- Research Article
- Open Access

# On the Stability of a Generalized Quadratic and Quartic Type Functional Equation in Quasi-Banach Spaces

- M. Eshaghi Gordji
^{1}, - S. Abbaszadeh
^{1}and - Choonkil Park
^{2}Email author

**2009**:153084

https://doi.org/10.1155/2009/153084

© M. Eshaghi Gordji et al. 2009

**Received:**31 May 2009**Accepted:**9 September 2009**Published:**12 October 2009

## Abstract

## Keywords

- Banach Space
- Functional Equation
- Real Vector Space
- Nonnegative Real Number
- Quadratic Functional Equation

## 1. Introduction

for all Moreover, if is continuous in for each fixed then is -linear. In 1978, Th. M. Rassias [3] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded. The functional equation

The generalized Hyers-Ulam stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where is a normed space and is a Banach space (see [8]). Cholewa [9] noticed that the theorem of Skof is still true if relevant domain is replaced by an abelian group. In [10] , Czerwik proved the generalized Hyers-Ulam stability of the functional equation (1.3). Grabiec [11] has generalized these results mentioned above.

In [12], Park and Bae considered the following quartic functional equation:

In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all . It is easy to show that satisfies the functional equation (1.5), which is called a quartic functional equation (see also [13]).

In addition, Kim [14] has obtained the generalized Hyers-Ulam stability for a mixed type of quartic and quadratic functional equation between two real linear Banach spaces. Najati and Zamani Eskandani [15] have established the general solution and the generalized Hyers-Ulam stability for a mixed type of cubic and additive functional equation, whenever is a mapping between two quasi-Banach spaces (see also [16, 17]).

Now we introduce the following functional equation for fixed integers with :

in quasi-Banach spaces. It is easy to see that the function is a solution of the functional equation (1.6). In the present paper we investigate the general solution of the functional equation (1.6) when is a mapping between vector spaces, and we establish the generalized Hyers-Ulam stability of this functional equation whenever is a mapping between two quasi-Banach spaces.

We recall some basic facts concerning quasi-Banach space and some preliminary results.

Definition 1.1 (See [18, 19]).

Let be a real linear space. A quasinorm is a real-valued function on satisfying the following.

(1) for all and if and only if

(3)There is a constant such that for all

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

for all . In this case, a quasi-Banach space is called a -Banach space.

Given a -norm, the formula gives us a translation invariant metric on X. By the Aoki-Rolewicz theorem [19] (see also [18]), each quasi-norm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. In [20], Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see [3, 21]) in quasi-Banach spaces.

## 2. General Solution

Throughout this section, and will be real vector spaces. We here present the general solution of (1.6).

Lemma 2.1.

If a mapping satisfies the functional equation (1.6), then is a quadratic and quartic mapping.

Proof.

Now we show that (2.15) is a quadratic and quartic functional equation. To get this, we show that the mappings , defined by , and , defined by , are quadratic and quartic, respectively.

for all . Thus the mapping is quadratic.

for all . Thus is a quartic mapping.

Theorem 2.2.

Proof.

for all where the mapping is symmetric multi-additive and is bi-additive. By a simple computation, one can show that the mappings and satisfy the functional equation (1.6), so the mapping f satisfies (1.6).

## 3. Generalized Hyers-Ulam Stability of (1.6)

From now on, let and be a quasi-Banach space with quasi-norm and a -Banach space with -norm , respectively. Let be the modulus of concavity of . In this section, using an idea of G vruta [22], we prove the stability of (1.6) in the spirit of Hyers, Ulam, and Rassias. For convenience we use the following abbreviation for a given mapping :

for all . Let . We will use the following lemma in this section.

Lemma 3.1 (see [15]).

Theorem 3.2.

Proof.

for all Thus (3.7) follows from (3.4) and (3.38).

for all Hence the mapping satisfies (1.6). By Lemma 2.1, the mapping is quadratic. Hence (3.40) implies that the mapping is quadratic.

for all . Using (3.44) and (3.42), we get as desired.

Theorem 3.3.

Proof.

The proof is similar to the proof of Theorem 3.2.

Corollary 3.4.

Proof.

In Theorem 3.2, putting for all , we get the desired result.

Corollary 3.5.

Proof.

In Theorem 3.2, putting for all , we get the desired result.

Theorem 3.6.

Proof.

for all Thus (3.60) follows from (3.58) and (3.70).

for all Hence the mapping satisfies (1.6). By Lemma 2.1, the mapping is quartic. Therefore, (3.75) implies that the mapping is quartic.

Theorem 3.7.

Proof.

The proof is similar to the proof of Theorem 3.6.

Corollary 3.8.

Proof.

In Theorem 3.6, putting for all , we get the desired result.

