- 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
- 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 ? 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].
Theorem 1 A.
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.
Now we will give the Hyers-Ulam stability for a bi-Jensen mapping.
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