# Stability of Quartic Functional Equations in the Spaces of Generalized Functions

- Young-Su Lee
^{1}Email author and - Soon-Yeong Chung
^{2}

**2009**:838347

**DOI: **10.1155/2009/838347

© Y.-S. Lee and S.-Y. Chung. 2009

**Received: **6 August 2008

**Accepted: **14 January 2009

**Published: **10 February 2009

## Abstract

We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.

## 1. Introduction

Lee et al. [12] obtained the general solution of (1.3) and proved the Hyers-Ulam-Rassias stability of this equation. Also Park [13] investigated the stability problem of (1.3) in the orthogonality normed space.

for fixed integer with . In the cases of in (1.4), homogeneity property of quartic functional equations does not hold. We dispense with this cases henceforth, and assume that . In Section 2, we show that for each fixed integer with , (1.4) is equivalent to (1.3). Moreover, using the idea of Găvruţa [14], we prove the Hyers-Ulam-Rassias stability of (1.4) in Section 3. Finally, making use of the pullbacks and the heat kernels, we reformulate and prove the Hyers-Ulam-Rassias stability of (1.4) in the spaces of some generalized functions such as of tempered distributions and of Fourier hyperfunctions in Section 4.

## 2. General Solution of (1.4)

Stability problems of quadratic functional equations can be found in [16–19]. Similarly, a function satisfies the quartic functional equation (1.3) if and only if there exists a symmetric biquadratic function such that for all (see [12]). We now present the general solution of (1.4) in the class of functions between real vector spaces.

Theorem 2.1.

A mapping satisfies the functional equation (1.3) if and only if for each fixed integer with , a mapping satisfies the functional equation (1.4).

Proof.

which proves the validity of (1.4) for . For a negative integer , replacing by one can easily prove the validity of (1.4). Therefore (1.3) implies (1.4) for any fixed integer with .

for each fixed integer . Replacing by in (2.15), and comparing (1.4) with (2.15) we have . Thus (2.14) implies (1.3). This completes the proof.

## 3. Stability of (1.4)

Now we are going to prove the Hyers-Ulam-Rassias stability for quartic functional equations. Let be a real vector space and let be a Banach space.

Theorem 3.1.

for all . Also, if for each fixed the mapping from to is continuous, then for all .

Proof.

for all . Letting , we must have for all . This completes the proof.

Corollary 3.2.

Corollary 3.3.

## 4. Stability of (1.4) in Generalized Functions

In this section, we reformulate and prove the stability theorem of the quartic functional equation (1.4) in the spaces of some generalized functions such as of tempered distributions and of Fourier hyperfunctions. We first introduce briefly spaces of some generalized functions. Here we use the multi-index notations, , , and , for , , where is the set of non-negative integers and .

Definition 4.1 (see [20, 21]).

for all . The set of all tempered distributions is denoted by .

Imposing growth conditions on in (4.1) a new space of test functions has emerged as follows.

Definition 4.2 (see [22]).

for some positive constants depending only on . We say that as if as for some , and denote by the strong dual of and call its elements Fourier hyperfunctions.

where . Here denotes the pullbacks of generalized functions. Also denotes the Euclidean norm and the inequality in (4.6) means that for all test functions defined on . We refer to (see [20, Chapter VI]) for pullbacks and to [21, 23–26] for more details of and .

If , the right side of (4.6) does not define a distribution. Thus, the inequality (4.6) makes no sense in this case. Also, if , it is not known whether Hyers-Ulam-Rassias stability of (1.4) holds even in the classical case. Thus, we consider only the case or .

holds for convolution. Semigroup property will be useful to convert inequality (3.3) into the classical functional inequality defined on upper-half plane. Moreover, the following result called heat kernel method holds [27].

- (i)
- (ii)

Conversely, every -solution of the heat equation satisfying the growth condition (4.11) can be uniquely expressed as for some . Similarly, we can represent Fourier hyperfunctions as initial values of solutions of the heat equation as a special case of the results (see [28]). In this case, the estimate (4.11) is replaced by the following.

We are now in a position to state and prove the main result of this paper.

Theorem 4.3.

Proof.

exists.

for all . Letting , we have for all . This proves the uniqueness.

This completes the proof.

As an immediate consequence, we have the following corollary.

Corollary 4.4.

## Declarations

### Acknowledgments

The first author was supported by the second stage of the Brain Korea 21 Project, The Development Project of Human Resources in Mathematics, KAIST, in 2009. The second author was supported by the Special Grant of Sogang University in 2005.

