On the Asymptoticity Aspect of Hyers-Ulam Stability of Quadratic Mappings

Abstract

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.

1. Introduction

The question concerning the stability of group homomorphisms was posed by Ulam [1]. Hyers [2] solved the case of approximately additive mappings on Banach spaces. Aoki [3] provided a generalization of the Hyers' theorem for additive mappings. In [4], Rassias generalized the result of Hyers for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The result of Rassias has been generalized by Gvruţa [6] 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].

It is easy to see that the function defined by with an arbitrary constant is a solution of the functional equation

(1..1)

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 [38] for functions , where is a normed space and is a Banach space. Cholewa [11] noticed that the result of Skof holds (with the same proof) if is replaced by an abelian group . In [12], Czerwik generalized the result of Skof by allowing growth of the form for the norm of , where and . In 1998, Jung [39] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [40–42]). Rassias [43] investigated the Hyers-Ulam stability of mixed type mappings on restricted domains. In [44], 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.

Theorem 2.1.

Given a real normed vector space and a real Banach space , let and with be fixed. If a mapping satisfies the inequality

(2.1)

for all such that , where , then there exists a unique quadratic mapping such that

(2.2)

for all with and . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

Proof.

Letting in (2.1), we get

(2.3)

for all with . If we put with and in (2.1), we obtain

(2.4)

It follows from (2.3) and (2.4) that

(2.5)

for all with . Replacing by in (2.5), we infer the inequality

(2.6)

for all with and all integers . Therefore,

(2.7)

for all with and all integers . It follows from (2.7) that the sequence converges for all with . Let us denote for all with . It is clear that

(2.8)

for all with . Letting and in (2.7), we get

(2.9)

for all with .

Now, suppose that such that , then by (2.1) and the definition of , we obtain

(2.10)

We have to extend the mapping to the whole space . Given any with , let denote the largest integer such that . Consider the mapping defined by and

(2.11)

Let with and let . We have two cases.

Case 1.

If , we have from (2.8) that

(2.12)

Case 2.

If , then is the largest integer satisfying , and we have

(2.13)

Therefore, for all with . From the definition of and (2.8), it follows that for all . Now, suppose that with and choose a positive integer such that . By the definition of and its property, we have

(2.14)

So by the definition of , we have

(2.15)

for all with . Since , (2.15) holds true for . Let with . It follows from (2.1) and (2.15) that

(2.16)

Letting in (2.16), we get for all with . Since , the same is true for . So, is even and this implies that (2.16) is true for all . Therefore, is quadratic. By the definition when , thus (2.2) follows from (2.9). To prove the uniqueness of , let be another quadratic mapping satisfying (2.2) for all . Let with and choose a positive integer such that , then

(2.17)

for all . Since and are quadratic, we get

(2.18)

for all . Therefore, . Since , we have for all . The proof of our last assertion follows from the proof of Theorem  1 in [12].

We now introduce one of the fundamental results of fixed point theory by Margolis and Diaz.

Theorem 2.2 (see [22]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . If there exists a nonnegative integer such that for some , then the following are true:

(1)the sequence converges to a fixed point of ,

(2) is the unique fixed point of in

(2.19)

(3) for all .

By using the idea of Cădariu and Radu [45], we applied a fixed point method to the investigation of the generalized Hyers-Ulam stability of the functional equation (1.1) on a restricted domain.

Theorem 2.3.

Given a real normed vector space and a real Banach space , let be fixed and let be a mapping which satisfies the inequality (2.1) for all , where is a function such that

(2.20)

for all , where is a constant number, then there exists a unique quadratic mapping such that

(2.21)

for all with , where

(2.22)

and for all . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

Proof.

It follows from (2.20) that

(2.23)

for all . Let . Letting for in (2.1), we get the following inequalities:

(2.24)

(2.25)

(2.26)

(2.27)

It follows from (2.24) and (2.25) that

(2.28)

By (2.26) and (2.27), we have

(2.29)

It follows from (2.25) and (2.29) that

(2.30)

Using (2.28) and (2.30), we have

(2.31)

By (2.24), we get

(2.32)

Hence, we obtain from (2.31) and (2.32) that

(2.33)

So, it follows from (2.28) and (2.33) that

(2.34)

for all . Let . We introduce a generalized metric on as follows:

(2.35)

We assert that is a generalized complete metric space. Let be a Cauchy sequence in and be given, then there exists an integer such that for all . This implies that for all and all . Therefore, is a Cauchy sequence in for all . Since is a Banach space, converges for all . Thus, we can define a function by

(2.36)

Since

(2.37)

for all and all , we get for all . That is, the Cauchy sequence converges to in . Hence, is complete. We now consider the mapping defined by

(2.38)

Let and let be an arbitrary constant with . From the definition of , we have

(2.39)

for all . By the assumption (2.20) and the last inequality, we have

(2.40)

for all . So . That is, is a strictly contractive on . It follows from (2.34) that . Therefore, according to Theorem 2.2, there exists a function such that the sequence converges to and . Indeed,

(2.41)

and , for all . Also, is the unique fixed point of in the set and

(2.42)

By (2.1), (2.23) and using the definition of , we get

(2.43)

for all . We will define a mapping such that . Similar to the proof of Theorem 2.1 for a given with , let denote the largest integer such that . Consider the mapping defined by and

(2.44)

Let with and let . We have two cases.

