- Research Article
- Open Access
Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces
© J.-H. Bae and W.-G. Park 2010
- Received: 16 October 2009
- Accepted: 30 January 2010
- Published: 2 February 2010
- Banach Space
- Functional Equation
- Nonnegative Integer
- Cauchy Sequence
- Affine Mapping
In 1940, Ulam proposed the general Ulam stability problem (see ).
In 1941, this problem was solved by Hyers  in the case of Banach space. Thereafter, we call that type the Hyers-Ulam stability.
Throughout this paper, let and be vector spaces. A mapping is called an additive mapping (respectively, an affine mapping) if satisfies the Cauchy functional equation (respectively, the Jensen functional equation ). Aoki  and Rassias [4, 5] extended the Hyers-Ulam stability by considering variables for Cauchy equation. Using the method introduced in , Jung  obtained a result for Jensen equation. It also has been generalized to the function case by Găvruta  and Jung  for Cauchy equation, and by Lee and Jun  for Jensen equation.
The pair is called a quasi-normed space if is a quasi-norm on . The smallest possible is called the modulus of concavity of . A quasi-Banach space is a complete quasi-normed space. A quasi-norm is called a -norm ( ) if
The authors  obtained the solutions of (1.1) and (1.2) as follows.
Theorem 1 A.
Theorem 1 B.
In this paper, we investigate the generalized Hyers-Ulam stability of (1.1) and (1.2).
On the other hand it follows from (2.1), (2.5), (2.6), and (2.13) that
We will use the following lemma in order to prove Theorem 2.3.
Lemma 2.2 (see ).
On the other hand it follows from (2.18), (2.21), and (2.37) that
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