- Research Article
- Open Access

# Hyers-Ulam Stability of a Bi-Jensen Functional Equation on a Punctured Domain

- GwangHui Kim
^{1}Email author and - Yang-Hi Lee
^{2}

**2010**:476249

https://doi.org/10.1155/2010/476249

© G. H. Kim and Y.-H. Lee. 2010

**Received:**16 November 2009**Accepted:**15 February 2010**Published:**21 February 2010

## Abstract

We obtain the Hyers-Ulam stability of a bi-Jensen functional equation: and simultaneously . And we get its stability on the punctured domain.

## Keywords

- Banach Space
- Positive Integer
- Functional Equation
- Additive Mapping
- Cauchy Sequence

## 1. Introduction

In 1940, Ulam [1] raised a question concerning the stability of homomorphisms: 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 ? The case of approximately additive mappings was solved by Hyers [2] under the assumption that and are Banach spaces. In 1949, 1950, and 1978, Bourgin [3], Aoki [4], and Rassias [5] gave a generalization of it under the conditions bounded by variables. Since then, the further generalization has been extensively investigated by a number of mathematicians, such as Găvruta, Rassias, and so forth, [6–25].

Throughout this paper, let be a normed space and a Banach space. A mapping is called a Jensen mapping if satisfies the functional equation For a given mapping , we define

for all . A mapping is called a bi-Jensen mapping if satisfies the functional equations and .

In 2006, Bae and Park [26] obtained the generalized Hyers-Ulam stability of a bi-Jensen mapping. The following result is a special case of Theorem 6 in [26].

Theorem 1 A.

for all .

In Theorem A, they did not show that there exist a and a unique bi-Jensen mapping such that for all . In 2008, Jun et al. [7, 8] improved Bae and Park's results.

In Section 2, we show that there exists a unique bi-Jensen mapping such that for all . In Section 3, we investigate the Hyers-Ulam stability of a bi-Jensen functional equation on the punctured domain.

## 2. Stability of a Bi-Jensen Functional Equation

From Lemma 1 in [8], we get the following lemma.

Lemma.

for all and .

Now we will give the Hyers-Ulam stability for a bi-Jensen mapping.

Theorem 2.2.

for all .

Proof.

for all and . As , we may conclude that for all . Thus the bi-Jensen mapping is unique.

Example 2.3.

for all . Then satisfy (1.4) for all . In addition, satisfy (2.2) for all and also satisfy (2.2) for all . But we get . Hence the condition is necessary to show that the mapping is unique.

## 3. Stability of a Bi-Jensen Functional Equation on the Punctured Domain

Let be a subset of . and are punctured domain on the spaces and , respectively.

Throughout this paper, for a given mapping , let be the mappings defined by

for all .

Lemma.

holds for all .

Proof.

for all and .

Let be the set defined by for each . From the above equalities, we can define by

From the definition of , we get the equalities

hold for all and as we desired, where .

Corollary 3.2.

for all .

Example 3.3.

and let be the mapping defined by for all . Then the mappings satisfy the conditions of Corollary 3.2 with .

Now, we prove the Hyers-Ulam stability of a bi-Jensen functional equation on the punctured domain .

Theorem 3.4.

for all .

Proof.

for all and . Hence is a bi-Jensen mapping satisfying (3.28).

Now, let be another bi-Jensen mapping satisfying (3.28) with . By Lemma 2.1, we have

for all and . As , we may conclude that for all . By Lemma 3.1, as we desired.

Example 3.5.

Let be the mapping defined by for all . Then satisfies the conditions in Theorem 3.4, and is a bi-Jensen mapping satisfying (3.28) but .

Corollary 3.6.

for all .

Proof.

for all .

## Authors’ Affiliations

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