- Research Article
- Open Access

# Approximately Quadratic Mappings on Restricted Domains

- Abbas Najati
^{1}and - Soon-Mo Jung
^{2}Email author

**2010**:503458

https://doi.org/10.1155/2010/503458

© Abbas Najati and Soon-Mo Jung. 2010

**Received:**16 September 2010**Accepted:**20 December 2010**Published:**23 December 2010

## Abstract

We introduce a generalized quadratic functional equation , where , are nonzero real numbers with . We show that this functional equation is quadratic if , are rational numbers. We also investigate its stability problem on restricted domains. These results are applied to study of an asymptotic behavior of these generalized quadratic mappings.

## Keywords

- Functional Equation
- Linear Space
- Quadratic Mapping
- Additive Mapping
- Asymptotic Property

## 1. Introduction

*Under what conditions does there exist a group homomorphism near an approximate group homomorphism?* This question concerning the stability of group homomorphisms was posed by Ulam [1]. The case of approximately additive mappings was solved by Hyers [2] on Banach spaces. In 1950 Aoki [3] provided a generalization of the Hyers' theorem for additive mappings and in 1978 Th. M. Rassias [4] generalized the Hyers' theorem for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The result of Rassias' theorem has been generalized by Găvruţa [6] who permitted the Cauchy difference to be bounded by a general control function. This stability concept is also applied to the case of other functional equations. For more results on the stability of functional equations, see [7–24]. We also refer the readers to the books in [25–29].

where , are nonzero real numbers with . 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 [13, 25, 27]).

We prove that the functional equations (1.1) and (1.2) are equivalent if , are nonzero rational numbers. The functional equation (1.1) is a spacial case of (1.2). Indeed, for the case in (1.2), we get (1.1).

In 1983 Skof [30] was the first author to solve the Hyers-Ulam problem for additive mappings on a restricted domain (see also [31–33]). In 1998 Jung [34] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [35–37]). J. M. Rassias [38] investigated the Hyers-Ulam stability of mixed type mappings on restricted domains.

## 2. Solutions of (1.2)

In this section we show that the functional equation (1.2) is equivalent to the quadratic equation (1.1). That is, every solution of (1.2) is a quadratic function. We recall that , are nonzero real numbers with .

Theorem 2.1.

Let and be real vector spaces and be an odd function satisfying (1.2). If is a rational number, then .

Proof.

for all , in . It easily follows from (2.6) that is additive, that is, for all . So if is a rational number, then for all in . Therefore, it follows from (2.1) that for all in . Since , are nonzero, we infer that .

Theorem 2.2.

Let and be real vector spaces and be an even function satisfying (1.2). Then satisfies (1.1).

Proof.

for all , in . Replacing by in (2.12), we get that for all . So satisfies (1.1).

Theorem 2.3.

Let be a function between real vector spaces and . If is a rational number, then satisfies (1.2) if and only if satisfies (1.1).

Proof.

Let and be the odd and the even parts of . Suppose that satisfies (1.2). It is clear that and satisfy (1.2). By Theorems 2.1 and 2.2, and satisfies (1.1). Since , we conclude that satisfies (1.1).

Proposition 2.4.

Proof.

for all . Therefore is an inner product space (see [14]).

Proposition 2.5.

for all , in , where and . Then .

Proof.

for all . Using (2.18) and (2.20), we get for all . Hence and (2.18) implies that . Finally, follows from (2.19).

Corollary 2.6.

## 3. Stability of (1.2) on Restricted Domains

In this section, we investigate the Hyers-Ulam stability of the functional equation (1.2) on a restricted domain. As an application we use the result to the study of an asymptotic behavior of that equation. It should be mentioned that Skof [39] was the first author who treats the Hyers-Ulam stability of the quadratic equation. Czerwik [8] proved a Hyers-Ulam-Rassias stability theorem on the quadratic equation. As a particular case he proved the following theorem.

Theorem 3.1.

for all , then there exists a unique quadratic mapping such that for all . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

We recall that , are nonzero real numbers with .

Theorem 3.2.

Proof.

Case 1.

Case 2.

for all with . This completes the proof by letting .

Theorem 3.3.

Proof.

Theorem 3.4.

Proof.

The result follows from Theorems 3.1 and 3.3.

Skof [39] has proved an asymptotic property of the additive mappings and Jung [34] has proved an asymptotic property of the quadratic mappings (see also [36]). We prove such a property also for the quadratic mappings.

Corollary 3.5.

holds true.

Proof.

for all . Since is a monotonically decreasing sequence, the quadratic mapping satisfies (3.20) for all . The uniqueness of implies for all . Hence, by letting in (3.20), we conclude that is quadratic.

Corollary 3.6.

Let be rational. An even mapping is quadratic if and only if the asymptotic condition (3.18) holds true.

## Declarations

### Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2010-0007143).

