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# Approximately Quadratic Mappings on Restricted Domains

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 503458 (2010)

## Abstract

We introduce a generalized quadratic functional equation , where , are nonzero real numbers with . We show that this functional equation is quadratic if , are rational numbers. We also investigate its stability problem on restricted domains. These results are applied to study of an asymptotic behavior of these generalized quadratic mappings.

## 1. Introduction

*Under what conditions does there exist a group homomorphism near an approximate group homomorphism?* This question concerning the stability of group homomorphisms was posed by Ulam [1]. The case of approximately additive mappings was solved by Hyers [2] on Banach spaces. In 1950 Aoki [3] provided a generalization of the Hyers' theorem for additive mappings and in 1978 Th. M. Rassias [4] generalized the Hyers' theorem for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The result of Rassias' theorem has been generalized by Găvruţa [6] who permitted the Cauchy difference to be bounded by a general control function. This stability concept is also applied to the case of other functional equations. For more results on the stability of functional equations, see [7–24]. We also refer the readers to the books in [25–29].

It is easy to see that the quadratic function is a solution of each of the following functional equations:

where , are nonzero real numbers with . So, it is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a function between real vector spaces and is quadratic if and only if there exists a unique symmetric biadditive function such that for all (see [13, 25, 27]).

We prove that the functional equations (1.1) and (1.2) are equivalent if , are nonzero rational numbers. The functional equation (1.1) is a spacial case of (1.2). Indeed, for the case in (1.2), we get (1.1).

In 1983 Skof [30] was the first author to solve the Hyers-Ulam problem for additive mappings on a restricted domain (see also [31–33]). In 1998 Jung [34] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [35–37]). J. M. Rassias [38] investigated the Hyers-Ulam stability of mixed type mappings on restricted domains.

## 2. Solutions of (1.2)

In this section we show that the functional equation (1.2) is equivalent to the quadratic equation (1.1). That is, every solution of (1.2) is a quadratic function. We recall that , are nonzero real numbers with .

Theorem 2.1.

Let and be real vector spaces and be an odd function satisfying (1.2). If is a rational number, then .

Proof.

Since is odd, . Letting (resp., ) in (1.2), we get

for all . Replacing by in (1.2) and adding the obtained functional equation to (1.2), we get

for all . Replacing by in (2.2) and using (2.1), we have

for all . Again if we replace by in (2.3) and use (2.1), we get

for all . Applying (1.2) and using the oddness of , we have

for all , in . So it follows from (2.4) and (2.5) that

for all , in . It easily follows from (2.6) that is additive, that is, for all . So if is a rational number, then for all in . Therefore, it follows from (2.1) that for all in . Since , are nonzero, we infer that .

Theorem 2.2.

Let and be real vector spaces and be an even function satisfying (1.2). Then satisfies (1.1).

Proof.

Letting in (1.2), we get . Replacing by in (1.2), we get

for all . Replacing by in (2.7) and using the evenness of , we get

for all , in . Adding (2.7) to (2.8), we obtain

for all . Replacing by in (2.7), we get

for all , in . Using (2.7) in (2.10), by a simple computation, we get

for all , in . Putting in (2.11), we get that for all . Therefore, it follows from (2.11) that

for all , in . Replacing by in (2.12), we get that for all . So satisfies (1.1).

Theorem 2.3.

Let be a function between real vector spaces and . If is a rational number, then satisfies (1.2) if and only if satisfies (1.1).

Proof.

Let and be the odd and the even parts of . Suppose that satisfies (1.2). It is clear that and satisfy (1.2). By Theorems 2.1 and 2.2, and satisfies (1.1). Since , we conclude that satisfies (1.1).

Conversely, let satisfy (1.1). Then there exists a unique symmetric biadditive function such that for all (see [13]). Therefore

for all . So satisfies (1.2).

Proposition 2.4.

Let be a linear space with the norm . is an inner product space if and only if there exists a real number such that

for all , where .

Proof.

