Research

# Stability of quadratic functional equations in generalized functions

Young-Su Lee

Author Affiliations

Advances in Difference Equations 2013, 2013:72  doi:10.1186/1687-1847-2013-72

 Received: 16 August 2012 Accepted: 4 March 2013 Published: 22 March 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we consider the following generalized quadratic functional equation with n-independent variables in the spaces of generalized functions:

Making use of the fundamental solution of the heat equation, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the Dirac sequence of regularizing functions, we extend these results to the space of distributions.

MSC: 39B82, 46F05.

##### Keywords:
quadratic functional equation; stability; generalized function; heat kernel; Gauss transform; distribution

### 1 Introduction

In 1940, Ulam [1] raised a question concerning the stability of group homomorphisms as follows: The case of approximately additive mappings was solved by Hyers [2] under the assumption that is a Banach space. In 1978, Rassias [3] generalized Hyers’ result to the unbounded Cauchy difference. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [4-6]).

Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?

Quadratic functional equations are used to characterize the inner product spaces. Note that a square norm on an inner product space satisfies the parallelogram equality

for all vectors x, y. By virtue of this equality, the following functional equation is induced:

(1.1)

It is easy to see that the quadratic function on a real field, where a is an arbitrary constant, is a solution of (1.1). Thus, it is natural that (1.1) is called a quadratic functional equation. It is well known that a function f between real vector spaces satisfies (1.1) if and only if there exists a unique symmetric biadditive function B such that (see [4-7]). The biadditive function B is given by

The Hyers-Ulam stability for quadratic functional equation (1.1) was proved by Skof [8]. Thereafter, many authors studied the stability problems of (1.1) in various settings (see [9-11]). In particular, the Hyers-Ulam-Rassias stability of (1.1) was proved by Czerwik [12].

Recently, Eungrasamee et al.[13] considered the following functional equation:

(1.2)

where n is a positive integer with . They proved that (1.2) is equivalent to (1.1). Also, they proved the Hyers-Ulam-Rassias stability of this equation.

In this paper, we solve the general solutions and the stability problems of (1.2) in the spaces of generalized functions such as of tempered distributions, of Fourier hyperfunctions and of distributions. Using the notions as in [14-19], we reformulate (1.2) and the related inequality in the spaces of generalized functions as follows:

(1.3)

(1.4)

where , and are the functions defined by

Here, ∘ denotes the pullback of generalized functions and the inequality in (1.4) means that for all test functions φ. We refer to [20] for pullbacks and to [14-17] for more details on the spaces of generalized functions. Recently, Chung [14,15,17] solved the general solutions and the stability problems of (1.1) in the spaces of generalized functions. As a matter of fact, our approaches are based on the methods as in [14-17].

This paper is organized as follows. In Section 2, we solve the general solutions and the stability problems of (1.2) in the spaces of and . We prove that every solution u in (or ) of equation (1.3) has the form

where . Also, we prove that every solution u in (or ) of inequality (1.4) can be written uniquely in the form

where and μ is a bounded measurable function such that . Subsequently, in Section 3, making use of the Dirac sequence of regularizing functions, we extend these results to the space .

### 2 Stability in

We first introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, denotes the set of nonnegative integers.

Definition 2.1[20,21]

We denote by the set of all infinitely differentiable functions φ in ℝ satisfying

(2.1)

for all . A linear functional u on is said to be a tempered distribution if there exists a constant and a nonnegative integer N such that

for all . The set of all tempered distributions is denoted by .

If , then each derivative of φ decreases faster than for all as . It is easy to see that the function , where belongs to but , is not a member of . For example, every polynomial , where , defines a tempered distribution by

Note that tempered distributions are generalizations of -functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform, but not all distributions have one. Imposing the growth condition on in (2.1), we get a new space of test functions as follows.

Definition 2.2[22]

We denote by the set of all infinitely differentiable functions φ in ℝ such that

(2.2)

for some positive constants A, B depending only on φ. The strong dual of , denoted by , is called the Fourier hyperfunction.

It is easy to see the following topological inclusions:

(2.3)

To solve the general solution and the stability problem of (1.2) in the space , we employ the heat kernel which is the fundamental solution to the heat equation,

Since for each , belongs to the space , the convolution

is well defined for all , which is called the Gauss transform of u. It is well known that the semigroup property of the heat kernel

holds for convolution. The semigroup property will be very useful for converting equation (1.3) into the classical functional equation defined on an upper-half plane. We also use the following famous result, the so-called heat kernel method, which is stated as follows.

Theorem 2.3[23]

Let. Then its Gauss transformis a-solution of the heat equation

satisfying the following:

(i) There exist positive constantsC, MandNsuch that

(2.4)

(ii) asin the sense that for every,

Conversely, every-solutionof the heat equation satisfying the growth condition (2.4) can be uniquely expressed asfor some.

Similarly, we can represent Fourier hyperfunctions as initial values of solutions of the heat equation as a special case of the results as in [24]. In this case, the estimate (2.4) is replaced by the following:

For every , there exists a positive constant such that

We are now going to solve the general solutions and the stability problems of (1.2) in the spaces of and . Here, we need the following lemma which will be crucial in the proof of the main theorem.

Lemma 2.4Suppose thatis a continuous function satisfying

(2.5)

for all, . Then the solutionfhas the form

for some constants.

Proof Define a function for all , . Then h satisfies for all and

(2.6)

for all , . Putting in (2.6) gives

(2.7)

for all , . Letting in (2.7) yields

for all , . This means that is independent of t. Thus we can write . It follows from (2.6) that satisfies

(2.8)

for all . Putting in (2.8), we see that satisfies the quadratic functional equation

for all . Given the continuity, the function q has the form

for some constant .

