- Research Article
- Open Access

# Stability of a Bi-Jensen Functional Equation II

- Kil-Woung Jun
^{1}Email author, - Il-Sook Jung
^{1}and - Yang-Hi Lee
^{2}Email author

**2009**:976284

https://doi.org/10.1155/2009/976284

© Kil-Woung Jun et al. 2009

**Received: **7 October 2008

**Accepted: **24 January 2009

**Published: **5 February 2009

## Abstract

## Keywords

## 1. Introduction

Găvruta [4] provided a further generalization of Rassias' theorem in which he replaced the bound by a general function.

Throughout this paper, let and be a normed space and a Banach space, respectively. A mapping is called a Cauchy mapping (resp., a Jensen mapping) if satisfies the functional equation (resp.,

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

Bae and Park [5] obtained the generalized Hyers-Ulam stability of a bi-Jensen mapping. Jun et al. [6] improved the results of Bae and Park in the spirit of Rassias.

In this paper, we investigate the stability of a bi-Jensen functional equation , in the spirit of Găvruta.

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

Jun et al. [7] established the basic properties of a bi-Jensen mapping in the following lemma.

Lemma 2.1.

Now we have the stability of a bi-Jensen mapping.

Theorem 2.2.

Proof.

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

Remark 2.3.

for all
. Then,
, and
satisfy (2.2), (2.3), (2.4) for all
. In addition,
satisfy (2.5) for all
and
also satisfy (2.5) for all
. But we get
. Hence, the condition
is necessary to show that the mapping *F* is unique.

We have another stability result applying for several cases.

Theorem 2.4.

Proof.

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

Theorem 2.5.

Proof.

Theorem 2.6.

*Let*be two functions satisfying (2.2) and (2.26) for all . Let be a mapping satisfying (2.4) for all . Then, there exists a bi-Jensen mapping satisfying

Theorem 2.7.

*Let*be two functions satisfying (2.3) and (2.25) for all . Let be a mapping satisfying (2.4) for all . Then, there exists a bi-Jensen mapping satisfying

Applying Theorems 2.2–2.7, we easily get the following corollaries.

Corollary 2.8.

Proof.

Applying Theorem 2.2 (Theorems 2.4 and 2.5, resp.) for the case ( and , resp.), we obtain the desired result.

Corollary 2.9.

Proof.

Applying Theorems 2.6 and 2.7, we obtain the desired result.

## Authors’ Affiliations

## References

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## Copyright

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.