## Authors’ Affiliations

## References

- Ulam SM:
*Problems in Modern Mathematics*. John Wiley & Sons, New York, NY, USA; 1964:xvii+150.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**(4):222-224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle 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/S0002-9939-1978-0507327-1MATHMathSciNetView ArticleGoogle Scholar - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge, NJ, USA; 2002:x+410.MATHView ArticleGoogle Scholar - Faĭziev VA, Rassias ThM, Sahoo PK:
**The space of**( ,**)-additive mappings on semigroups.***Transactions of the American Mathematical Society*2002,**354**(11):4455-4472. 10.1090/S0002-9947-02-03036-2MATHMathSciNetView ArticleGoogle Scholar - Hyers DH, Rassias ThM:
**Approximate homomorphisms.***Aequationes Mathematicae*1992,**44**(2-3):125-153. 10.1007/BF01830975MATHMathSciNetView ArticleGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications*.*Volume 34*. Birkhäuser, Boston, Mass, USA; 1998:vi+313.Google Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar - Rassias ThM:
**On the stability of functional equations and a problem of Ulam.***Acta Applicandae Mathematicae*2000,**62**(1):23-130. 10.1023/A:1006499223572MATHMathSciNetView ArticleGoogle Scholar - Rassias JM:
**Solution of the Ulam stability problem for quartic mappings.***Glasnik Matematicki Series III*1999,**34**(2):243-252.MATHGoogle 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.MATHMathSciNetView ArticleGoogle Scholar - Lee SH, Im SM, Hwang IS:
**Quartic functional equations.***Journal of Mathematical Analysis and Applications*2005,**307**(2):387-394. 10.1016/j.jmaa.2004.12.062MATHMathSciNetView ArticleGoogle Scholar - Park C-G:
**On the stability of the orthogonally quartic functional equation.***Bulletin of the Iranian Mathematical Society*2005,**31**(1):63-70.MATHGoogle 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.1994.1211MATHMathSciNetView ArticleGoogle Scholar - Aczél J, Dhombres J:
*Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications*.*Volume 31*. Cambridge University Press, Cambridge, UK; 1989:xiv+462.View ArticleGoogle Scholar - Borelli C, Forti GL:
**On a general Hyers-Ulam stability result.***International Journal of Mathematics and Mathematical Sciences*1995,**18**(2):229-236. 10.1155/S0161171295000287MATHMathSciNetView ArticleGoogle 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/BF02941618MATHMathSciNetView ArticleGoogle Scholar - Lee JR, An JS, Park C:
**On the stability of quadratic functional equations.***Abstract and Applied Analysis*2008,**2008:**-8.Google Scholar - Skof F:
**Local properties and approximation of operators.***Rendiconti del Seminario Matematico e Fisico di Milano*1983,**53:**113-129. 10.1007/BF02924890MATHMathSciNetView ArticleGoogle Scholar - Hörmander L:
*The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Grundlehren der Mathematischen Wissenschaften*.*Volume 256*. Springer, Berlin, Germany; 1983:ix+391.Google Scholar - Schwartz L:
*Théorie des Distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X*. Hermann, Paris, France; 1966:xiii+420.Google Scholar - Chung J, Chung S-Y, Kim D:
**A characterization for Fourier hyperfunctions.***Publications of the Research Institute for Mathematical Sciences*1994,**30**(2):203-208. 10.2977/prims/1195166129MATHMathSciNetView ArticleGoogle Scholar - Chung J, Lee S:
**Some functional equations in the spaces of generalized functions.***Aequationes Mathematicae*2003,**65**(3):267-279. 10.1007/s00010-003-2657-yMATHMathSciNetView ArticleGoogle Scholar - Chung J:
**Stability of functional equations in the spaces of distributions and hyperfunctions.***Journal of Mathematical Analysis and Applications*2003,**286**(1):177-186. 10.1016/S0022-247X(03)00468-2MATHMathSciNetView ArticleGoogle Scholar - Chung J, Chung S-Y, Kim D:
**The stability of Cauchy equations in the space of Schwartz distributions.***Journal of Mathematical Analysis and Applications*2004,**295**(1):107-114. 10.1016/j.jmaa.2004.03.009MATHMathSciNetView ArticleGoogle Scholar - Lee Y-S:
**Stability of a quadratic functional equation in the spaces of generalized functions.***Journal of Inequalities and Applications*2008,**2008:**-12.Google Scholar - Matsuzawa T:
**A calculus approach to hyperfunctions. III.***Nagoya Mathematical Journal*1990,**118:**133-153.MATHMathSciNetGoogle Scholar - Kim KW, Chung S-Y, Kim D:
**Fourier hyperfunctions as the boundary values of smooth solutions of heat equations.***Publications of the Research Institute for Mathematical Sciences*1993,**29**(2):289-300. 10.2977/prims/1195167274MATHMathSciNetView ArticleGoogle Scholar

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