Corollary 3.9.

Proof.

In Theorem 3.6, putting for all , we get the desired result.

Theorem 3.10.

for all where and are defined in Theorems 3.2 and 3.7, respectively.

Proof.

for all So we obtain (3.92) by letting and for all

for all (3.62) implies that for all Thus But is only a quartic function and is only a quadratic function.

Therefore, we have and this completes the uniqueness property of and We can prove the other results similarly.

Corollary 3.11.

for all where , and are defined as in Corollaries 3.4 and 3.8.

Corollary 3.12.

## Declarations

### Acknowledgment

The third and corresponding author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). Also, the second author would like to thank the office of gifted students at Semnan University for its financial support.

## Authors’ Affiliations

## References

- Ulam SM:
*Problems in Modern Mathematics*. John Wiley & Sons, New York, NY, USA; 1940.MATHGoogle Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences of the United States of America*1941,**27:**222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72**(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar - Aczèl J, Dhombres J:
*Functional Equations in Several Variables*.*Volume 31*. Cambridge University Press, Cambridge, UK; 1989.View ArticleMATHGoogle Scholar - Amir D:
*Characterizations of Inner Product Spaces*.*Volume 20*. Birkhäuser, Basel, Switzerland; 1986.MATHGoogle Scholar - Jordan P, von Neumann J:
**On inner products in linear, metric spaces.***Annals of Mathematics. Second Series*1935,**36**(3):719–723. 10.2307/1968653MathSciNetView ArticleMATHGoogle Scholar - Kannappan Pl:
**Quadratic functional equation and inner product spaces.***Results in Mathematics*1995,**27**(3–4):368–372.MathSciNetView ArticleMATHGoogle Scholar - Skof F:
**Proprietà locali e approssimazione di operatori.***Rendiconti del Seminario Matematico e Fisico di Milano*1983,**53:**113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar - Cholewa PW:
**Remarks on the stability of functional equations.***Aequationes Mathematicae*1984,**27**(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar - Czerwik S:
**On the stability of the quadratic mapping in normed spaces.***Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*1992,**62:**59–64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar - Grabiec A:
**The generalized Hyers-Ulam stability of a class of functional equations.***Publicationes Mathematicae Debrecen*1996,**48**(3–4):217–235.MathSciNetMATHGoogle Scholar - Park W-G, Bae J-H:
**On a bi-quadratic functional equation and its stability.***Nonlinear Analysis: Theory, Methods & Applications*2005,**62**(4):643–654. 10.1016/j.na.2005.03.075MathSciNetView ArticleMATHGoogle Scholar - Chung JK, Sahoo PK:
**On the general solution of a quartic functional equation.***Bulletin of the Korean Mathematical Society*2003,**40**(4):565–576.MathSciNetView ArticleMATHGoogle Scholar - Kim H-M:
**On the stability problem for a mixed type of quartic and quadratic functional equation.***Journal of Mathematical Analysis and Applications*2006,**324**(1):358–372. 10.1016/j.jmaa.2005.11.053MathSciNetView ArticleMATHGoogle Scholar - Najati A, Eskandani GZ:
**Stability of a mixed additive and cubic functional equation in quasi-Banach spaces.***Journal of Mathematical Analysis and Applications*2008,**342**(2):1318–1331. 10.1016/j.jmaa.2007.12.039MathSciNetView ArticleMATHGoogle Scholar - Eshaghi Gordji M: Stability of a functional equation deriving from quartic and additive functions. to appear in Bulletin of the Korean Mathematical Society to appear in Bulletin of the Korean Mathematical SocietyGoogle Scholar
- Eshaghi Gordji M, Ebadian A, Zolfaghari S:
**Stability of a functional equation deriving from cubic and quartic functions.***Abstract and Applied Analysis*2008,**2008:**-17.Google Scholar - Benyamini Y, Lindenstrauss J:
*Geometric Nonlinear Functional Analysis. Vol. 1*.*Volume 48*. American Mathematical Society, Providence, RI, USA; 2000.MATHGoogle Scholar - Rolewicz S:
*Metric Linear Spaces*. 2nd edition. PWN—Polish Scientific, Warsaw, Poland; 1984.MATHGoogle Scholar - Tabor J:
**Stability of the Cauchy functional equation in quasi-Banach spaces.***Annales Polonici Mathematici*2004,**83**(3):243–255. 10.4064/ap83-3-6MathSciNetView ArticleMATHGoogle Scholar - Gajda Z:
**On stability of additive mappings.***International Journal of Mathematics and Mathematical Sciences*1991,**14**(3):431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar - Găvruţa P:
**A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.***Journal of Mathematical Analysis and Applications*1994,**184**(3):431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.