Abstract and Applied AnalysisVolume 2011 (2011), Article ID 502903, 15 pagesdoi:10.1155/2011/502903

Research Article

Stability in Generalized Functions

Young-Su Lee

Department of Mathematics, Sogang University, Seoul 121-741, Republic of Korea

Received 31 July 2011; Revised 17 September 2011; Accepted 21 September 2011

Academic Editor: Dumitru Baleanu

Copyright © 2011 Young-Su Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the following additive functional equation with -independent variables: in the spaces of generalized functions. Making use of the heat kernels, 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 mollifiers, we extend these results to the space of distributions.

1. Introduction

The most famous functional equation is the Cauchy equation any solution of which is called additive. It is well known that every measurable solution of (1.1) is of the form for some constant . In 1941, Hyers proved the stability theorem for (1.1) as follows.

Theorem 1.1 (see [1]). Let be a normed vector space, a Banach space. Suppose that satisfies the inequality for all , then there exists the unique additive mapping such that for all .

The above stability theorem was motivated by Ulam [2]. As noted in the above theorem, the stability problem of the functional equations means how the solution of the inequality differs from the solution of the original equation. Forti [3] noticed that the theorem of Hyers is still true if is replaced by an arbitrary semigroup. In 1950 Aoki [4] and in 1978 Rassias [5] generalized Hyers’ result to the unbounded Cauchy difference. Thereafter, many authors studied the stability problems for (1.1) in various settings (see [6, 7]).

During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [8–17]). Among them, the following additive functional equation with -independent variables: was proposed by Nakmahachalasint [18], where is a positive integer with and . He solved the general solutions and the stability problems for the above equation. Actually, he proved that (1.4) is equivalent to (1.1).

In this paper, in a similar manner as in [19–23], we solve the general solutions and the stability problems for (1.4) in the spaces of generalized functions such as the space of tempered distributions, the space of Fourier hyperfunctions, and the space of distributions. Making use of the pullbacks, we first reformulate (1.4) and the related inequality in the spaces of generalized functions as follows: where , , and are the functions defined by Here denotes the pullback of generalized functions, and the inequality in (1.6) means that for all test functions .

In Section 2, we will prove that every solution in or of (1.5) has the form where . Also, we shall figure out that every solution in or of the inequality (1.6) can be written uniquely in the form where is a bounded measurable function such that . Subsequently, in Section 3, these results are extended to the space .

2. Stability in

We first introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the -dimensional notations, , , , and , for , , where is the set of nonnegative integers and .

Definition 2.1 (see [24, 25]). We denote by the Schwartz space of all infinitely differentiable functions in satisfying for all . A linear functional on is said to be tempered distribution if there exist a constant and a nonnegative integer such that for all . The set of all tempered distributions is denoted 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. Imposing the growth condition on in (2.1), a new space of test functions has emerged as follows.

Definition 2.2 (see [26]). We denote by the set of all infinitely differentiable functions in such that for some positive constants depending only on . The strong dual of , denoted by , is called the Fourier hyperfunction.

It can be verified that the seminorm (2.3) is equivalent to for some constants . It is easy to see the following topological inclusions: Taking the inclusions (2.5) into account, it suffices to consider the space . In order to solve the general solutions and the stability problems for (1.4) in the spaces and , we employ the -dimensional heat kernel, fundamental solution of the heat equation, Since for each , belongs to the space , the convolution is well defined for all in , which is called the Gauss transform of . Subsequently, the semigroup property of the heat kernel is very useful to convert (1.5) into the classical functional equation defined on upper-half plane. We also use the following famous result, so-called heat kernel method, which states as follows.

Theorem 2.3 (see [27]). Let , then its Gauss transform is a -solution of the heat equation satisfying the following: (i) there exist positive constants , , and such that (ii) as in the sense that for every , Conversely, every -solution of the heat equation satisfying the growth condition (2.10) can be uniquely expressed as for some .

Similarly, we can represent Fourier hyperfunctions as a special case of the results as in [28]. In this case, the estimate (2.10) is replaced by the following.

For every , there exists a positive constant such that

We need the following lemma in order to solve the general solutions for the additive functional equation in the spaces of and . In what follows, we denote and .

Lemma 2.4. Suppose that is a continuous function satisfying for all , , then the solution has the form for some .

Proof. Putting in (2.13) yields for all . In view of (2.15), we see that exists. Letting in (2.15) gives . Setting and letting , in (2.13), we have for all , . Substituting with and letting , , in (2.13), we obtain from (2.17) that for all , . Putting in (2.18) yields which shows that is independent with respect to . For that reason, we see from (2.18) that satisfies for all . Replacing by and by in (2.20), we have for all . Given the continuity, we obtain for some .

