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# Stability of a Bi-Jensen Functional Equation II

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 976284 (2009)

## Abstract

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

## 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 1978, Rassias [3] gave a generalization of Hyers' theorem by allowing the Cauchy difference to be controlled by a sum of powers like

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 a given mapping , we define

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 . Let be a mapping such that

for all . Then, there exists a unique bi-Jensen mapping such that

for all with , where

The mapping is given by

for all .

Proof.

Let be the maps defined by

for all . By (2.4), we get

for all and . For given integers (),

for all . By (2.2) and (2.3), the sequences , , and are Cauchy sequences for all . Since is complete, the sequences , , and converge for all . Define by

for all . Putting and taking in (2.10), one can obtain the inequalities

for all . By (2.4) and the definitions of and , we get

for all . So is a bi-Jensen mapping satisfying (2.5), where is given by

Now, let be another bi-Jensen mapping satisfying (2.5) with . By Lemma 2.1, we have

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

Remark 2.3.

Let be the functions defined by

for all . Let be the bi-Jensen mappings defined by

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 . Let be a mapping satisfying (2.4) for all . Then, there exists a unique bi-Jensen mapping satisfying

for all . The mapping is given by

for all .

Proof.

By (2.4) and the similar method in Theorem 2.2, we define the maps by

for all . By (2.4) and the definitions of , , and , we get

for all . By the similar method in Theorem 2.2, is a bi-Jensen mapping satisfying (2.19), where is given by

Now, let be another bi-Jensen mapping satisfying (2.19). Using Lemma 2.1, , and , we have

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 . Let be a mapping satisfying (2.4) for all . Then, there exists a bi-Jensen mapping satisfying

for all , where the mapping is given by

for all .

Proof.

We can obtain as in Theorem 2.4 and as in Theorem 2.2. Hence, is a bi-Jensen mapping satisfying (2.27), where is given by

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 , where the mapping is given by

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 , where the mapping is given by

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 . Then, there exists a unique bi-Jensen 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 . Then, there exists a unique bi-Jensen mapping such that

for all .

Proof.

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

## References

Ulam SM:

*A Collection of Mathematical Problems*. Interscience, New York, NY, USA; 1968.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.27.4.222Rassias ThM:

**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72**(2):297–300. 10.1090/S0002-9939-1978-0507327-1Găvruta 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.1994.1211Bae 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.Jun K-W, Jung I-S, Lee Y-H:

**Stability of a bi-Jensen functional equation.**preprint preprintJun 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.

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Jun, KW., Jung, IS. & Lee, YH. Stability of a Bi-Jensen Functional Equation II.
*J Inequal Appl* **2009**, 976284 (2009). https://doi.org/10.1155/2009/976284

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