- Mohammad B. Moghimi
^{1}, - Abbas Najati
^{1}Email author and - Choonkil Park
^{2}

**2009**:256165

https://doi.org/10.1155/2009/256165

© Mohammad B. Moghimi et al. 2009

**Received: **18 May 2009

**Accepted: **31 July 2009

**Published: **19 August 2009

## Abstract

## 1. Introduction and Preliminaries

*Under what conditions does there exist a group homomorphism near an approximately group homomorphism?*In 1941, Hyers [2] considered the case of approximately additive functions , where and are Banach spaces and satisfies

*Hyers inequality*

for all . Aoki [3] and Th. M. Rassias [4] provided a generalization of the Hyers' theorem for additive mappings and for linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also [5]).

Theorem 1.1 (Th. M. Rassias).

for all . If then inequality (1.2) holds for and (1.4) for . Also, if for each the mapping is continuous in , then is -linear.

*quadratic functional equation*is a functional equation of the following form:

*quadratic mapping.*It is well known that a mapping between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mapping such that for all (see [16, 21, 26, 27]. The biadditive mapping is given by

The Hyers-Ulam stability problem for the quadratic functional equation (1.5) was studied by Skof [28] for mappings where is a normed space and is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if we replace by an Abelian group. Czerwik [9] proved the generalized Hyers-Ulam stability of the quadratic functional equation (1.5). Grabiec [11] has generalized these results mentioned above. Jun and Lee [14] proved the generalized Hyers-Ulam stability of a Pexiderized quadratic functional equation.

Let
be a set. A function
is called a *generalized metric* on
if
satisfies

We recall the following theorem by Margolis and Diaz.

Theorem 1.2 (see [29]).

for all nonnegative integers or there exists a non-negative integer such that

- (1)
- (2)
- (3)
- (4)

in -algebras. A systematic study of fixed point theorems in nonlinear analysis is due to Hyers et al. [30] and Isac and Rassias [13].

## 2. Solutions of (1.8)

Theorem 2.1.

Let be a linear space. If a mapping satisfies and the functional equation (1.8), then is quadratic.

Proof.

Remark 2.2.

A quadratic mapping does not satisfy (1.8) in general. Let be the mapping defined by for all It is clear that is quadratic and that does not satisfy (1.8).

Corollary 2.3.

Let be a linear space. If a mapping satisfies the functional equation (1.8), then there exists a symmetric biadditive mapping such that for all

## 3. Generalized Hyers-Ulam Stability of (1.8) in -Algebras

In this section, we use a fixed point method (see [7, 15, 17]) to investigate the stability problem of the functional equation (1.8) in -algebras.

for all where is a linear space.

Theorem 3.1.

Moreover, if is continuous in for each fixed , then is -quadratic, that is, for all and all

Proof.

is a generalized complete metric space [7].

for all By Theorem 2.1, the function is quadratic.

Moreover, if is continuous in for each fixed , then by the same reasoning as in the proof of [4] is -quadratic.

Corollary 3.2.

for all . Moreover, if is continuous in for each fixed , then is -quadratic.

The following theorem is an alternative result of Theorem 3.1 and we will omit the proof.

Theorem 3.3.

for all , where is defined as in Theorem 3.1. Moreover, if is continuous in for each fixed , then is -quadratic.

Corollary 3.4.

for all . Moreover, if is continuous in for each fixed , then is -quadratic.

For the case we use the Gajda's example [31] to give the following counterexample (see also [9]).

Example 3.5.

which contradicts (3.33).

## Declarations

### Acknowledgment

The third author was supported by Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00041).

