Ulam Stability and Non-Stability of Additive Functional Equation in IFN-Spaces and 2-Banach Spaces by Different Methods

Abstract

This paper introduces a new dimension of an additive functional equation and obtains its general solution. The main goal of this study is to examine the Ulam stability of this equation in IFN-spaces (intuitionistic fuzzy normed spaces) with the help of direct and fixed point approaches and 2-Banach spaces. Also, we use an appropriate counterexample to demonstrate that the stability of this equation fails in a particular case.

1. Introduction

The study of stability problems for functional equations is one of the essential research areas in mathematics, which originated in issues related to applied mathematics. The first question concerning the stability of homomorphisms was given by Ulam [1] as follows.

Given a group , a metric group with the metric , and a mapping from and , does exist such thatfor all . If such a mapping exists, then does a homomorphism exist such thatfor all ? Ulam defined such a problem in 1940 and solved it the following year for the Cauchy functional equationby the way of Hyers [2]. The consequence of Hyers becomes stretched out by Aoki [3] with the aid of assuming the unbounded Cauchy contrasts. Hyers theorem for additive mapping was investigated by Rassias [4], and then Rassias results were generalized by Gavruta [5].

As of late, Nakmahachalasint [6] gave the overall answer and HUR (briefly, Hyers–Ulam–Rassias) stability of finite variable functional equation; furthermore, Khodaei and Rassias [7] examined the stability of generalized additive functions in several variables. The stability result of additive functional equations was examined by means of Najati and Moghimi [8], Shin et al. [9], and Gordji [10]. Stability problems of various functional equations have been investigated by many researchers, and there are various interesting results about this problem (see [11–14]).

Zadeh [15] established the concept of fuzzy sets, which is a tool for demonstrating weakness and ambiguity in several scientific and technological problems. The possibility of IFN-spaces, from the start, has been presented in [16]. Saadati [17] have examined the modified intuitionistic fuzzy metric spaces and proven some fixed point theorems in these spaces.

The IFN-spaces and IF2N-spaces (briefly, intuitionistic fuzzy 2-normed spaces) have been studied by a number of researchers [18–20]. Furthermore, several researchers have discussed the generalized Ulam–Hyers stability of various functional equations in IFN-spaces (see [21–24]).

In this current work, we present a new kind of additive functional equation:where is a fixed integer, and obtain its general solution. The main goal of this study is to examine the Ulam–Hyers stability of this equation in IFN-spaces with the help of direct and fixed point approaches and 2-Banach spaces by using the direct approach. Also, we use an appropriate counterexample to demonstrate that the stability of equation (4) fails in a particular case.

2. General Solution

Theorem 1. If a mappingbetween two real vector spacesandsatisfies functional equation (4), then the functionis additive.

Proof. Setting in (4), we have . Replacing by in (4), we get for all . Hence, is an odd function. Replacing by in (4), we havefor all . Replacing by in (5), we havefor all . Again, replacing with in (6), we getfor all . In general, for any non-negative integer , we havefor all . Replacing by in (4), we obtain (3) for all .

Remark 1. If a mapping between two real vector spaces and satisfies functional equation (3), then the function satisfies additive functional equation (4), for all .

For our notational handiness, we define a mapping byfor all .

3. Stability Results in IFN-Spaces

We can recall some basic notions and preliminaries from [25] and using the alternative fixed point theorem which are important results in fixed point theory [26].

Definition 1 (see [25]). Consider a membership degree and non-membership degree of an intuitionistic fuzzy set from to such that for all and . The triple is called as an Intuitionistic Fuzzy Normed-space (briefly, IFN-space) if a vector space , a continuous -representable and satisfying and , (IFN1) . (IFN2) if and only if . (IFN3) , for all . (IFN4) .In this case, is called an intuitionistic fuzzy norm, where .

Definition 2 (see [25]). A sequence in is called as a Cauchy sequence if for every and , there exists such thatfor all .

Remark 2. In an intuitionistic fuzzy normed space, every convergent sequence is a Cauchy sequence.
If every Cauchy sequence is convergent, then the intuitionistic fuzzy normed space is called as complete.

Definition 3 (see [25]). A mapping between two IFN-spaces and is continuous at if for every converging to in , the sequence converges to . If is continuous at each point , then the mapping is called as a continuous mapping on .

