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

*Journal of Inequalities and Applications*
**volumeÂ 2010**, ArticleÂ number:Â 151547 (2010)

## Abstract

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

## 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 GÄƒvruta [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

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.

Suppose that a mapping satisfies the inequalities

for all . Then the limits

exist for all and the mappings and are Cauchy-Jensen mappings satisfying

for all .

Proof.

Letting and replacing by in (2.5) then,

for all . Replacing by in the above inequality and dividing by , we get

for all and all nonnegative integers . Since is a -Banach space, we have

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

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

for all . So for all .

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

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

respectively, for all . By two above inequalities,

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]).

Let and let be nonnegative real numbers. Then

Theorem 2.3.

Suppose that a mapping satisfies and the inequality

for all . Then the limit exists for all and the mapping is the unique Cauchy-Jensen mapping satisfying

where

for all .

Proof.

Letting in (2.21), we get

for all . Putting and in (2.24), we get

for all . Replacing by and by in (2.24), we get

for all . By (2.25) and (2.26), we have

for all . Setting and in (2.24), we get

for all . By (2.27) and the above inequality, we get

for all . Replacing by in (2.26), we get

for all . By (2.29) and the above inequality, we have

for all . Replacing by in the above inequality, we get

for all . By (2.25) and the above inequality, we get

where

for all . Replacing by and by in the above inequality and dividing ,we get

for all and all nonnegative integers . Since is a -norm, we have

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

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

for all . So for all .

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

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

for all . Hence for all . So it follows from (2.22) and (2.37) that

for all . So .

## References

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Bae, JH., Park, WG. Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces.
*J Inequal Appl* **2010**, 151547 (2010). https://doi.org/10.1155/2010/151547

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