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  • Research Article
  • Open Access

Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces

Journal of Inequalities and Applications20102010:151547

https://doi.org/10.1155/2010/151547

  • Received: 16 October 2009
  • Accepted: 30 January 2010
  • Published:

Abstract

We obtain the generalized Hyers-Ulam stability of the Cauchy-Jensen functional equation .

Keywords

  • Banach Space
  • Functional Equation
  • Nonnegative Integer
  • Cauchy Sequence
  • Affine Mapping

1. Introduction

In 1940, Ulam proposed the general Ulam stability problem (see [1]).

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 ?

In 1941, this problem was solved by Hyers [2] in the case of Banach space. Thereafter, we call that type the Hyers-Ulam stability.

Throughout this paper, let and be vector spaces. A mapping is called an additive mapping (respectively, an affine mapping) if satisfies the Cauchy functional equation (respectively, the Jensen functional equation ). Aoki [3] and Rassias [4, 5] extended the Hyers-Ulam stability by considering variables for Cauchy equation. Using the method introduced in [3], Jung [6] obtained a result for Jensen equation. It also has been generalized to the function case by Găvruta [7] and Jung [8] for Cauchy equation, and by Lee and Jun [9] for Jensen equation.

Definition 1.1.

A mapping is called a Cauchy-Jensen mapping if satisfies the system of equations
(11)

When , the function given by is a solution of (1.1). In particular, letting , we get a function given by .

For a mapping , consider the functional equation

(12)

Definition 1.2 (see [10, 11]).

Let be a real linear space. A quasi-norm is real-valued function on X 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 .

The pair is called a quasi-normed space if is a quasi-norm on . The smallest possible is called the modulus of concavity of . A quasi-Banach space is a complete quasi-normed space. A quasi-norm is called a -norm ( ) if

(13)

for all . In this case, a quasi-Banach space is called a -Banach space.

The authors [12] obtained the solutions of (1.1) and (1.2) as follows.

Theorem 1 A.

A mapping satisfies (1.1) if and only if there exist a biadditive mapping and an additive mapping such that for all .

Theorem 1 B.

A mapping satisfies (1.1) if and only if it satisfies (1.2).

In this paper, we investigate the generalized Hyers-Ulam stability of (1.1) and (1.2).

2. Stability of (1.1) and (1.2)

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

Let and be two functions such that

(21)
(22)

for all , and

(23)
(24)

for all .

Theorem 2.1.

Suppose that a mapping satisfies the inequalities
(25)
(26)
for all . Then the limits
(27)
exist for all and the mappings and are Cauchy-Jensen mappings satisfying
(28)
(29)

for all .

Proof.

Letting and replacing by in (2.5) then,
(210)
for all . Replacing by in the above inequality and dividing by , we get
(211)
for all and all nonnegative integers . Since is a -Banach space, we have
(212)
for all and all nonnegative integers and with . Therefore we conclude from (2.3) and (2.12) that the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . So one can define the mapping by
(213)
for all . Letting and passing the limit in (2.12), we get (2.8). Now, we show that is a Cauchy-Jensen mapping. It follows from (2.1), (2.11), and (2.13) that
(214)

for all . So for all .

On the other hand it follows from (2.1), (2.5), (2.6), and (2.13) that

(215)
for all . Thus is a Cauchy-Jensen mapping. Next, setting in (2.6) and replacing by and by in (2.6), one can obtain that
(216)
respectively, for all . By two above inequalities,
(217)

for all . By the same method as above, one can find a Cauchy-Jensen mapping which satisfies (2.9). In fact, for all .

From now on, let be a function such that

(218)
(219)

for all .

We will use the following lemma in order to prove Theorem 2.3.

Lemma 2.2 (see [13]).

Let and let be nonnegative real numbers. Then
(220)

Theorem 2.3.

Suppose that a mapping satisfies and the inequality
(221)
for all . Then the limit exists for all and the mapping is the unique Cauchy-Jensen mapping satisfying
(222)
where
(223)

for all .

Proof.

Letting in (2.21), we get
(224)
for all . Putting and in (2.24), we get
(225)
for all . Replacing by and by in (2.24), we get
(226)
for all . By (2.25) and (2.26), we have
(227)
for all . Setting and in (2.24), we get
(228)
for all . By (2.27) and the above inequality, we get
(229)
for all . Replacing by in (2.26), we get
(230)
for all . By (2.29) and the above inequality, we have
(231)
for all . Replacing by in the above inequality, we get
(232)
for all . By (2.25) and the above inequality, we get
(233)
where
(234)
for all . Replacing by and by in the above inequality and dividing ,we get
(235)
for all and all nonnegative integers . Since is a -norm, we have
(236)
for all and all nonnegative integers and with . Therefore we conclude from (2.18) and (2.36) that the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . So one can define the mapping by
(237)
for all . Letting , passing the limit in (2.36), and applying lemma, we get (2.22). Now, we show that is a Cauchy-Jensen mapping. By lemma, we infer that for all . It follows from (2.18), (2.35), and the above equality that
(238)

for all . So for all .

On the other hand it follows from (2.18), (2.21), and (2.37) that

(239)
for all . Hence the mapping satisfies (1.2). To prove the uniqueness of , let be another Cauchy-Jensen mapping satisfying (2.22). It follows from (2.19) that
(240)
for all . Hence for all . So it follows from (2.22) and (2.37) that
(241)

for all . So .

Authors’ Affiliations

(1)
College of Liberal Arts, Kyung Hee University, Yongin, 446-701, South Korea
(2)
Division of Computational Sciences in Mathematics, National Institute for Mathematical Sciences, 385-16 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea

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Copyright

© J.-H. Bae and W.-G. Park 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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