From the above lemma, we can solve the general solutions for the additive functional equation in the spaces of and .

Theorem 2.5. Every solution in (or , resp.) of (1.5) has the form for some .

Proof. Convolving the tensor product of the heat kernels on both sides of (1.5), we have where is the Gauss transform of . Thus, (1.5) is converted into the following classical functional equation: for all , . It follows from Lemma 2.4 that the solution of (2.25) has the form for some . Letting in (2.26), we finally obtain the general solution for (1.5).

We are going to solve the stability problems for the additive functional equation in the spaces of and .

Lemma 2.6. Suppose that is a continuous function satisfying for all , , then there exists a unique such that for all , .

Proof. Putting in (2.27) yields for all . In view of (2.29), we see that exists. Letting in (2.29) gives Setting and letting , in (2.27), we have for all , . Substituting and letting , in (2.13), we obtain for all , . Adding (2.33) to (2.32) yields for all , . Letting , in (2.29) gives for all . Combining (2.31), (2.34), and (2.35), we have for all , . Making use of induction argument in (2.36), we obtain for all , , . Replacing , by , in (2.37), respectively, and dividing the result by , we see that for , Since the right-hand side of (2.38) tends to 0 as , the sequence is a Cauchy sequence which converges uniformly. Thus, we may define for all , . Now, we verify from (2.27) that the function satisfies for all , . As observed in Lemma 2.4, the continuous solution of (2.40) has the form for some . It follows from (2.37) that is the unique function in satisfying for all , .

From the above lemma, we have the following stability theorem for the additive functional equation in the spaces of and .

Theorem 2.7. Suppose that in (or , resp.) satisfies the inequality (1.6), then there exists a unique such that

Proof. Convolving the tensor product of the heat kernels on both sides of (1.6), we have for all , , where is the Gauss transform of . By Lemma 2.6, we have for all , . Letting in (2.45), we obtain the conclusion.

3. Stability in

In this section, we shall extend the previous results to the space of distributions. Recall that a distribution is a linear functional on of infinitely differentiable functions on with compact supports such that for every compact set , there exist constants and satisfying for all with supports contained in . The set of all distributions is denoted by . It is well known that the following topological inclusions hold: As we see in [19, 20, 23], by the semigroup property of the heat kernel, (1.5) can be controlled easily in the spaces and . But we cannot employ the heat kernel in the space . For that reason, instead of the heat kernel, we use the regularizing functions. We denote by the function on satisfying where It is easy to see that is an infinitely differentiable function supported in the set with . For each , we define , then has all the properties of except that the support of is contained in the ball of radius with center at 0. If , then for each , is a smooth function in and in the sense of distributions, that is, for every , For each , the function is called a regularization of , and the transform which maps to is called a mollifier. Making use of the mollifiers, we can solve the general solution for the additive functional equation in the space as follows.

Theorem 3.1. Every solution in of (1.5) has the form for some .

Proof. Convolving the tensor product of the regularizing functions on both sides of (1.5), we have Thus, (1.5) is converted into the following functional equation: for all , . In view of (3.8), it is easy to see that, for each fixed , exists. Putting and letting in (3.8) yield . Setting and letting , in (3.8) give for all , . Substituting with and letting , , in (3.8), we obtain from (3.10) that for all , . Letting in (3.11) yields for all , . Putting in (3.12) gives for all , . Applying (3.13) to (3.11), we see that satisfies which is equivalent to the Cauchy equation (1.1) for all . Since is a smooth function in view of (3.13), it follows that for some . Letting in (3.13), we finally obtain the general solution for (1.5).

Now, we shall extend the stability theorem for the additive equation mentioned in the previous section to the space .

Theorem 3.2. Suppose that in satisfies the inequality (1.6), then there exists a unique such that

Proof. It suffices to show that every distribution satisfying (1.6) belongs to the space . Convolving the tensor product on both sides of (1.6), we have for all , . In view of (3.16), it is easy to see that for each fixed , exists. Putting and letting in (3.16) yield Setting and letting , in (3.16), we have for all , . Substituting with and letting , , in (3.16), we have for all , . It follows from (3.19) that the inequality (3.20) can be rewritten as for all , . Letting in (3.21) yields for all , . Using (3.19) we may write the inequality (3.22) as for all , . Putting in (3.23) and dividing the result by 2 give for all . From (3.23) and (3.24), we have which is equivalent to for all . Thus, by virtue of the result as in [29], there exists a unique function satisfying such that for all . It follows from (3.18), (3.24), and (3.28) that for all , . Letting in (3.29), we obtain Inequality (3.30) implies that belongs to . Thus, we conclude that .

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