Stability of a Bi-Jensen Functional Equation II

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.