# 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

## 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

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