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  • Research Article
  • Open Access

Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces

Journal of Inequalities and Applications20102010:472721

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

  • Received: 27 April 2010
  • Accepted: 18 June 2010
  • Published:

Abstract

We obtain the generalized Hyers-Ulam stability of the bi-quadratic functional equation in quasinormed spaces.

Keywords

  • Banach Space
  • Functional Equation
  • Stability Problem
  • Additive Mapping
  • Product Space

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 ?

Hyers [2] proved the stability problem of additive mappings in Banach spaces. Rassias [3] provided a generalization of Hyers theorem which allows the Cauchy difference to be unbounded: let be a mapping from a normed vector space into a Banach space subject to the inequality
(1.1)
for all , where and are constants with and . The above inequality provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept. G vruţa [4] provided a further generalization of Hyers-Ulam theorem. A square norm on an inner product space satisfies the important parallelogram equality:
(1.2)
The functional equation
(1.3)

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.

A mapping is called bi-quadratic if satisfies the system of the following equations:
(1.4)

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

For a mapping , consider the functional equation:
(1.5)

Definition 1.2 (see [9, 10]).

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

(i) for all and if and only if .

(ii) for all and all .

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

It follows from the condition (iii) that
(1.6)

for all and 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 quasinormed space. A quasinorm is called a -norm ( ) if
(1.7)

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 [1318]).

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 .

Let and be two functions such that
(2.1)

for all .

Let be two functions satisfying
(2.2)
(2.3)

for all .

Theorem 2.1.

Let be a mapping such that
(2.4)
(2.5)
and let and for all . Then there exist two bi-quadratic mappings such that
(2.6)
(2.7)

for all .

Proof.

Letting in (2.4), we get
(2.8)
for all . Thus we have
(2.9)
for all . Replacing by in the above inequality, we obtain
(2.10)
for all . Since is a -Banach space, for given integers , we see that
(2.11)
for all . By (2.2) and (2.11), the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define by
(2.12)
for all . Putting and taking in (2.11), one can obtain the inequality (2.6). By (2.4) and (2.5), we get
(2.13)

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

Next, setting in (2.5),
(2.14)

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.

Let be a mapping such that
(2.15)
and let and for all . Then there exist two bi-quadratic mappings such that
(2.16)

for all .

Proof.

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

From now on, let be a function such that
(2.17)
(2.18)

for all .

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

Lemma 2.3 (see [20]).

Let and let be nonnegative real numbers. Then
(2.19)

Theorem 2.4.

Let be a mapping such that
(2.20)
and let and for all . Then there exists a unique bi-quadratic mapping such that
(2.21)

for all .

Proof.

Letting and in (2.20), we have
(2.22)
for all . Thus we obtain
(2.23)
for all and all . Replacing by in the above inequality, we see that
(2.24)
for all and all . By Lemma 2.3, for given integers , we get
(2.25)
for all . By (2.18) and (2.25), the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define by
(2.26)

for all .

By (2.20), we have
(2.27)
for all and all . Letting and using (2.17), we see that satisfies (1.5). By Theorem B, we obtain that is bi-quadratic. Setting and taking in (2.25), one can obtain the inequality (2.21). If is another bi-quadratic mapping satisfying (2.21), we obtain
(2.28)

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

Corollary 2.5.

Let be a nonnegative real number. Let be a mapping such that
(2.29)
and let and for all . Then there exists a unique bi-quadratic mapping such that
(2.30)

for all .

Proof.

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

Authors’ Affiliations

(1)
Department of Mathematics Education, College of Education, Mokwon University, Daejeon, 302-729, Republic of Korea
(2)
College of Liberal Arts, Kyung Hee University, Yongin, 446-701, Republic of Korea

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Copyright

© W.-G. Park and J.-H. Bae. 2010

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.

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