- Research Article
- Open Access
A Fixed Point Approach to the Stability of a Quadratic Functional Equation in -Algebras
© Mohammad B. Moghimi et al. 2009
- Received: 18 May 2009
- Accepted: 31 July 2009
- Published: 19 August 2009
We use a fixed point method to investigate the stability problem of the quadratic functional equation in -algebras.
- 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).
for all . If then inequality (1.2) holds for and (1.4) for . Also, if for each the mapping is continuous in , then is -linear.
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.
Let be a set. A function is called a generalized metric on if satisfies
if and only if ;
for all ;
We recall the following theorem by Margolis and Diaz.
Theorem 1.2 (see ).
for all nonnegative integers or there exists a non-negative integer such that
for all ;
the sequence converges to a fixed point of ;
is the unique fixed point of in the set ;
for all .
Let be a linear space. If a mapping satisfies and the functional equation (1.8), then is quadratic.
for all Hence is quadratic.
A quadratic mapping does not satisfy (1.8) in general. Let be the mapping defined by for all It is clear that is quadratic and that does not satisfy (1.8).
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
for all where is a linear space.
Moreover, if is continuous in for each fixed , then is -quadratic, that is, for all and all
is a generalized complete metric space .
Let be the mapping defined by
for all By Theorem 2.1, the function is quadratic.
Moreover, if is continuous in for each fixed , then by the same reasoning as in the proof of  is -quadratic.
for all . Moreover, if is continuous in for each fixed , then is -quadratic.
The following theorem is an alternative result of Theorem 3.1 and we will omit the proof.
for all , where is defined as in Theorem 3.1. Moreover, if is continuous in for each fixed , then is -quadratic.
for all . Moreover, if is continuous in for each fixed , then is -quadratic.
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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