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# Fuzzy Stability of Quadratic Functional Equations

*Advances in Difference Equations*
**volume 2010**, Article number: 412160 (2010)

## Abstract

The fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. In this paper, we prove the generalized Hyers-Ulam stability of the following quadratic functional equations and in fuzzy Banach spaces.

## 1. Introduction and Preliminaries

Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [2–4]. In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [7]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [8].

We use the definition of fuzzy normed spaces given in [5, 9, 10] to investigate a fuzzy version of the generalized Hyers-Ulam stability for the quadratic functional equations

in the fuzzy normed vector space setting, where are nonzero real numbers with .

Definition 1.1 (see [5, 9, 10]).

Let be a real vector space. A function is called a *fuzzy norm* on if, for all and all ,

for ,

if and only if for all ,

if ,

,

is a nondecreasing function of and ,

for , is continuous on .

The pair is called a *fuzzy normed vector space*.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [9, 10].

Definition 1.2 (see [5, 9, 10]).

Let be a fuzzy normed vector space. A sequence in is said to *be convergent* or *converges* if there exists an such that for all . In this case, is called the *limit* of the sequence and we denote it by -.

Definition 1.3 (see [5, 9, 10]).

Let be a fuzzy normed vector space. A sequence in is called *Cauchy* if for each and each there exists an such that, for all and all , we have .

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be *complete* and the fuzzy normed vector space is called a *fuzzy Banach space*.

We say that a mapping between fuzzy normed vector spaces and is continuous at a point if, for each sequence converging to in , the sequence converges to . If is continuous at each , then is said to be *continuous* on (see [8]).

The stability problem of functional equations is originated from a question of Ulam [11] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [13] for additive mappings and by Th. M. Rassias [14] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [14] has provided a lot of influence in the development of what we call *generalized Hyers-Ulam stability* of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [15] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

A square norm on an inner product space satisfies the parallelogram equality

The functional equation

is called a *quadratic functional equation*. In particular, every solution of the quadratic functional equation is said to be a *quadratic mapping*. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [16] for mappings , where is a normed space and is a Banach space. Cholewa [17] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. In [18], Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [19–31]).

This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces. In Section 3, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.

Throughout this paper, assume that is a vector space and that is a fuzzy Banach space. Let be nonzero real numbers with .

## 2. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.1)

In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.1) in fuzzy Banach spaces.

Theorem 2.1.

Let be a function such that

for all . Let be a mapping with such that

uniformly on . Then - exists for each and defines a quadratic mapping such that if for some

for all , then

for all .

Furthermore, the quadratic mapping is a unique mapping such that

uniformly on .

Proof.

For a given , by (2.2), we can find some such that

for all . By induction on , we show that

for all , all and all .

Letting in (2.6), we get

for all and all . So we get (2.7) for .

Assume that (2.7) holds for . Then

This completes the induction argument. Letting and replacing and by and in (2.7), respectively, we get

for all integers .

It follows from (2.1) and the equality

that for a given there is an such that

for all and . Now we deduce from (2.10) that

for all and all . Thus the sequence is Cauchy in . Since is a fuzzy Banach space, the sequence converges to some . So we can define a mapping by -; namely, for each and , .

Let . Fix and . Since , there is an such that for all . Hence for all , we have

The first four terms on the right-hand side of the above inequality tend to 1 as , and the fifth term is greater than

which is greater than or equal to . Thus

for all . Since for all , by , for all . Thus the mapping is quadratic, that is, for all .

Now let, for some positive and , (2.3) hold. Let

for all . Let . By the same reasoning as in the beginning of the proof, one can deduce from (2.3) that

for all positive integers . Let . We have

Combining (2.18) and (2.19) and the fact that , we observe that

for large enough . Thanks to the continuity of the function , we see that . Letting , we conclude that

To end the proof, it remains to prove the uniqueness assertion. Let be another quadratic mapping satisfying (2.5). Fix . Given that , by (2.5) for and , we can find some such that

for all and all . Fix some and find some integer such that

for all . Since

we have

It follows that for all . Thus for all .

