- Research Article
- Open Access
Hyers-Ulam Stability of a Bi-Jensen Functional Equation on a Punctured Domain
© G. H. Kim and Y.-H. Lee. 2010
- Received: 16 November 2009
- Accepted: 15 February 2010
- Published: 21 February 2010
We obtain the Hyers-Ulam stability of a bi-Jensen functional equation: and simultaneously . And we get its stability on the punctured domain.
- Banach Space
- Positive Integer
- Functional Equation
- Additive Mapping
- Cauchy Sequence
In 1940, Ulam  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  under the assumption that and are Banach spaces. In 1949, 1950, and 1978, Bourgin , Aoki , and Rassias  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
for all . A mapping is called a bi-Jensen mapping if satisfies the functional equations and .
Theorem 1 A.
for all .
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.
From Lemma 1 in , we get the following lemma.
for all and .
Now we will give the Hyers-Ulam stability for a bi-Jensen mapping.
for all .
for all and . As , we may conclude that for all . Thus the bi-Jensen mapping is unique.
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.
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
for all .
holds for all .
for all and .
Let be the set defined by for each . From the above equalities, we can define by
From the definition of , we get the equalities
hold for all and as we desired, where .
for all .
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 .
for all .
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
for all and . As , we may conclude that for all . By Lemma 3.1, as we desired.
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 .
for all .
for all .
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