and let be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all ?

If the answer is affirmative, we would say that the equation of homomorphism is stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has significantly influenced the development of what we now call the Hyers-Ulam-Rassias stability of functional equations. Since then several stability problems for various functional equations have been investigated in [5–20]. Quite recently, the stability problem for Pexiderized quadratic functional equation, Jensen functional equation, cubic functional equation, functional equations associated with inner product spaces, and additive functional equation was considered in [21–26], respectively, in the intuitionistic fuzzy normed spaces; while the idea of intuitionistic fuzzy normed space was introduced in [27] and further studied in [28–34] to deal with some summability problems. Quite recently, Alotaibi and Mohiuddine [35] established the stability of a cubic functional equation in random 2-normed spaces, while the notion of random 2-normed spaces was introduced by Goleţ [36] and further studied in [37–39].

By modifying the definition of intuitionistic fuzzy normed space [27], in this paper, we introduce the notion of non-Archimedean intuitionistic fuzzy normed space and also establish Hyers-Ulam-Rassias-type stability results concerning the Cauchy, Pexiderized Cauchy, Jensen, and Pexiderized Jension functional equations in this new setup. This work indeed presents a relationship between four various disciplines: the theory of fuzzy spaces, the theory of non-Archimedean spaces, the theory of Hyers-Ulam-Rassias stability, and the theory of functional equations.

#### 2. Non-Archimedean Intuitionistic Fuzzy Normed Space

In this section, we introduce the concept of non-Archimedean intuitionistic fuzzy normed space and further define the notions of convergence and Cauchy sequences in this new framework. We will assume throughout this paper that the symbols , and will denote the set of all natural, real, complex, and rational numbers, respectively.

A *valuation* is a map from a field into such that is the unique element having the valuation, , and the triangle inequality holds, that is, , for all . We say that a field is *valued* if carries a valuation. The usual absolute values of and are examples of valuations.

Let us consider a valuation which satisfies stronger condition than the triangle inequality. If the triangle inequality is replaced by , for all then, a map is called *non-Archimedean* or *ultrametric valuation*, and field is called a *non-Archimedean field*. Clearly and , for all . A trivial example of a non-Archimedean valuation is the map taking everything but into and .

Let be a vector space over a field with a non-Archimedean valuation . A non-Archimedean normed space is a pair , where is such that (i) if and only if , (ii) for , and (iii)the strong triangle inequality, , for .

In [40], Hensel discovered the -adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean spaces is -adic numbers.

*Example 2.1. *Let be a prime number. For any nonzero rational number such that and are coprime to the prime number , define the -adic absolute value . Then is a non-Archimedean norm on . The completion of with respect to is denoted by and is called the -adic number field.

A binary operation is said to be a *continuous t*-*norm* if it satisfies the following conditions.

(a) is associative and commutative, (b) is continuous, (c) for all , and (d) whenever and for each .

A binary operation is said to be a *continuous t*-*conorm* if it satisfies the following conditions.

(a') is associative and commutative, (b') is continuous, (c') for all , and (d') whenever and for each .

*Definition 2.2. *The five-tuple is said to be an *non-Archimedean intuitionistic fuzzy normed space* (for short, non-Archimedean IFN space) if is a vector space over a non-Archimedean field , is a *continuous t-norm*, is a *continuous t-conorm*, and are functions from to satisfying the following conditions. For every and (i) , (ii) , (iii) if and only if , (iv) for each , (v) , (vi) is continuous, (vii) and , (viii) , (ix) if and only if , for each , (xi) , (xii) is continuous, and (xiii) and .

In this case is called a non-Archimedean intuitionistic fuzzy norm.

*Example 2.3. *Let be a non-Archimedean normed space, and for all . For all , every and , consider the following:
Then is a non-Archimedean intuitionistic fuzzy normed space.

*Definition 2.4. *Let be a non-Archimedean intuitionistic fuzzy normed space. Then, a sequence is said to be (i)*convergent* in or simply -*convergent* to if for every and , there exists such that and for all . In this case we write - and is called the -limit of . (ii)Cauchy in or simply -*Cauchy* if for every and , there exists such that and for all . A non-Archimedean IFN-space is said to be *complete* if every -Cauchy is -*convergent*. In this case is called *non-Archimedean intuitionistic fuzzy Banach space*.

#### 3. Stability of Cauchy Functional Equation

In this section, we determine stability result concerning the Cauchy functional equation in non-Archimedean intuitionistic fuzzy normed space.

Theorem 3.1. *Let be a linear space over a non-Archimedean field and let be a non-Archimedean IFN space. Suppose that is a function such that for some and some positive integer with **
for all and . Let be a non-Archimedean intuitionistic fuzzy Banach space over and let be a -approximately Cauchy mapping in the sense that
**
for all and . Then there exists a unique additive mapping such that
**
for all and , where
*

*Proof. *By induction on we will show that for each and
Putting in (3.2), we obtain
for all and . This proves (3.5) for . Let (3.5) hold for some . Replacing by in (3.2), we get
for each and . Thus
for each and . Hence (3.5) holds for all . In particular
Replacing by in (3.9) and using (3.1), we get
for all , and . Therefore
for all , and . Since
so (3.11) shows that is a Cauchy sequence in non-Archimedean intuitionistic fuzzy Banach space . Therefore, we can define a mapping by . Hence
For each , and
where and . It follows from (3.13) and (3.14) that
for each and for sufficiently large ; that is, (3.3) holds. Also, from (3.1), (3.2), and (3.13), we have
for all , and for large . Since
which shows that is additive. Now if is another additive mapping such that
for all and . Then, for all and , we have
Therefore
Hence for all .

