Open Access

Stability of a Bi-Jensen Functional Equation II

Journal of Inequalities and Applications20092009:976284

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

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

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
(1.1)
for all , then there is a homomorphism with
(1.2)
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
(1.3)

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
(1.4)

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:
(2.1)

for all and .

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

Theorem 2.2.

Let be two functions satisfying
(2.2)
(2.3)
for all . Let be a mapping such that
(2.4)
for all . Then, there exists a unique bi-Jensen mapping such that
(2.5)
for all with , where
(2.6)
The mapping is given by
(2.7)

for all .

Proof.

Let be the maps defined by
(2.8)
for all . By (2.4), we get
(2.9)
for all and . For given integers ( ),
(2.10)
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
(2.11)
for all . Putting and taking in (2.10), one can obtain the inequalities
(2.12)
for all . By (2.4) and the definitions of and , we get
(2.13)
for all . So is a bi-Jensen mapping satisfying (2.5), where is given by
(2.14)
Now, let be another bi-Jensen mapping satisfying (2.5) with . By Lemma 2.1, we have
(2.15)

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
(2.16)
for all . Let be the bi-Jensen mappings defined by
(2.17)

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
(2.18)
for all . Let be a mapping satisfying (2.4) for all . Then, there exists a unique bi-Jensen mapping satisfying
(2.19)
for all . The mapping is given by
(2.20)

for all .

Proof.

By (2.4) and the similar method in Theorem 2.2, we define the maps by
(2.21)
for all . By (2.4) and the definitions of , , and , we get
(2.22)
for all . By the similar method in Theorem 2.2, is a bi-Jensen mapping satisfying (2.19), where is given by
(2.23)
Now, let be another bi-Jensen mapping satisfying (2.19). Using Lemma 2.1, , and , we have
(2.24)

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
(2.25)
(2.26)
for all . Let be a mapping satisfying (2.4) for all . Then, there exists a bi-Jensen mapping satisfying
(2.27)
for all , where the mapping is given by
(2.28)

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
(2.29)

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
(2.30)
for all , where the mapping is given by
(2.31)

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
(2.32)
for all , where the mapping is given by
(2.33)

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
(2.34)
for all . Then, there exists a unique bi-Jensen mapping such that
(2.35)

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
(2.36)
for all . Then, there exists a unique bi-Jensen mapping such that
(2.37)

for all .

Proof.

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

Authors’ Affiliations

(1)
Department of Mathematics, Chungnam National University
(2)
Department of Mathematics Education, Gongju National University of Education

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Copyright

© Kil-Woung Jun et al. 2009

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