Hyers-Ulam Stability of a Bi-Jensen Functional Equation on a Punctured Domain

Abstract

We obtain the Hyers-Ulam stability of a bi-Jensen functional equation: and simultaneously . And we get its stability on the punctured domain.

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 1949, 1950, and 1978, Bourgin [3], Aoki [4], and Rassias [5] gave a generalization of it under the conditions bounded by variables. Since then, the further generalization has been extensively investigated by a number of mathematicians, such as Găvruta, Rassias, and so forth, [6–25].

Throughout this paper, let be a normed space and a Banach space. A mapping is called a Jensen mapping if satisfies the functional equation For a given mapping , we define

(1.3)

for all . A mapping is called a bi-Jensen mapping if satisfies the functional equations and .

In 2006, Bae and Park [26] obtained the generalized Hyers-Ulam stability of a bi-Jensen mapping. The following result is a special case of Theorem 6 in [26].

Theorem 1 A.

Let and let be a mapping such that

(1.4)

for all . Then there exist two bi-Jensen mappings such that

(1.5)

for all .

In Theorem A, they did not show that there exist a and a unique bi-Jensen mapping such that for all . In 2008, Jun et al. [7, 8] improved Bae and Park's results.

In Section 2, we show that there exists a unique bi-Jensen mapping such that for all . In Section 3, we investigate the Hyers-Ulam stability of a bi-Jensen functional equation on the punctured domain.

2. Stability of a Bi-Jensen Functional Equation

From Lemma 1 in [8], we get the following lemma.

Lemma.

Let be a bi-Jensen mapping. Then

(2.1)

for all and .

Now we will give the Hyers-Ulam stability for a bi-Jensen mapping.

Theorem 2.2.

Let and let be a mapping satisfying (1.4) for all . Then there exists a unique bi-Jensen mapping such that

(2.2)

for all with . In particular, the mapping is given by

(2.3)

for all .

Proof.

Let be the map defined by

(2.4)

for all and . By (1.4), we get

(2.5)

for all and . For given integers with , we obtain

(2.6)

for all . By the above inequality, the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define by

(2.7)

for all . Putting and taking in (2.6), we obtain the inequality

(2.8)

for all . By (1.4) and the definition of , we get

(2.9)

for all . So is a bi-Jensen mapping satisfying (2.2). Now, let be another bi-Jensen mapping satisfying (2.2) with . By Lemma 2.1, we have

(2.10)

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

Example 2.3.

Let be the bi-Jensen mappings defined by

(2.11)

for all . Then satisfy (1.4) for all . In addition, satisfy (2.2) for all and also satisfy (2.2) for all . But we get . Hence the condition is necessary to show that the mapping is unique.

3. Stability of a Bi-Jensen Functional Equation on the Punctured Domain

Let be a subset of . and are punctured domain on the spaces and , respectively.

Throughout this paper, for a given mapping , let be the mappings defined by

(3.1)

for all .

Lemma.

Let be a subset of satisfying the following condition: for every , there exists a positive integer such that for all integer with , and such that for all integer with . Let be a mapping such that

(3.2)

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

(3.3)

for all . Moreover, the equality

(3.4)

holds for all .

Proof.

Note that , , , and for all . Let for any . From (3.2), we get the equality

(3.5)

for all , and we know that the equality

(3.6)

holds for all . From (3.2), we have

(3.7)

for all . From the above equalities, we obtain the equalities

(3.8)

(3.9)

(3.10)

(3.11)

for all and .

Let be the set defined by for each . From the above equalities, we can define by

(3.12)

From the definition of , we get the equalities

(3.13)

for all . By (3.10), we get the equality

(3.14)

for all and , where . And also we get the equality

(3.15)

for all and , where . Hence the equality

(3.16)

holds for all . From (3.8), (3.9), (3.10), and the definition of , we easily get

(3.17)

for all . And we obtain

(3.18)

for all with , where and . From this, we have

(3.19)

for all with , where . From the above equalities, we get

(3.20)

for all . By the similar method, we have

(3.21)

for all . Hence is a bi-Jensen mapping. Let be another bi-Jensen mapping satisfying

(3.22)

for all . Using the above equality, we show that the equalities

(3.23)

hold for all and as we desired, where .

Corollary 3.2.

Let be a mapping such that

(3.24)

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

(3.25)

for all .

Example 3.3.

Let be the mapping defined by

(3.26)

and let be the mapping defined by for all . Then the mappings satisfy the conditions of Corollary 3.2 with .

Now, we prove the Hyers-Ulam stability of a bi-Jensen functional equation on the punctured domain .

Theorem 3.4.

Let and . Let be a mapping such that

(3.27)

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

(3.28)

holds for all with . The mapping is given by

(3.29)

for all .

Proof.

By (3.27), we get

(3.30)

for all and . For given integers (), we have

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

for all . The sequences , , and are Cauchy sequences for all . Since is complete, the above sequences converge for all . From (3.34) and (3.35), we have

(3.36)

for all . Using the inequalities (3.31)–(3.35) and the above equality, we can define the mappings by

(3.37)

for all . By (3.27) and the definition of , we obtain

(3.38)

for all . Since and

(3.39)

for all with , where , we have

(3.40)

for all . Similarly, the equalities

(3.41)

hold for all . By Lemma 3.1, There exist bi-Jensen mappings such that

(3.42)

for all . Since the equalities

(3.43)

hold, are bi-Jensen mappings. Putting and taking in (3.31), (3.32), and (3.33), one can obtain the inequalities

(3.44)

for all . By (3.30) and the above equalities, we get

(3.45)

for all , where is given by

(3.46)

and . By (3.45), we get the inequalities

(3.47)

for all and , where , and the inequalities

(3.48)

for all and . Hence is a bi-Jensen mapping satisfying (3.28).

Now, let be another bi-Jensen mapping satisfying (3.28) with . By Lemma 2.1, we have

(3.49)

for all and . As , we may conclude that for all . By Lemma 3.1, as we desired.

Example 3.5.

Let be the mapping defined by

(3.50)

Let be the mapping defined by for all . Then satisfies the conditions in Theorem 3.4, and is a bi-Jensen mapping satisfying (3.28) but .

Corollary 3.6.

Let be a mapping satisfying (3.13) and (3.27) for all . Then there exists a bi-Jensen mapping such that

(3.51)

for all .

Proof.

Let be as in the proof of Theorem 3.4. By (3.30), we obtain

(3.52)

for . From the above inequalities and (3.45), we get the inequality

(3.53)

for all .