Corollary 3.2. *Let be a linear space over non-Archimedean field and let be a non-Archimedean normed space. Suppose that a function satisfies
**
for all , where and is an integer with . If a map satisfies
**
for all , then there exists a unique additive mapping satisfies
*

*Proof. *Consider the non-Archimedean intuitionistic fuzzy norm
on . Let and let the function be defined by
Then is a non-Archimedean intuitionistic fuzzy norm on . The result follows from the fact that (3.21), (3.22), and (3.23) are equivalent to (3.1), (3.2), and (3.3), respectively.

*Example 3.3. *Let be a linear space over non-Archimedean field and let be a non-Archimedean normed space. Suppose that a function satisfies
for all and . Suppose that there exists an integer such that . Since , by applying Corollary 3.2 for , we observe that (3.21) holds for . Inequality (3.23) assures the existence of a unique additive mapping such that
for all .

#### 4. Stability of Pexiderized Cauchy Functional Equation

The functional equation is said to be Pexiderized Cauchy, where , , and are mappings between linear spaces. In the case , it is called Cauchy functional equation.

Theorem 4.1. *Let be a linear space over a non-Archimedean field and let be a non-Archimedean intuitionistic fuzzy Banach space. Suppose that , , and are mappings from to with . Suppose that is a function from to a non-Archimedean IFN space such that
**
for all and . If
**
for some positive real number and some positive integer with , then there exists a unique additive mapping such that
**
for all and , where
*

*Proof. *Put in (4.1). Then, for all and
For , (4.1) becomes
for all and . Combining (4.1), (4.7), and (4.8), we obtain
for each and . Replacing and by and , respectively, in Theorem 3.1, we can find that there exists a unique additive mapping that satisfies (4.3). From (4.3) and (4.7), we see that
for all and , which proves (4.4). Similarly, we can prove (4.5).

Corollary 4.2. *Let be a linear space over a non-Archimedean field and let be a non-Archimedean IFN space. Let be a non-Archimedean intuitionistic fuzzy Banach space. Suppose that , and are functions from to such that , and there is an integer with and satisfies
**
for all and for some fixed and with . Then there exists a unique additive mapping such that
**
for all and .*

*Proof. *Let the function be defined by for all and is a fixed unit vector in . Then (4.1) holds. Since
for each and . If and , then holds. It follows from Theorem 4.1 that there exists a unique additive mapping such that (4.3)–(4.5) hold.

#### 5. Stability of Jensen Functional Equation

The stability problem for the Jensen functional equation was first proved by Kominek [13] and since then several generalizations and applications of this notion have been investigated by various authors, namely, Jung [12], Mohiuddine [23], Parnami and Vasudeva [41], and many others. The Jensen functional equation is , where is a mapping between linear spaces. It is easy to see that a mapping between linear spaces with satisfies the Jensen equation if and only if it is additive (cf. [41]).

Theorem 5.1. *Let be a linear space over a non-Archimedean field and let be a non-Archimedean IFN space. Suppose that is a function such that for some and some positive integer with satisfies (3.1). Suppose that is a non-Archimedean intuitionistic fuzzy Banach space. If a map satisfies
**
for all and , then there exists a unique additive mapping such that
**
for all and , where
*

*Proof. *Suppose that for all . Then
for all and . Replacing by and by in (5.5), then, for all and , we have
From (5.5) and (5.6), we conclude that
for all and . Proceeding the same lines as in the proof of Theorem 3.1, one can show that there exists a unique additive mapping such that
for all and .

#### 6. Stability of Pexiderized Jensen Functional Equation

The functional equation is said to be Pexiderized Jensen, where , , and are mappings between linear spaces. In the case , it is called Jensen functional equation.

Theorem 6.1. *Let be a linear space over a non-Archimedean field and let be a non-Archimedean intuitionistic fuzzy Banach space. Suppose that , , and are mappings from to with . Let be non-Archimedean IFN space. Suppose that is a function such that for some , and some positive integer with satisfies (3.1) and inequality
**
for all and . Then there exists a unique additive mapping such that
**
for all and , where
*

*Proof. *Put in (6.1). Then, for all and
Replacing by in (6.1), we get
for all and . Again replacing by as well as by in (6.1), we get
for all and . It follows from (6.1) and (6.6)–(6.8) that
Thus, for all and ,
Proceeding the same argument used in Theorem 5.1 shows that there exists a unique additive mapping such that (6.2) holds. Therefore
for all and . Put in (6.1), we get
for all and . It follows from (6.11) and (6.12) that (6.3) holds. Similarly we can show that (6.4) holds.

Corollary 6.2. *Let be a non-Archimedean normed space. Suppose that such that , and there is an integer with and satisfies
**
for all . Then there exists a unique additive mapping such that
**
for all .*

* Proof. *Let the function be defined by
on . It is easy to see that is a non-Archimedean intuitionistic fuzzy Banach space. Consider the non-Archimedean intuitionistic fuzzy norm
Then is a non-Archimedean intuitionistic fuzzy norm on . It is easy to see that (4.1) holds for and satisfies (3.1). Therefore the condition of Theorem 6.1 is fulfilled. Hence there exists a unique additive mapping such that (6.14) holds.

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