- Research Article
- Open Access

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

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

**2010**:151547

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

© J.-H. Bae and W.-G. Park 2010

**Received:**16 October 2009**Accepted:**30 January 2010**Published:**2 February 2010

## 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.

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

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

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

for all , and

for all .

Theorem 2.1.

for all .

Proof.

for all . So for all .

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

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

for all .

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

Lemma 2.2 (see [13]).

Theorem 2.3.

for all .

Proof.

for all . So for all .

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

for all . So .

## Authors’ Affiliations

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