Corollary 2.2.

Let and let be a real number with . Let be a mapping with such that

uniformly on . Then - exists for each and defines a quadratic mapping such that if for some

for all , then

for all .

Furthermore, the quadratic mapping is a unique mapping such that

uniformly on .

Proof.

Define and apply Theorem 2.1 to get the result.

Similarly, we can obtain the following. We will omit the proof.

Theorem 2.3.

Let be a function such that

for all . Let be a mapping satisfying (2.2) and . Then - exists for each and defines a quadratic mapping such that if for some

for all , then

for all .

Furthermore, the quadratic mapping is a unique mapping such that

uniformly on .

Corollary 2.4.

Let and let be a real number with . Let be a mapping satisfying (2.26) and . Then - exists for each and defines a quadratic mapping such that if for some

for all , then

for all .

Furthermore, the quadratic mapping is a unique mapping such that

uniformly on .

Proof.

Define and apply Theorem 2.3 to get the result.

## 3. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.2)

In this section, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.2) in fuzzy Banach spaces.

Lemma 3.1.

Let and be real vector spaces. If a mapping satisfies and

for all , then the mapping is quadratic, that is,

for all .

Proof.

Assume that satisfies (3.1).

Letting in (3.1), we get

for all .

Letting in (3.1), we get

for all . Replacing by in (3.4), we get

for all . It follows from (3.4) and (3.5) that for all . So

for all . Thus

for all . Replacing and by and in (3.7), respectively, we get

for all , as desired.

Theorem 3.2.

Let be a function such that

for all . Let be a mapping with such that

uniformly on . Then - exists for each and defines a quadratic mapping such that if for some

for all , then

for all .

Furthermore, the quadratic mapping is a unique mapping such that

uniformly on .

Proof.

For a given , by (3.10), we can find some such that

for all . By induction on , we show that

for all , all , and all .

Letting in (3.14), we get

for all and all . So we get (3.15) for .

Assume that (3.15) holds for . Then

This completes the induction argument. Letting and replacing and by and in (3.15), respectively, we get

for all integers .

It follows from (3.9) and the equality

that for a given there is an such that

for all and . Now we deduce from (3.18) that

for each and all . Thus the sequence is Cauchy in . Since is a fuzzy Banach space, the sequence converges to some . So we can define a mapping by -; namely, for each and , .

Let . Fix and . Since , there is an such that for all . Hence for each , we have

The first four terms on the right-hand side of the above inequality tend to 1 as , and the fifth term is greater than

which is greater than or equal to . Thus

for all . Since for all , by , for all . By Lemma 3.1, the mapping is quadratic.

Now let, for some positive and , (3.18) hold. Let

for all . Let . By the same reasoning as in the beginning of the proof, one can deduce from (3.18) that

for all positive integers . Let . We have

Combining (3.26) and (3.27) and the fact that , we observe that

for large enough . Thanks to the continuity of the function , we see that . Letting , we conclude that

To end the proof, it remains to prove the uniqueness assertion. Let be another quadratic mapping satisfying (3.1) and (3.13). Fix . Given that , by (3.13) for and , we can find some such that

for all and all . Fix some and find some integer such that

for all . Since

we have

It follows that for all . Thus for all .

Corollary 3.3.

Let and let be a real number with if and with if . Let be a mapping with such that

uniformly on . Then - exists for each and defines a quadratic mapping such that if for some

for all , then

for all .

Furthermore, the quadratic mapping is a unique mapping such that

uniformly on .

Proof.

Define and apply Theorem 3.2 to get the result.

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### Keywords

- Functional Equation
- Quadratic Mapping
- Unique Mapping
- Quadratic Functional Equation
- Fuzzy Normed Space