# On the Asymptoticity Aspect of Hyers-Ulam Stability of Quadratic Mappings

- A Rahimi
^{1}, - A Najati
^{2}and - J-H Bae
^{3}Email author

**2010**:454875

https://doi.org/10.1155/2010/454875

© A. Rahimi et al. 2010

**Received: **30 September 2010

**Accepted: **27 December 2010

**Published: **5 January 2011

## Abstract

We investigate the Hyers-Ulam stability of the quadratic functional equation on restricted domains. Applying these results, we study of an asymptotic behavior of these quadratic mappings.

## 1. Introduction

The question concerning the stability of group homomorphisms was posed by Ulam [1]. Hyers [2] solved the case of approximately additive mappings on Banach spaces. Aoki [3] provided a generalization of the Hyers' theorem for additive mappings. In [4], Rassias generalized the result of Hyers for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The result of Rassias has been generalized by G vruţa [6] who permitted the norm of the Cauchy difference to be bounded by a general control function under some conditions. This stability concept is also applied to the case of various functional equations by a number of authors. For more results on the stability of functional equations, see [7–32]. We also refer the readers to the books [33–37].

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 [21, 33, 35]).

A stability theorem for the quadratic functional equation (1.1) was proved by Skof [38] for functions , where is a normed space and is a Banach space. Cholewa [11] noticed that the result of Skof holds (with the same proof) if is replaced by an abelian group . In [12], Czerwik generalized the result of Skof by allowing growth of the form for the norm of , where and . In 1998, Jung [39] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [40–42]). Rassias [43] investigated the Hyers-Ulam stability of mixed type mappings on restricted domains. In [44], the authors considered the asymptoticity of Hyers-Ulam stability close to the asymptotic derivability.

## 2. Stability of (1.1) on Restricted Domains

In this section, we investigate the Hyers-Ulam stability of the functional equation (1.1) on a restricted domain. As an application, we use the result to the study of an asymptotic behavior of that equation.

Theorem 2.1.

for all with and . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

Proof.

Let with and let . We have two cases.

Case 1.

Case 2.

for all . Therefore, . Since , we have for all . The proof of our last assertion follows from the proof of Theorem 1 in [12].

We now introduce one of the fundamental results of fixed point theory by Margolis and Diaz.

Theorem 2.2 (see [22]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . If there exists a nonnegative integer such that for some , then the following are true:

(1)the sequence converges to a fixed point of ,

By using the idea of Cădariu and Radu [45], we applied a fixed point method to the investigation of the generalized Hyers-Ulam stability of the functional equation (1.1) on a restricted domain.

Theorem 2.3.

and for all . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

Proof.

Let with and let . We have two cases.

Case 1.

Case 2.

for all . Therefore, (2.23) implies that . Since , we have for all . Our last assertion is trivial in view of Theorem 2.1.

Corollary 2.4.

for all with and . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

Remark 2.5.

Using ideas from the papers [39, 43], we prove the generalized Hyers-Ulam stability of (1.1) on restricted domains. We first prove the following lemma.

Lemma 2.6.

Proof.

So, satisfies (2.57) for all .

Theorem 2.7.

Proof.

for all and all integers . It follows from (2.67) that the sequence converges for all . So, we can define the mapping by for all . Letting and in (2.67), we get (2.64).

For the case and in Theorem 2.7, it is obvious that our inequality (2.64) is sharper than the corresponding inequalities of Jung [39] and Rassias [43].

Skof [38] has proved an asymptotic property of the additive mappings, and Jung [39] has proved an asymptotic property of the quadratic mappings (see also [41]). Using the method in [39], the proof of the following corollary follows from Theorem 2.7 by letting and .

Corollary 2.8 (see [39]).

holds true.

## 3. -Asymptotically Quadratic Mappings

We apply our results to the study of -asymptotical derivatives. Let be a real normed vector space and let be a real Banach space . Let be arbitrary.

Definition 3.1.

A mapping
is called
*-asymptotically close* to a mapping
if and only if
.

Definition 3.2.

A mapping
is called
*-asymptotically derivable* if the mapping
is
-asymptotically close to a quadratic mapping
. In this case, we say that
is a
-asymptotical derivative of
.

Definition 3.3.

Definition 3.4.

A mapping
is called *quadratic outside a ball* if there exists
such that
for all
with
.

We have the following result.

Theorem 3.5.

If is quadratic outside a ball and is -asymptotically close to , then is -asymptotically quadratic.

The following result follows from Corollary 2.4.

Corollary 3.6.

If is quadratic outside a ball and is -asymptotically close to , then has a -asymptotical derivative.

## Authors’ Affiliations

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