Example 1. Letbe a normed space. Letfor all;andbe membership and non-membership degree of an intuitionistic fuzzy set defined byThen, is an IFN-space.

Theorem 2 (see [26]). Letbe a generalized complete metric space and a strictly contractive mappingwith Lipschitz constant. Then, for all, eitheror there exists a positive integer such that(i).(ii)The sequenceconverges to a fixed pointof.(iii)is the unique fixed point ofin.(iv), for all.

3.1. Stability Results: Direct Technique

In this section, we assume that , , and are linear space, IFN-space, and complete IFN-space, respectively.

Theorem 3. If a mappingwith,for all and all . If a mapping satisfiesfor all and all , then the limitexists and there exists a unique additive mapping satisfying functional equation (4) andfor all and all .

Proof. Fix and all . Replacing by in (15), we haveReplacing by in (18) and using (IFN3), we obtainBy the inequality (13) and (IFN3) in (19), we haveClearly, we can show from inequality (20) thatReplacing by in (21), we getClearly,It follows from (22) and (23) thatfor all and . Replacing by in (24) and with the help of (13), we havefor every . Replacing by in (25), we haveUsing (IFN3) in (26), we obtainfor all . Since and , the Cauchy criterion for convergence in IFNS shows that is Cauchy sequence in . Since is a complete, this sequence converges to some point . Then, we can define the mapping bySetting in inequality (29), we obtainTaking the limit as in (29), we obtainNext, we want to prove that the function satisfies functional equation (4); replacing by in (15), we havefor all and all . Sincethe function satisfies functional equation (4). Thus, the function is additive. Finally, we want to prove that the function is unique; consider another additive mapping satisfying functional equations (4) and (17). Hence,for all and all . Aswe obtainThus,
Therefore, . Thus, the additive function is unique. This ends the proof.

Theorem 4. If a mappingwith,for alland all. If a mappingsatisfies (15), then the limit exists and there exists a unique additive mapping satisfying functional equation (4) andfor all and all .

Proof. Fix and all . Replacing by in (15), we haveFrom (39), we obtain thatReplacing by in (40), we getReplacing by in (41) and using (IFN3), we haveWith the help of inequality (36) and (IFN3) in (42), we obtain thatThe remaining part of the proof can be proven in the same way as Theorem 3.

Corollary 1. Let. If a mappingsuch thatfor alland all, then there exists a unique additive mappingsatisfyingfor all and all .

Proof. The proof holds from Theorems 3 and 4 by letting and .

Corollary 2. Letwith. If a mappingsuch thatfor alland all, then there exists a unique additive mappingsatisfyingfor all and all .

Proof. The proof holds from Theorems 3 and 4 by setting and .

Corollary 3. Letwith. If a mappingsuch thatfor alland all, then there exists a unique additive mappingsatisfyingfor all and all .

Proof. The proof holds from Theorems 3 and 4 by setting and .

Corollary 4. Letwith. If a mappingsuch thatfor all and all , then the mapping is additive.

Proof. The proof is valid from Theorems 3 and 4 by setting .

3.2. Stability Results: Fixed Point Technique

Before we begin, let us consider a constant such thatand is the set such that .

Theorem 5. Consider a mappingfor which there is a mappingwithsatisfying functional inequality (15). If there issuch thathas the propertythen there exists a unique additive mapping satisfying functional equation (4) andfor all and all .

Proof. Let be a general metric on :Clearly, is complete. Define a mapping by , for all . For , we haveThus, the function is strictly contractive on with (Lipschitz constant). Replacing by in (15), we haveUsing (IFN3) in (57), we haveUsing equation (53) for the case , we haveReplacing by in (57), we havefor all and all ; using (53) for the case , we haveWe can conclude from equations (59) and (61) thatBy the fixed point alternative in both cases, there is a fixed point of in such thatReplacing by in (15), we obtainfor all and all . By same manner of Theorem 3, we can show that the function satisfies functional equation (4). By Theorem 2, as is a unique fixed point of in , the function is unique such thatUsing fixed point alternative, we reachfor all and all . Hence, the proof of the theorem is now completed.

Corollary 5. Letwith. If a mappingsuch thatfor alland, then there exists a unique additive mappingsatisfyingfor all and all .

Proof. SetThen,Thus, (52) holds. But we have thathas the propertyHence,Now,From inequality (53), we can verify the following cases for conditions of .

Case 1. if .

Case 2. if .

Case 3. for if .

Case 4. for if .

