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Approximate Behavior of Bi-Quadratic Mappings in Quasinormed Spaces

Open Access
Research Article

Abstract

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

Keywords

Banach Space Functional Equation Stability Problem Additive Mapping Product Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 Open image in new window be a group and let Open image in new window be a metric group with the metric Open image in new window . Given Open image in new window , does there exist a Open image in new window such that if a mapping Open image in new window satisfies the inequality Open image in new window for all Open image in new window then there is a homomorphism Open image in new window with Open image in new window for all Open image in new window ?

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 Open image in new window be a mapping from a normed vector space Open image in new window into a Banach space Open image in new window subject to the inequality
for all Open image in new window , where Open image in new window and Open image in new window are constants with Open image in new window and Open image in new window . The above inequality provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept. G Open image in new window vruţa [4] provided a further generalization of Hyers-Ulam theorem. A square norm on an inner product space satisfies the important parallelogram equality:
The functional equation

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 Open image in new window , where Open image in new window is a normed space and Open image in new window is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain Open image in new window 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 Open image in new window -algebra.

Throughout this paper, let Open image in new window and Open image in new window be vector spaces.

Definition 1.1.

A mapping Open image in new window is called bi-quadratic if Open image in new window satisfies the system of the following equations:

When Open image in new window , the function Open image in new window given by Open image in new window is a solution of (1.4).

For a mapping Open image in new window , consider the functional equation:

Definition 1.2 (see [9, 10]).

Let Open image in new window be a real linear space. A quasinorm is real-valued function on Open image in new window satisfying the following

(i) Open image in new window for all Open image in new window and Open image in new window if and only if Open image in new window .

(ii) Open image in new window for all Open image in new window and all Open image in new window .

(iii)There is a constant Open image in new window such that Open image in new window for all Open image in new window .

It follows from the condition (iii) that

for all Open image in new window and all Open image in new window .

The pair Open image in new window is called a quasinormed space if Open image in new window is a quasinorm on Open image in new window . The smallest possible Open image in new window is called the modulus of concavity of Open image in new window . A quasi-Banach space is a complete quasinormed space. A quasinorm Open image in new window is called a Open image in new window -norm ( Open image in new window ) if

for all Open image in new window . In this case, a quasi-Banach space is called a Open image in new window -Banach space.

Given a Open image in new window -norm, the formula Open image in new window gives us a translation invariant metric on Open image in new window . By the Aoki-Rolewicz theorem [10] (see also [9]), each quasinorm is equivalent to some Open image in new window -norm. Since it is much easier to work with Open image in new window -norms, henceforth we restrict our attention mainly to Open image in new window -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, 14, 15, 16, 17, 18]).

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

Theorem A.

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

Theorem B.

A mapping Open image in new window 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 Open image in new window is a quasinormed space with quasinorm Open image in new window and that Open image in new window is a Open image in new window -Banach space with Open image in new window -norm Open image in new window . Let Open image in new window be the modulus of concavity of Open image in new window .

for all Open image in new window .

Let Open image in new window be two functions satisfying

for all Open image in new window .

Theorem 2.1.

Let Open image in new window be a mapping such that
and let Open image in new window and Open image in new window for all Open image in new window . Then there exist two bi-quadratic mappings Open image in new window such that

for all Open image in new window .

Proof.

Letting Open image in new window in (2.4), we get
for all Open image in new window . Replacing Open image in new window by Open image in new window in the above inequality, we obtain
for all Open image in new window . Since Open image in new window is a Open image in new window -Banach space, for given integers Open image in new window , we see that
for all Open image in new window . By (2.2) and (2.11), the sequence Open image in new window is a Cauchy sequence for all Open image in new window . Since Open image in new window is complete, the sequence Open image in new window converges for all Open image in new window . Define Open image in new window by
for all Open image in new window . Putting Open image in new window and taking Open image in new window in (2.11), one can obtain the inequality (2.6). By (2.4) and (2.5), we get

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

Next, setting Open image in new window in (2.5),

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

Corollary 2.2.

Let Open image in new window be a mapping such that
and let Open image in new window and Open image in new window for all Open image in new window . Then there exist two bi-quadratic mappings Open image in new window such that

for all Open image in new window .

Proof.

In Theorem 2.1, putting Open image in new window and Open image in new window for all Open image in new window , we get the desired result.

From now on, let Open image in new window be a function such that

for all Open image in new window .

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

Lemma 2.3 (see [20]).

Let Open image in new window and let Open image in new window be nonnegative real numbers. Then

Theorem 2.4.

Let Open image in new window be a mapping such that
and let Open image in new window and Open image in new window for all Open image in new window . Then there exists a unique bi-quadratic mapping Open image in new window such that

for all Open image in new window .

Proof.

for all Open image in new window . Thus we obtain
for all Open image in new window and all Open image in new window . By Lemma 2.3, for given integers Open image in new window , we get
for all Open image in new window . By (2.18) and (2.25), the sequence Open image in new window is a Cauchy sequence for all Open image in new window . Since Open image in new window is complete, the sequence Open image in new window converges for all Open image in new window . Define Open image in new window by

for all Open image in new window .

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

for all Open image in new window . Hence the mapping Open image in new window is the unique bi-quadratic mapping, as desired.

Corollary 2.5.

Let Open image in new window be a nonnegative real number. Let Open image in new window be a mapping such that
and let Open image in new window and Open image in new window for all Open image in new window . Then there exists a unique bi-quadratic mapping Open image in new window such that

for all Open image in new window .

Proof.

In Theorem 2.4, putting Open image in new window for all Open image in new window , we get the desired result.

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Copyright information

© 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.

Authors and Affiliations

  1. 1.Department of Mathematics Education, College of EducationMokwon UniversityDaejeonRepublic of Korea
  2. 2.College of Liberal ArtsKyung Hee UniversityYonginRepublic of Korea

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