# 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

**DOI: **10.1155/2009/976284

© Kil-Woung Jun et al. 2009

**Received: **7 October 2008

**Accepted: **24 January 2009

**Published: **5 February 2009

## Abstract

We investigate the stability of the bi-Jensen functional equation II in the spirit of Găvruta.

## 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.

*Let*be a bi-Jensen mapping. Then, the following equalities hold:

for all and .

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

Theorem 2.2.

*Let*be two functions satisfying

for all .

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.

*Let*be two functions satisfying

for all .

Proof.

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

Theorem 2.5.

*Let*be two functions satisfying

for all .

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

for all .

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

for all .

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

Corollary 2.8.

*Let*and . Let be a mapping such that

for all .

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.

*Let*( ), and . Let be a mapping such that

for all .

Proof.

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

## Authors’ Affiliations

## References

- Ulam SM:
*A Collection of Mathematical Problems*. Interscience, New York, NY, USA; 1968.MATHGoogle Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences of the United States of America*1941,**27**(4):222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72**(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar - Găvruta P:
**A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.***Journal of Mathematical Analysis and Applications*1994,**184**(3):431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar - Bae J-H, Park W-G:
**On the solution of a bi-Jensen functional equation and its stability.***Bulletin of the Korean Mathematical Society*2006,**43**(3):499–507.MathSciNetView ArticleMATHGoogle Scholar - Jun K-W, Jung I-S, Lee Y-H:
**Stability of a bi-Jensen functional equation.**preprint preprint - Jun K-W, Lee Y-H, Oh J-H:
**On the Rassias stability of a bi-Jensen functional equation.***Journal of Mathematical Inequalities*2008,**2**(3):363–375.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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