# Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces

- Won-Gil Park
^{1}and - Jae-Hyeong Bae
^{2}Email author

**2010**:472721

https://doi.org/10.1155/2010/472721

© W.-G. Park and J.-H. Bae. 2010

**Received: **27 April 2010

**Accepted: **18 June 2010

**Published: **5 July 2010

## Abstract

## Keywords

## 1. Introduction

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems, containing the stability problem of homomorphisms as follows

*Let*
*be a group and let*
*be a metric group with the metric*
*. Given*
*, does there exist a*
*such that if a mapping*
*satisfies the inequality*
*for all*
*then there is a homomorphism*
*with*
*for all*
*?*

is called the quadratic functional equation whose solution is said to be a quadratic mapping. A generalized stability problem for the quadratic functional equation was proved by Skof [5] for mappings , where is a normed space and is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [7] proved the generalized stability of the quadratic functional equation, and Park [8] proved the generalized stability of the quadratic functional equation in Banach modules over a -algebra.

Throughout this paper, let and be vector spaces.

Definition 1.1.

When , the function given by is a solution of (1.4).

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

(i) for all and if and only if .

(iii)There is a constant such that for all .

*quasinormed space*if is a quasinorm on . The smallest possible is called the

*modulus of concavity*of . A

*quasi-Banach space*is a complete quasinormed space. A quasinorm is called a

*-norm*( ) if

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 . By the Aoki-Rolewicz theorem [10] (see also [9]), each quasinorm is equivalent to some -norm. Since it is much easier to work with -norms, henceforth we restrict our attention mainly to -norms. In [11], Tabor has investigated a version of Hyers-Rassias-Gajda theorem (see also [3, 12]) in quasi-Banach spaces. Since then, the stability problems have been investigated by many authors (see [13–18]).

The authors [19] solved the solutions of (1.4) and (1.5) as follows.

Theorem A.

A mapping satisfies (1.4) if and only if there exist a multi-additive mapping such that and for all .

Theorem B.

A mapping satisfies (1.4) if and only if it satisfies (1.5).

In this paper, we investigate the generalized Hyers-Ulam stability of (1.4) and (1.5) in quasi-Banach spaces.

## 2. Stability of (1.4) and (1.5) in Quasi-normed Spaces

Throughout this section, assume that is a quasinormed space with quasinorm and that is a -Banach space with -norm . Let be the modulus of concavity of .

Theorem 2.1.

Proof.

for all and all . Letting in the above two inequalities and using (2.1), is bi-quadratic.

for all . By the same method as above, define by for all . By the same argument as above, is a bi-quadratic mapping satisfying (2.7).

Corollary 2.2.

Proof.

In Theorem 2.1, putting and for all , we get the desired result.

We will use the following lemma in order to prove Theorem 2.4.

Lemma 2.3 (see [20]).

Theorem 2.4.

Proof.

for all . Hence the mapping is the unique bi-quadratic mapping, as desired.

Corollary 2.5.

Proof.

In Theorem 2.4, putting for all , we get the desired result.

## Authors’ Affiliations

## References

- Ulam SM:
*A Collection of Mathematical Problems*. Interscience, New York, NY, USA; 1968.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 - 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 - Skof F: Local properties and approximation of operators.
*Rendiconti del Seminario Matematico e Fisico di Milano*1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleMATHGoogle Scholar - Cholewa PW: Remarks on the stability of functional equations.
*Aequationes Mathematicae*1984, 27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar - Czerwik St: 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 - Park C: On the stability of the quadratic mapping in Banach modules.
*Journal of Mathematical Analysis and Applications*2002, 276(1):135–144. 10.1016/S0022-247X(02)00387-6MathSciNetView ArticleMATHGoogle Scholar - Benyamini Y, Lindenstrauss J:
*Geometric Nonlinear Functional Analysis. 1, American Mathematical Society Colloquium Publications*.*Volume 48*. American Mathematical Society, Providence, RI, USA; 2000:xii+488.MATHGoogle Scholar - Rolewicz S:
*Metric Linear Spaces*. 2nd edition. PWN-Polish Scientific, Warsaw, Poland; 1984:xi+459.Google 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 - Eshaghi Gordji M, Abbaszadeh S, Park C: On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces.
*Journal of Inequalities and Applications*2009, 2009:-26.Google Scholar - Najati A: Homomorphisms in quasi-Banach algebras associated with a Pexiderized Cauchy-Jensen functional equation.
*Acta Mathematica Sinica*2009, 25(9):1529–1542. 10.1007/s10114-009-7648-zMathSciNetView 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 - Najati A, Moradlou F: Stability of a quadratic functional equation in quasi-Banach spaces.
*Bulletin of the Korean Mathematical Society*2008, 45(3):587–600. 10.4134/BKMS.2008.45.3.587MathSciNetView ArticleMATHGoogle 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 - Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras.
*Bulletin des Sciences Mathématiques*2008, 132(2):87–96.View ArticleMathSciNetMATHGoogle 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 - Najati A, Moghimi MB: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces.
*Journal of Mathematical Analysis and Applications*2008, 337(1):399–415. 10.1016/j.jmaa.2007.03.104MathSciNetView 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.