- Research Article
- Open Access

- Dongseung Kang
^{1}Email author

**2010**:198098

https://doi.org/10.1155/2010/198098

© Dongseung Kang. 2010

**Received:**28 August 2009**Accepted:**25 January 2010**Published:**2 February 2010

## Abstract

## Keywords

- Banach Space
- Functional Equation
- Linear Space
- Normed Space
- Regularity Condition

## 1. Introduction

One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam [1] as follows. *Let*
*be a group and let*
*be a metric group with the metric*
*. Given*
*, does there exist a*
*such that if a function*
*satisfies the inequality*
*for all*
*, then there is a homomorphism*
*with*
*for all*
*?* In other words, we are looking for situations when the homomorphisms are stable; that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. In 1941, Hyers [2] considered the case of approximately additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. The famous Hyers stability result that appeared in [2] was generalized in the stability involving a sum of powers of norms by Aoki [3]. In 1978, Rassias [4] provided a generalization of Hyers Theorem which allows the Cauchy difference to be unbounded. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors [5–10]. In particular, Rassias [11] introduced the quartic functional equation

It is easy to see that is a solution of (1.1) by virtue of the identity

For this reason, (1.1) is called a quartic functional equation. Also Chung and Sahoo [12] determined the general solution of (1.1) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (1.1) if and only if where the function is symmetric and additive in each variable. Lee and Chung [13] introduced a quartic functional equation as follows:

Let be a real number with and let be either or We will consider the definition and some preliminary results of a quasi- -norm on a linear space.

Definition 1.1.

Let
be a linear space over a field
A *quasi-*
*-norm*
is a real-valued function on
satisfying the followings.

(1) for all and if and only if

(3) There is a constant such that for all

The pair
is called a *quasi-*
*-normed space* if
is a quasi-
-norm on
The smallest possible
is called the *modulus of concavity* of
A *quasi-Banach space* is a complete quasi-
-normed space.

A quasi- -norm is called a -norm ( if for all In this case, a quasi- -Banach space is called a -Banach space; see [14–16].

In this paper, we consider the following the generalized quartic functional equation:

for fixed integers and such that for all We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi- -normed spaces and then the stability by using a subadditive function for the generalized quartic function satisfying (1.4).

For the same reason as (1.1) and (1.2), we call (1.4) generalized quartic functional equation.

## 2. Quartic Functional Equations

Let be real vector spaces. In this section, we will investigate that the functional equation (1.1) is equivalent to the presented functional equation (1.4).

Lemma 2.1.

Proof.

for all Thus they are equivalent.

Theorem 2.2.

If a mapping satisfies the functional equation (1.4), then satisfies the functional equation (2.1).

Proof.

Corollary 2.3.

If a mapping satisfies the functional equation (1.1), then satisfies the functional equation (1.4).

## 3. Stabilities

Throughout this section, let be a quasi- -normed space and let be a quasi- -Banach space with a quasi- -norm . Let be the modulus of concavity of . We will investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.4). After then we will study the stability by using a subadditive function. For a given mapping and all fixed integers and with let

Theorem 3.1.

Proof.

for all By letting we immediately have the uniqueness of .

Theorem 3.2.

Proof.

If is replaced by in the inequality (3.5), then the proof follows from the proof of Theorem 3.1.

Now we will recall a subadditive function and then investigate the stability under the condition that the space is a -Banach space. The basic definitions of subadditive functions follow from [16].

A function having a domain and a codomain that are both closed under addition is called

(2)*a contractively subadditive function* if there exists a constant
with
such that

(3)*an expansively superadditive function* if there exists a constant
with
such that

Theorem 3.3.

Proof.

for all By letting we immediately have the uniqueness of .

Theorem 3.4.

Proof.

## Declarations

### Acknowledgments

The author would like to thank referees for their valuable suggestions and comments. The present research was conducted by the research fund of Dankook University in 2009.

## Authors’ Affiliations

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## Copyright

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