On the other hand, putting in (2.5) gives

(2.9)

for all . In view of (2.9), we see that

exists. In (2.9), let so that as . Then we have

(2.10)

for all . In the same way, in (2.10), let so that as . Then we have

(2.11)

for all . Applying the same way in (2.11) repeatedly, we obtain . Setting in (2.9) yields

for all . Given the continuity, we must have for some . Therefore, the solution f of (2.5) is of the form

for some constants . □

According to the above lemma, we solve the general solution of (1.2) in the spaces of and . From the inclusions (2.3), it suffices to consider the space instead of .

Theorem 2.5Every solutionuin (orresp.) of the equation

(2.12)

has the form

where.

Proof Convolving the tensor product of the heat kernels on both sides of (2.12), we have

where is the Gauss transform of u. Thus, (2.12) is converted into the following classical functional equation:

(2.13)

for all , . It follows from Lemma 2.4 that the solution of (2.13) has the form

(2.14)

where . Letting in (2.14), we obtain

This completes the proof. □

From the above theorem, we have the following corollary immediately.

Corollary 2.6Every solutionuin (orresp.) of the equation

has the form

where.

We now solve the stability problem of (1.2) in the spaces of and .

Theorem 2.7Suppose that (orresp.) satisfies the inequality

(2.15)

Then there exists a unique quadratic functionsuch that

Proof Convolving the tensor product of the heat kernels on both sides of (2.15), we have

(2.16)

for all , , where f is the Gauss transform of u. Putting in (2.16) yields

(2.17)

for all . Then by the triangle inequality we have

for all . It follows from the continuity of f and the inequality above that

exists. Choose a sequence ,  , of positive numbers, which tends to 0 as , such that as . Letting in (2.17) gives

(2.18)

Setting , in (2.17) and using (2.18), we have

(2.19)

Using an induction argument in (2.19), we obtain

(2.20)

for all , . From (2.20) we see that is a Cauchy sequence, and hence

exists. Replacing by in (2.17), , dividing by and letting , we have

for all . As in the proof of Lemma 2.4, the function h satisfies

(2.21)

for all . Given the continuity, the function h is of the form for some constant . Letting in (2.20), we obtain

(2.22)

for all . It follows from (2.21) and (2.22) that

(2.23)

On the other hand, putting , and letting , in (2.16), we have

(2.24)

Using the iterative method in (2.24) gives

(2.25)

Adding (2.23) to (2.25) and letting , we obtain

(2.26)

From (2.26) we see that is a Cauchy sequence, and hence

exists. By the definition of G and (2.16), we have for all and

(2.27)

for all , . As in the proof of Lemma 2.4, we obtain

for some constant . Letting in (2.16) yields

(2.28)

for some constants . Letting in (2.28), we have

This completes the proof. □

Corollary 2.8Suppose that (orresp.) satisfies the inequality

Then there exists a unique quadratic functionsuch that

### 3 Stability in

In this section, we extend the previous results to the space of distributions. Recall that a distribution u is a linear functional on of infinitely differentiable functions on ℝ with compact supports such that for every compact set , there exist constants and satisfying

for all with supports contained in K. The set of all distributions is denoted by . It is well known that the following topological inclusions hold:

As we see in [14,16], by virtue of the semigroup property of the heat kernel, equation (1.3) can be controlled easily in the space . But we cannot employ the heat kernel in the space . Instead of the heat kernel, we use the function , , , where such that

For example, let

where

then it is easy to see is an infinitely differentiable function with support . Now we employ the function , . If , then for each , is a smooth function in ℝ and as in the sense of distributions, that is, for every ,

Theorem 3.1Every solutionuinof the equation

(3.1)

has the form

where.

Proof Convolving the tensor product of the regularizing functions on both sides of (3.1), we have

Thus, (3.1) is converted into the following functional equation:

(3.2)

for all , . In view of (3.2), it is easy to see that

exists. Putting and letting in (3.2) yields . Setting , , and letting , , in (3.2), we have

(3.3)

for all , . Letting in (3.3) gives

(3.4)

for all , . Putting in (3.4) yields

(3.5)

Applying (3.5) to (3.4), we see that f satisfies the following quadratic functional equation:

for all . Since f is a smooth function, in view of (3.5), it follows that , where . Thus, from (3.5), we have

(3.6)

Letting in (3.6), we obtain

This completes the proof. □

In a similar manner, we have the following corollary immediately.

Corollary 3.2Every solutionuinof the equation

has the form

where.

We are now going to state and prove the main result of this paper.

Theorem 3.3Suppose thatsatisfies the inequality

(3.7)

Then there exists a unique quadratic functionsuch that

Proof It suffices to show that every distribution satisfying (3.7) belongs to the space . Convolving the tensor product on both sides of (3.7), we have

(3.8)

for all , . In view of (3.8), it is easy to see that for each fixed x,

exists. Putting and letting in (3.8) yields

(3.9)

Setting , , , letting , , in (3.8) and using (3.9), we have

(3.10)

Putting in (3.10) and dividing the result by 2, we obtain

(3.11)

Letting in (3.11) gives

(3.12)

From (3.10), (3.11) and (3.12) we have

for all . According to the result as in [8], there exists a unique quadratic function satisfying

such that

(3.13)

It follows from (3.12) and (3.13) that

(3.14)

Letting in (3.14), we have

(3.15)

Inequality (3.15) implies that belongs to . Thus, we conclude that . □

From the above theorem, we have the following corollary immediately.

Corollary 3.4Suppose thatsatisfies the inequality

Then there exists a unique quadratic functionsuch that

### Competing interests

The author declares that he has no competing interests.

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