Case 5. for if .

Case 6. for if .

4. Stability Results in 2-Banach Spaces

In 1960, Gahler [27, 28] developed the concept of linear 2-normed spaces.

Definition 4. Consider a linear space over with dimension and consider a mapping with the following conditions:(a) if and only if and are linearly dependent.(b),(c),(d),for all and .Then, the function is called as a 2-norm on and the pair is called as a linear 2-normed space. A typical example of 2-normed space is with 2-norm defined as the area of the triangle with the vertices 0, , and is a typical example of a 2-normed space.

As a result of (d), it follows that

Thus, are continuous mappings of into for any fixed .

Definition 5. A sequence in a linear 2-normed space is known as a Cauchy sequence if there exist two points such that and are linearly independent.

Definition 6. A sequence in a linear 2-normed space is called as a convergent sequence if there exists an element such thatfor all . If converges to , then we denote as and say that is the limit point of . We also write in this instance

Definition 7. A 2-Banach space is a linear 2-normed space in which every Cauchy sequence is convergent.

Lemma 1 (see [29]). Letbe a linear 2-normed space. Ifandfor every, then.

Lemma 2 (see [29]). For a convergent sequencein a linear 2-normed space,for every .

Park studied approximate additive mappings, approximate Jensen mappings, and approximate quadratic mappings in 2-Banach spaces in his paper [29]. In [30], Park examined the superstability of the Cauchy functional inequality and the Cauchy–Jensen functional inequality in 2-Banach spaces under certain conditions.

In this section, we let be a normed linear space and be a 2-Banach space.

Theorem 6. Letbe a function such thatfor all . If a mapping such that and for all . Then, there exists a unique additive mapping satisfyingfor all .

Proof. Replacing by in (87), we getfor all . Replacing by in (90) and dividing both sides by , we havefor all and all non-negative integers . Hence,for all and all non-negative integers and with . Therefore, it follows from (15) and (19) that the sequence is Cauchy in for every . Since is complete, the sequence converges in for all . Thus, we may define a mapping byfor all . Therefore,for all . Letting and taking the limit as in (94), we have (89). Next, we want to prove that the function is additive. From inequalities (86), (87), and (94) and Lemma 2,for all . By Lemma 1,for all . Hence, according to Theorem 1, the mapping is additive.
To prove that the function is unique, we consider another additive mapping satisfying (89). Then,for all . By Lemma 1, for all . Therefore, .

Remark 3. A theorem analogous to (93) can be formulated, in which the sequenceis defined with appropriate assumptions for .

Corollary 6. Letbe a mapping such thatand(i).(ii)for all.

If a mapping with andfor all , then there exists a unique additive mapping satisfyingfor all .

Proof. Letfor all . It follows from (i) thatBy using Theorem 6, we obtain (96).

Corollary 7. Letbe a positive real number such thatand letbe a homogeneous mapping with degree. If a mappingwithandfor all , then there exists a unique additive mapping satisfyingfor all .

Proof. Letfor all . By using Theorem 6, we have (104).

Corollary 8. Letsuch thatand letbe a homogeneous mapping with degree. If a mappingwithandfor all , then there exists a unique additive mapping satisfyingfor all .

Proof. Letfor all . By using Theorem 6, we have (110).

Corollary 9. Letsuch that. If a mappingwithandfor all , then there exists a unique additive mapping satisfyingfor all .

We use an appropriate example to demonstrate that the stability of the functional equation (4) fails in the singular case. We provide the following counterexample, which shows the instability in a particular condition in Corollary 9 of functional equation (4), inspired by Gajda’s excellent example in [31].

Remark 4. If a mapping satisfies (4), then the following assertions hold:(1), and .(2) if the function is continuous.

Example 2. Let a mappingbe defined bywhereThen, the mapping satisfiesfor all , but there does not exist an additive mapping satisfyingwhere and are constants.

5. Conclusion

In this work, a new dimensional additive functional (equation (4)) has been introduced. We primarily found its solution and examined Hyers–Ulam stability in IFN-spaces using the direct approach in Section 3.1 and the fixed point approach in Section 3.2. In Section 4, we investigated the Hyers–Ulam stability in 2-Banach space by using the direct method. Also, we provided the counterexample, which shows the instability in a particular condition in Corollary 9 of equation (4), by the way of Gajda.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this study. All authors have read and approved the final version of the manuscript.