Let be a function defined by . If is an inner product space, then satisfies (2.14) for all . Conversely, let and the (even) function satisfy (2.14). So satisfies (1.2). By Theorem 2.3, the function satisfies (1.1), that is,

for all . Therefore is an inner product space (see [14]).

Proposition 2.5.

Let and be a linear space with the norm . Suppose that

for all , in , where and . Then .

Proof.

Setting in (2.16), we get

for all in . If we take with in (2.17), we get that . Letting in (2.16), we get

for all in . Letting in (2.16), we get

for all in . Since , it follows from (2.17) and (2.19) that

for all . Using (2.18) and (2.20), we get for all . Hence and (2.18) implies that . Finally, follows from (2.19).

Corollary 2.6.

Let be a linear space with the norm . is an inner product space if and only if there exists a real number and such that

for all , where .

## 3. Stability of (1.2) on Restricted Domains

In this section, we investigate the Hyers-Ulam stability of the functional equation (1.2) on a restricted domain. As an application we use the result to the study of an asymptotic behavior of that equation. It should be mentioned that Skof [39] was the first author who treats the Hyers-Ulam stability of the quadratic equation. Czerwik [8] proved a Hyers-Ulam-Rassias stability theorem on the quadratic equation. As a particular case he proved the following theorem.

Theorem 3.1.

Let be fixed. If a mapping satisfies the inequality

for all , then there exists a unique quadratic mapping such that for all . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

We recall that , are nonzero real numbers with .

Theorem 3.2.

Let and be given. Assume that an even mapping satisfies the inequality

for all with . Then there exists such that satisfies

for all with .

Proof.

Let with . Then, since , we get . So it follows from (3.2) that

for all with . So

for all with .

Let with . We have two cases.

Case 1.

. Then .

Case 2.

. Then we have . So

Therefore we get that from Cases 1 and 2. Hence by (3.4) we have

for all with . Set . Then

for all with . From (3.4) and (3.5), we get the following inequalities:

Using (3.7) and the above inequalities, we get

for all with . If with , then . So it follows from (3.10) that

Letting in (3.11), we get

for all with . Letting (and with ) in (3.11), we get . Therefore it follows from (3.11) and (3.12) that

for all with . Since is even, the inequality (3.13) holds for all with . Therefore

for all with . This completes the proof by letting .

Theorem 3.3.

Let and be given. Assume that an even mapping satisfies the inequality (3.2) for all with . Then satisfies

for all .

Proof.

By Theorem 3.2 there exists such that satisfies (3.3) for all with and (see the proof of Theorem 3.2). Using Theorem 2 of [38], we get that

all .

Theorem 3.4.

Let and be given. Assume that an even mapping satisfies the inequality (3.2) for all with . Then there exists a unique quadratic mapping such that and

for all .

Proof.

The result follows from Theorems 3.1 and 3.3.

Skof [39] has proved an asymptotic property of the additive mappings and Jung [34] has proved an asymptotic property of the quadratic mappings (see also [36]). We prove such a property also for the quadratic mappings.

Corollary 3.5.

An even mapping satisfies (1.2) if and only if the asymptotic condition

holds true.

Proof.

By the asymptotic condition (3.18), there exists a sequence monotonically decreasing to 0 such that

for all with . Hence, it follows from (3.19) and Theorem 3.4 that there exists a unique quadratic mapping such that

for all . Since is a monotonically decreasing sequence, the quadratic mapping satisfies (3.20) for all . The uniqueness of implies for all . Hence, by letting in (3.20), we conclude that is quadratic.

Corollary 3.6.

Let be rational. An even mapping is quadratic if and only if the asymptotic condition (3.18) holds true.

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## Acknowledgments

This research 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. 2010-0007143).

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Najati, A., Jung, SM. Approximately Quadratic Mappings on Restricted Domains.
*J Inequal Appl* **2010**, 503458 (2010). https://doi.org/10.1155/2010/503458

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DOI: https://doi.org/10.1155/2010/503458

### Keywords

- Functional Equation
- Linear Space
- Quadratic Mapping
- Additive Mapping
- Asymptotic Property