- Research Article
- Open Access
Approximately Quadratic Mappings on Restricted Domains
© Abbas Najati and Soon-Mo Jung. 2010
- Received: 16 September 2010
- Accepted: 20 December 2010
- Published: 23 December 2010
We introduce a generalized quadratic functional equation , where , are nonzero real numbers with . We show that this functional equation is quadratic if , are rational numbers. We also investigate its stability problem on restricted domains. These results are applied to study of an asymptotic behavior of these generalized quadratic mappings.
- Functional Equation
- Linear Space
- Quadratic Mapping
- Additive Mapping
- Asymptotic Property
Under what conditions does there exist a group homomorphism near an approximate group homomorphism? This question concerning the stability of group homomorphisms was posed by Ulam . The case of approximately additive mappings was solved by Hyers  on Banach spaces. In 1950 Aoki  provided a generalization of the Hyers' theorem for additive mappings and in 1978 Th. M. Rassias  generalized the Hyers' theorem for linear mappings by allowing the Cauchy difference to be unbounded (see also ). The result of Rassias' theorem has been generalized by Găvruţa  who permitted the Cauchy difference to be bounded by a general control function. This stability concept is also applied to the case of other functional equations. For more results on the stability of functional equations, see [7–24]. We also refer the readers to the books in [25–29].
where , are nonzero real numbers with . 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 [13, 25, 27]).
We prove that the functional equations (1.1) and (1.2) are equivalent if , are nonzero rational numbers. The functional equation (1.1) is a spacial case of (1.2). Indeed, for the case in (1.2), we get (1.1).
In 1983 Skof  was the first author to solve the Hyers-Ulam problem for additive mappings on a restricted domain (see also [31–33]). In 1998 Jung  investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [35–37]). J. M. Rassias  investigated the Hyers-Ulam stability of mixed type mappings on restricted domains.
In this section we show that the functional equation (1.2) is equivalent to the quadratic equation (1.1). That is, every solution of (1.2) is a quadratic function. We recall that , are nonzero real numbers with .
Let and be real vector spaces and be an odd function satisfying (1.2). If is a rational number, then .
for all , in . It easily follows from (2.6) that is additive, that is, for all . So if is a rational number, then for all in . Therefore, it follows from (2.1) that for all in . Since , are nonzero, we infer that .
Let and be real vector spaces and be an even function satisfying (1.2). Then satisfies (1.1).
for all , in . Replacing by in (2.12), we get that for all . So satisfies (1.1).
Let be a function between real vector spaces and . If is a rational number, then satisfies (1.2) if and only if satisfies (1.1).
Let and be the odd and the even parts of . Suppose that satisfies (1.2). It is clear that and satisfy (1.2). By Theorems 2.1 and 2.2, and satisfies (1.1). Since , we conclude that satisfies (1.1).
for all . So satisfies (1.2).
for all , where .
for all . Therefore is an inner product space (see ).
for all , in , where and . Then .
for all . Using (2.18) and (2.20), we get for all . Hence and (2.18) implies that . Finally, follows from (2.19).
for all , where .
In this section, we investigate the Hyers-Ulam stability of the functional equation (1.2) on a restricted domain. As an application we use the result to the study of an asymptotic behavior of that equation. It should be mentioned that Skof  was the first author who treats the Hyers-Ulam stability of the quadratic equation. Czerwik  proved a Hyers-Ulam-Rassias stability theorem on the quadratic equation. As a particular case he proved the following theorem.
for all , then there exists a unique quadratic mapping such that for all . Moreover, if is measurable or if is continuous in for each fixed , then for all and .
We recall that , are nonzero real numbers with .
for all with .
for all with .
Let with . We have two cases.
for all with . This completes the proof by letting .
for all .
for all .
The result follows from Theorems 3.1 and 3.3.
Skof  has proved an asymptotic property of the additive mappings and Jung  has proved an asymptotic property of the quadratic mappings (see also ). We prove such a property also for the quadratic mappings.
for all . Since is a monotonically decreasing sequence, the quadratic mapping satisfies (3.20) for all . The uniqueness of implies for all . Hence, by letting in (3.20), we conclude that is quadratic.
Let be rational. An even mapping is quadratic if and only if the asymptotic condition (3.18) holds true.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2010-0007143).
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