Case 1.

. Since for all , we have

(2.45)

Case 2.

If , then is the largest integer satisfying , and we have

(2.46)

Therefore, for all with . Using for all and the definition of , we get that for all . Now, suppose that with and choose a positive integer such that . By the definition of and its property, we have

(2.47)

So by the definition of , we have

(2.48)

for all with . Since , (2.48) holds true for . Let with , , . It follows from (2.1), (2.23), and (2.48) that

(2.49)

Since and for all , we conclude that (2.49) is true for all . Let with . Putting in (2.49), we get . Therefore, by letting in (2.49), we get for all with . Since , the same is true for . So, is even and this implies that (2.49) is true for all . Therefore, is quadratic. To prove the uniqueness of , let be another quadratic mapping satisfying (2.21), for all . Let with and choose a positive integer such that , then

(2.50)

for all . Since and are quadratic, we get

(2.51)

for all . Therefore, (2.23) implies that . Since , we have for all . Our last assertion is trivial in view of Theorem 2.1.

Corollary 2.4.

Given a real normed vector space and a real Banach space , let and with be fixed. Suppose that a mapping satisfies the inequality (2.1) for all , then there exists a unique quadratic mapping such that

(2.52)

for all with and . Moreover, if is measurable or if is continuous in for each fixed , then for all and .

Remark 2.5.

We may replace the condition (2.20) by

(2.53)

for all and . Using the direct method, there exists a unique quadratic mapping such that

(2.54)

for all with . For the case , where and , we have

(2.55)

Using ideas from the papers [39, 43], we prove the generalized Hyers-Ulam stability of (1.1) on restricted domains. We first prove the following lemma.

Lemma 2.6.

Given a real normed vector space and a real Banach space , let and be fixed. If a mapping satisfies the inequality

(2.56)

for all with , then

(2.57)

for all , where

(2.58)

Proof.

Assume that . If , then we choose a with . Otherwise, let

(2.59)

It is clear that and

(2.60)

Also

(2.61)

Therefore,

(2.62)

So, we get

(2.63)

So, satisfies (2.57) for all .

Theorem 2.7.

Given a real normed vector space and a real Banach space , let and with be given. Assume that a mapping satisfies the inequality (2.56) for all with , then there exists a unique quadratic mapping such that and

(2.64)

for all .

Proof.

By Lemma 2.6, satisfies (2.57) for all . Letting in (2.57), we get

(2.65)

for all , where

(2.66)

We can use the argument given in the proof of Theorem 2.1 to arrive the inequality

(2.67)

for all and all integers . It follows from (2.67) that the sequence converges for all . So, we can define the mapping by for all . Letting and in (2.67), we get (2.64).

For the case and in Theorem 2.7, it is obvious that our inequality (2.64) is sharper than the corresponding inequalities of Jung [39] and Rassias [43].

Skof [38] has proved an asymptotic property of the additive mappings, and Jung [39] has proved an asymptotic property of the quadratic mappings (see also [41]). Using the method in [39], the proof of the following corollary follows from Theorem 2.7 by letting and .

Corollary 2.8 (see [39]).

Given a real normed vector space and a real Banach space , a mapping satisfies (1.1) if and only if the asymptotic condition

(2.68)

holds true.

3. -Asymptotically Quadratic Mappings

We apply our results to the study of -asymptotical derivatives. Let be a real normed vector space and let be a real Banach space . Let be arbitrary.

Definition 3.1.

A mapping is called -asymptotically close to a mapping if and only if .

Definition 3.2.

A mapping is called -asymptotically derivable if the mapping is -asymptotically close to a quadratic mapping . In this case, we say that is a -asymptotical derivative of .

Definition 3.3.

A mapping is called -asymptotically quadratic if and only if, for every , there exists such that

(3.1)

for all with .

Definition 3.4.

A mapping is called quadratic outside a ball if there exists such that for all with .

We have the following result.

Theorem 3.5.

If is quadratic outside a ball and is -asymptotically close to , then is -asymptotically quadratic.

The following result follows from Corollary 2.4.

Corollary 3.6.

If is quadratic outside a ball and is -asymptotically close to , then has a -asymptotical derivative.