## Authors’ Affiliations

## References

- Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8*. Interscience, New York, NY, USA; 1960:xiii+150.Google 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 - 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/00210064MathSciNetView 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 - Bourgin DG: Classes of transformations and bordering transformations.
*Bulletin of the American Mathematical Society*1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7MathSciNetView 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 - 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 - 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/S0002-9947-02-03036-2MathSciNetView ArticleMATHGoogle Scholar - Forti GL: An existence and stability theorem for a class of functional equations.
*Stochastica*1980, 4(1):23–30. 10.1080/17442508008833155MathSciNetView ArticleMATHGoogle Scholar - Forti GL: Hyers-Ulam stability of functional equations in several variables.
*Aequationes Mathematicae*1995, 50(1–2):143–190. 10.1007/BF01831117MathSciNetView 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 - Kannappan P: Quadratic functional equation and inner product spaces.
*Results in Mathematics*1995, 27(3–4):368–372.MathSciNetView ArticleMATHGoogle Scholar - Jordan P, von Neumann J: On inner products in linear, metric spaces.
*Annals of Mathematics*1935, 36(3):719–723. 10.2307/1968653MathSciNetView ArticleMATHGoogle Scholar - Hyers DH, Rassias ThM: Approximate homomorphisms.
*Aequationes Mathematicae*1992, 44(2–3):125–153. 10.1007/BF01830975MathSciNetView ArticleMATHGoogle 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/S0161171296000324MathSciNetView ArticleMATHGoogle 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.MathSciNetView ArticleMATHGoogle Scholar - Najati A: Hyers-Ulam stability of an -Apollonius type quadratic mapping.
*Bulletin of the Belgian Mathematical Society. Simon Stevin*2007, 14(4):755–774.MathSciNetMATHGoogle Scholar - Najati A, Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation.
*Journal of Mathematical Analysis and Applications*2007, 335(2):763–778. 10.1016/j.jmaa.2007.02.009MathSciNetView ArticleMATHGoogle Scholar - Najati A, Park C: The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between -algebras.
*Journal of Difference Equations and Applications*2008, 14(5):459–479. 10.1080/10236190701466546MathSciNetView ArticleMATHGoogle 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)00386-4MathSciNetView ArticleMATHGoogle 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-AMathSciNetView ArticleMATHGoogle 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:1006499223572MathSciNetView ArticleMATHGoogle 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.2000.7046MathSciNetView ArticleMATHGoogle 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 ArticleMATHGoogle Scholar - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle 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.MATHGoogle 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 (Ed):
*Functional Equations, Inequalities and Applications*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.MATHGoogle Scholar - Skof F: Sull' approssimazione delle applicazioni localmente -additive.
*Atti della Accademia delle Scienze di Torino*1983, 117: 377–389.MathSciNetGoogle Scholar - Hyers DH, Isac G, Rassias ThM: On the asymptoticity aspect of Hyers-Ulam stability of mappings.
*Proceedings of the American Mathematical Society*1998, 126(2):425–430. 10.1090/S0002-9939-98-04060-XMathSciNetView ArticleMATHGoogle Scholar - Jung S-M: Hyers-Ulam-Rassias stability of Jensen's equation and its application.
*Proceedings of the American Mathematical Society*1998, 126(11):3137–3143. 10.1090/S0002-9939-98-04680-2MathSciNetView ArticleMATHGoogle Scholar - Jung S-M, Moslehian MS, Sahoo PK: Stability of a generalized Jensen equation on restricted domains.
*Journal of Mathematical Inequalities*2010, 4: 191–206.MathSciNetView ArticleMATHGoogle Scholar - Jung S-M: On the Hyers-Ulam stability of the functional equations that have the quadratic property.
*Journal of Mathematical Analysis and Applications*1998, 222(1):126–137. 10.1006/jmaa.1998.5916MathSciNetView ArticleMATHGoogle Scholar - Jung S-M: Stability of the quadratic equation of Pexider type.
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*2000, 70: 175–190. 10.1007/BF02940912View ArticleMathSciNetMATHGoogle Scholar - Jung S-M, Kim B: On the stability of the quadratic functional equation on bounded domains.
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*1999, 69: 293–308. 10.1007/BF02940881MathSciNetView ArticleMATHGoogle Scholar - Jung S-M, Sahoo PK: Hyers-Ulam stability of the quadratic equation of Pexider type.
*Journal of the Korean Mathematical Society*2001, 38(3):645–656.MathSciNetMATHGoogle Scholar - Rassias JM: On the Ulam stability of mixed type mappings on restricted domains.
*Journal of Mathematical Analysis and Applications*2002, 276(2):747–762. 10.1016/S0022-247X(02)00439-0MathSciNetView ArticleMATHGoogle Scholar - Skof F: Proprieta' locali e approssimazione di operatori.
*Rendiconti del Seminario Matematico e Fisico di Milano*1983, 53(1):113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle 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.