- Research Article
- Open Access
© Mohammad B. Moghimi et al. 2009
- Received: 18 May 2009
- Accepted: 31 July 2009
- Published: 19 August 2009
- Banach Space
- Functional Equation
- Positive Element
- Group Homomorphism
- Real Vector Space
for all . Aoki  and Th. M. Rassias  provided a generalization of the Hyers' theorem for additive mappings and for linear mappings, respectively, by allowing the Cauchy difference to be unbounded (see also ).
Theorem 1.1 (Th. M. Rassias).
The Hyers-Ulam stability problem for the quadratic functional equation (1.5) was studied by Skof  for mappings where is a normed space and is a Banach space. Cholewa  noticed that the theorem of Skof is still true if we replace by an Abelian group. Czerwik  proved the generalized Hyers-Ulam stability of the quadratic functional equation (1.5). Grabiec  has generalized these results mentioned above. Jun and Lee  proved the generalized Hyers-Ulam stability of a Pexiderized quadratic functional equation.
We recall the following theorem by Margolis and Diaz.
Theorem 1.2 (see ).
Let be a linear space. If a mapping satisfies and the functional equation (1.8), then is quadratic.
Let be a linear space. If a mapping satisfies the functional equation (1.8), then there exists a symmetric biadditive mapping such that for all
is a generalized complete metric space .
Moreover, if is continuous in for each fixed , then by the same reasoning as in the proof of  is -quadratic.
The following theorem is an alternative result of Theorem 3.1 and we will omit the proof.
which contradicts (3.33).
The third author was supported by Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00041).
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