- 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
We obtain the generalized Hyers-Ulam stability of the Cauchy-Jensen functional equation .
- Banach Space
- Functional Equation
- Nonnegative Integer
- Cauchy Sequence
- Affine Mapping
In 1940, Ulam proposed the general Ulam stability problem (see ).
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 ?
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.
When , the function given by is a solution of (1.1). In particular, letting , we get a function given by .
For a mapping , consider the functional equation
Let be a real linear space. A quasi-norm is real-valued function on X satisfying the following.
(i) for all and if and only if .
(ii) for all and all .
(iii)There is a constant such that for all .
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
for all . In this case, a quasi-Banach space is called a -Banach space.
The authors  obtained the solutions of (1.1) and (1.2) as follows.
Theorem 1 A.
A mapping satisfies (1.1) if and only if there exist a biadditive mapping and an additive mapping such that for all .
Theorem 1 B.
A mapping satisfies (1.1) if and only if it satisfies (1.2).
In this paper, we investigate the generalized Hyers-Ulam stability of (1.1) and (1.2).
Throughout this section, assume that is a quasi-normed space with quasi-norm and that is a -Banach space with -norm . Let be the modulus of concavity of .
Let and be two functions such that
for all , and
for all .
for all .
for all . So for all .
On the other hand it follows from (2.1), (2.5), (2.6), and (2.13) that
for all . By the same method as above, one can find a Cauchy-Jensen mapping which satisfies (2.9). In fact, for all .
From now on, let be a function such that
for all .
We will use the following lemma in order to prove Theorem 2.3.
Lemma 2.2 (see ).
for all .
for all . So for all .
On the other hand it follows from (2.18), (2.21), and (2.37) that
for all . So .
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