On the Asymptoticity Aspect of Hyers-Ulam Stability of Quadratic Mappings
© A. Rahimi et al. 2010
Received: 30 September 2010
Accepted: 27 December 2010
Published: 5 January 2011
We investigate the Hyers-Ulam stability of the quadratic functional equation on restricted domains. Applying these results, we study of an asymptotic behavior of these quadratic mappings.
The question concerning the stability of group homomorphisms was posed by Ulam . Hyers  solved the case of approximately additive mappings on Banach spaces. Aoki  provided a generalization of the Hyers' theorem for additive mappings. In , Rassias generalized the result of Hyers for linear mappings by allowing the Cauchy difference to be unbounded (see also ). The result of Rassias has been generalized by G vruţa  who permitted the norm of the Cauchy difference to be bounded by a general control function under some conditions. This stability concept is also applied to the case of various functional equations by a number of authors. For more results on the stability of functional equations, see [7–32]. We also refer the readers to the books [33–37].
So, it is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a function between real vector spaces and is quadratic if and only if there exists a unique symmetric biadditive function such that for all (see [21, 33, 35]).
A stability theorem for the quadratic functional equation (1.1) was proved by Skof  for functions , where is a normed space and is a Banach space. Cholewa  noticed that the result of Skof holds (with the same proof) if is replaced by an abelian group . In , Czerwik generalized the result of Skof by allowing growth of the form for the norm of , where and . In 1998, Jung  investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [40–42]). Rassias  investigated the Hyers-Ulam stability of mixed type mappings on restricted domains. In , the authors considered the asymptoticity of Hyers-Ulam stability close to the asymptotic derivability.
2. Stability of (1.1) on Restricted Domains
In this section, we investigate the Hyers-Ulam stability of the functional equation (1.1) on a restricted domain. As an application, we use the result to the study of an asymptotic behavior of that equation.
for all . Therefore, . Since , we have for all . The proof of our last assertion follows from the proof of Theorem 1 in .
We now introduce one of the fundamental results of fixed point theory by Margolis and Diaz.
Theorem 2.2 (see ).
Skof  has proved an asymptotic property of the additive mappings, and Jung  has proved an asymptotic property of the quadratic mappings (see also ). Using the method in , the proof of the following corollary follows from Theorem 2.7 by letting and .
Corollary 2.8 (see ).
We have the following result.
The following result follows from Corollary 2.4.
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