Approximately Quadratic Mappings on Restricted Domains

Abstract

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.

1. Introduction

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 [1]. The case of approximately additive mappings was solved by Hyers [2] on Banach spaces. In 1950 Aoki [3] provided a generalization of the Hyers' theorem for additive mappings and in 1978 Th. M. Rassias [4] generalized the Hyers' theorem for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The result of Rassias' theorem has been generalized by Găvruţa [6] 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].

It is easy to see that the quadratic function is a solution of each of the following functional equations:

(1.1)

(1.2)

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 [30] was the first author to solve the Hyers-Ulam problem for additive mappings on a restricted domain (see also [31–33]). In 1998 Jung [34] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [35–37]). J. M. Rassias [38] investigated the Hyers-Ulam stability of mixed type mappings on restricted domains.

2. Solutions of (1.2)

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 .

Theorem 2.1.

Let and be real vector spaces and be an odd function satisfying (1.2). If is a rational number, then .

Proof.

Since is odd, . Letting (resp., ) in (1.2), we get

(2.1)

for all . Replacing by in (1.2) and adding the obtained functional equation to (1.2), we get

(2.2)

for all . Replacing by in (2.2) and using (2.1), we have

(2.3)

for all . Again if we replace by in (2.3) and use (2.1), we get

(2.4)

for all . Applying (1.2) and using the oddness of , we have

(2.5)

for all , in . So it follows from (2.4) and (2.5) that

(2.6)

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 .

Theorem 2.2.

Let and be real vector spaces and be an even function satisfying (1.2). Then satisfies (1.1).

Proof.

Letting in (1.2), we get . Replacing by in (1.2), we get

(2.7)

for all . Replacing by in (2.7) and using the evenness of , we get

(2.8)

for all , in . Adding (2.7) to (2.8), we obtain

(2.9)

for all . Replacing by in (2.7), we get

(2.10)

for all , in . Using (2.7) in (2.10), by a simple computation, we get

(2.11)

for all , in . Putting in (2.11), we get that for all . Therefore, it follows from (2.11) that

(2.12)

for all , in . Replacing by in (2.12), we get that for all . So satisfies (1.1).

Theorem 2.3.

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).

Proof.

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).

Conversely, let satisfy (1.1). Then there exists a unique symmetric biadditive function such that for all (see [13]). Therefore

(2.13)

for all . So satisfies (1.2).

Proposition 2.4.

Let be a linear space with the norm . is an inner product space if and only if there exists a real number such that

(2.14)

for all , where .

Proof.

Let be a function defined by . If is an inner product space, then satisfies (2.14) for all . Conversely, let and the (even) function satisfy (2.14). So satisfies (1.2). By Theorem 2.3, the function satisfies (1.1), that is,

(2.15)

for all . Therefore is an inner product space (see [14]).

Proposition 2.5.

Let and be a linear space with the norm . Suppose that

(2.16)

for all , in , where and . Then .

Proof.

Setting in (2.16), we get

(2.17)

for all in . If we take with in (2.17), we get that . Letting in (2.16), we get

(2.18)

for all in . Letting in (2.16), we get

(2.19)

for all in . Since , it follows from (2.17) and (2.19) that

(2.20)

for all . Using (2.18) and (2.20), we get for all . Hence and (2.18) implies that . Finally, follows from (2.19).

Corollary 2.6.

Let be a linear space with the norm . is an inner product space if and only if there exists a real number and such that

(2.21)

for all , where .

3. Stability of (1.2) on Restricted Domains

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 [39] was the first author who treats the Hyers-Ulam stability of the quadratic equation. Czerwik [8] proved a Hyers-Ulam-Rassias stability theorem on the quadratic equation. As a particular case he proved the following theorem.

Theorem 3.1.

Let be fixed. If a mapping satisfies the inequality

(3.1)

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 .

Theorem 3.2.

Let and be given. Assume that an even mapping satisfies the inequality

(3.2)

for all with . Then there exists such that satisfies

(3.3)

for all with .

Proof.

Let with . Then, since , we get . So it follows from (3.2) that

(3.4)

for all with . So

(3.5)

for all with .

Let with . We have two cases.

Case 1.

. Then .

Case 2.

. Then we have . So

(3.6)

Therefore we get that from Cases 1 and 2. Hence by (3.4) we have

(3.7)

for all with . Set . Then

(3.8)

for all with . From (3.4) and (3.5), we get the following inequalities:

(3.9)

Using (3.7) and the above inequalities, we get

(3.10)

for all with . If with , then . So it follows from (3.10) that

(3.11)

Letting in (3.11), we get

(3.12)

for all with . Letting (and with ) in (3.11), we get . Therefore it follows from (3.11) and (3.12) that

(3.13)

for all with . Since is even, the inequality (3.13) holds for all with . Therefore

(3.14)

for all with . This completes the proof by letting .

Theorem 3.3.

Let and be given. Assume that an even mapping satisfies the inequality (3.2) for all with . Then satisfies

(3.15)

for all .

Proof.

By Theorem 3.2 there exists such that satisfies (3.3) for all with and (see the proof of Theorem 3.2). Using Theorem 2 of [38], we get that

(3.16)

all .

Theorem 3.4.

Let and be given. Assume that an even mapping satisfies the inequality (3.2) for all with . Then there exists a unique quadratic mapping such that and

(3.17)

for all .

Proof.

The result follows from Theorems 3.1 and 3.3.

Skof [39] has proved an asymptotic property of the additive mappings and Jung [34] has proved an asymptotic property of the quadratic mappings (see also [36]). We prove such a property also for the quadratic mappings.

Corollary 3.5.

An even mapping satisfies (1.2) if and only if the asymptotic condition

(3.18)

holds true.

Proof.

By the asymptotic condition (3.18), there exists a sequence monotonically decreasing to 0 such that

(3.19)

for all with . Hence, it follows from (3.19) and Theorem 3.4 that there exists a unique quadratic mapping such that

(3.20)

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.

Corollary 3.6.

Let be rational. An even mapping is quadratic if and only if the asymptotic condition (3.18) holds true.