## Authors’ Affiliations

## References

- Ulam SM:
*A Collection of Mathematical Problems*. Interscience Publishers, New York, NY, USA; 1960:xiii+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:**222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar - Aoki T:
**On the stability of the linear transformation in Banach spaces.***Journal of the Mathematical Society of Japan*1950,**2:**64–66. 10.2969/jmsj/00210064MATHMathSciNetView 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 - Bourgin DG:
**Classes of transformations and bordering transformations.***Bulletin of the American Mathematical Society*1951,**57:**223–237. 10.1090/S0002-9904-1951-09511-7MATHMathSciNetView ArticleGoogle 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 - Cădariu L, Radu V:
**On the stability of the Cauchy functional equation: a fixed point approach.**In*Iteration Theory, Grazer Mathematische Berichte*.*Volume 346*. Karl-Franzens-Universitaet Graz, Graz, Austria; 2004:43–52.Google Scholar - Cholewa PW:
**Remarks on the stability of functional equations.***Aequationes Mathematicae*1984,**27**(1–2):76–86.MATHMathSciNetView 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 - 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 - Grabiec A:
**The generalized Hyers-Ulam stability of a class of functional equations.***Publicationes Mathematicae Debrecen*1996,**48**(3–4):217–235.MATHMathSciNetGoogle Scholar - Hyers DH, Rassias ThM:
**Approximate homomorphisms.***Aequationes Mathematicae*1992,**44**(2–3):125–153. 10.1007/BF01830975MATHMathSciNetView ArticleGoogle Scholar - Isac G, Rassias ThM:
**Stability of -additive mappings: applications to nonlinear analysis.***International Journal of Mathematics and Mathematical Sciences*1996,**19**(2):219–228. 10.1155/S0161171296000324MATHMathSciNetView ArticleGoogle Scholar - Jun K-W, Lee Y-H:
**On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality.***Mathematical Inequalities & Applications*2001,**4**(1):93–118.MATHMathSciNetView ArticleGoogle Scholar - Jung S-M, Kim T-S:
**A fixed point approach to the stability of the cubic functional equation.***Boletín de la Sociedad Matemática Mexicana*2006,**12**(1):51–57.MATHMathSciNetGoogle Scholar - Kannappan Pl:
**Quadratic functional equation and inner product spaces.***Results in Mathematics*1995,**27**(3–4):368–372.MATHMathSciNetView ArticleGoogle Scholar - Mirzavaziri M, Moslehian MS:
**A fixed point approach to stability of a quadratic equation.***Bulletin of the Brazilian Mathematical Society*2006,**37**(3):361–376. 10.1007/s00574-006-0016-zMATHMathSciNetView ArticleGoogle Scholar - Park C-G:
**On the stability of the linear mapping in Banach modules.***Journal of Mathematical Analysis and Applications*2002,**275**(2):711–720. 10.1016/S0022-247X(02)00386-4MATHMathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On a modified Hyers-Ulam sequence.***Journal of Mathematical Analysis and Applications*1991,**158**(1):106–113. 10.1016/0022-247X(91)90270-AMATHMathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On the stability of functional equations in Banach spaces.***Journal of Mathematical Analysis and Applications*2000,**251**(1):264–284. 10.1006/jmaa.2000.7046MATHMathSciNetView 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 - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge, NJ, USA; 2002:x+410.MATHView 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:
*Functional Equations, Inequalities and Applications*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.MATHView ArticleGoogle Scholar - Amir D:
*Characterizations of Inner Product Spaces, Operator Theory: Advances and Applications*.*Volume 20*. Birkhäuser, Basel, Switzerland; 1986:vi+200.View ArticleGoogle Scholar - Jordan P, von Neumann J:
**On inner products in linear, metric spaces.***Annals of Mathematics*1935,**36**(3):719–723. 10.2307/1968653MATHMathSciNetView ArticleGoogle 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 - Diaz JB, Margolis B:
**A fixed point theorem of the alternative, for contractions on a generalized complete metric space.***Bulletin of the American Mathematical Society*1968,**74:**305–309. 10.1090/S0002-9904-1968-11933-0MATHMathSciNetView ArticleGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Topics in Nonlinear Analysis & Applications*. World Scientific, River Edge, NJ, USA; 1997:xiv+699.MATHView ArticleGoogle Scholar - Gajda Z:
**On stability of additive mappings.***International Journal of Mathematics and Mathematical Sciences*1991,**14**(3):431–434. 10.1155/S016117129100056XMATHMathSciNetView ArticleGoogle Scholar

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