# A Functional Equation of Aczél and Chung in Generalized Functions

- Jae-Young Chung
^{1}Email author

**2008**:147979

**DOI: **10.1155/2008/147979

© Jae-Young Chung. 2008

**Received: **1 October 2008

**Accepted: **25 December 2008

**Published: **10 February 2009

## Abstract

We consider an -dimensional version of the functional equations of Aczél and Chung in the spaces of generalized functions such as the Schwartz distributions and Gelfand generalized functions. As a result, we prove that the solutions of the distributional version of the equation coincide with those of classical functional equation.

## 1. Introduction

*exponential polynomials*, that is, the functions of the form

where and 's are polynomials for all .

where (resp., ), and denotes the pullback, denotes the tensor product of generalized functions, and , , , , , , , . As in [1], we assume that and for all , , .

In [2], Baker previously treated (1.3). By making use of differentiation of distributions which is one of the most powerful advantages of the Schwartz theory, and reducing (1.3) to a system of differential equations, he showed that, for the dimension
, the solutions of (1.3) are *exponential polynomials*. We refer the reader to [2–6] for more results using this method of reducing given functional equations to differential equations.

In this paper, by employing tensor products of regularizing functions as in [7, 8], we consider the regularity of the solutions of (1.3) and prove in an elementary way that (1.3) can be reduced to the classical equation (1.1) of smooth functions. This method can be applied to prove the Hyers-Ulam stability problem for functional equation in Schwartz distribution [7, 8]. In the last section, we consider the Hyers-Ulam stability of some related functional equations. For some elegant results on the classical Hyers-Ulam stability of functional equations, we refer the reader to [6, 9–21].

## 2. Generalized Functions

In this section, we briefly introduce the spaces of generalized functions such as the Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions. Here we use the following notations: , , , , and , for , , where is the set of nonnegative integers and .

Definition 2.1.

for all with supports contained in . One denotes by the space of the Schwartz distributions on .

Definition 2.2.

The topology on the space
is defined by the seminorms
in the left-hand side of (2.2), and the elements of the dual space
of
are called *Gelfand-Shilov generalized functions*. In particular, one denotes
by
and calls its elements *Fourier hyperfunctions*.

for some .

We briefly introduce some basic operations on the spaces of the generalized functions.

Definition 2.3.

Definition 2.4.

The tensor product belongs to .

Definition 2.5.

Let , , and let be a smooth function such that for each the derivative is surjective. Then there exists a unique continuous linear map such that , when is a continuous function. One calls the pullback of by and simply is denoted by .

The differentiations, pullbacks, and tensor products of Fourier hyperfunctions and Gelfand generalized functions are defined in the same way as distributions. For more details of tensor product and pullback of generalized functions, we refer the reader to [9, 22].

## 3. Main Result

Theorem 3.1.

Let , , , be a solution of (1.3), and both and are linearly independent. Then, , , , , , where , , , a smooth solution of (1.1).

Proof.

It is obvious that is a smooth function. Also it follows from (3.27) that each , , converges locally and uniformly to the function as , which implies that the equality (3.28) holds in the sense of distributions. Finally, letting and in (3.5) we see that , are smooth solutions of (1.1). This completes the proof.

Combined with the result of Aczél and Chung [1], we have the following corollary as a consequence of the above result.

Corollary 3.2.

Every solution , , , of (1.3) for the dimension has the form of exponential polynomials.

Applying the proof of Theorem 3.1, we get the result for the space of Gelfand generalized functions.

## 4. Hyers-Ulam Stability of Related Functional Equations

where . In 1990, Székelyhidi [23] has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with stability questions for the sine and cosine functional equations. As the results, he proved that if , , is a bounded function on , then either there exist , not both zero, such that is a bounded function on , or else , , respectively. For some other elegant Hyers-Ulam stability theorems, we refer the reader to [6, 9–21].

and investigate the behavior of satisfying the inequality for each , where , , , denotes the pullback, denotes the tensor product of generalized functions as in Theorem 3.1, and means that for all .

As a result, we obtain the following theorems.

Theorem 4.1.

Let satisfy . Then, and satisfy one of the following items:

- (i)
, : arbitrary,

- (ii)
and are bounded measurable functions,

- (iii)
, ,

- (iv)
, ,

- (v)
, ,

- (vi)
, ,

where , , , and is a bounded measurable function.

Theorem 4.2.

Let satisfy . Then, and satisfy one of the following items:

- (i)
and are bounded measurable functions,

- (ii)
and is a bounded measurable function,

- (iii)
, ,

- (iv)
,

- (v)
, ,

- (vi)
, ,

where , , , and is a bounded measurable function.

Theorem 4.3.

Let satisfy . Then, and satisfy one of the following items:

- (i)
and is arbitrary,

- (ii)
and are bounded measurable functions,

- (iii)
, ,

- (iv)
, ,

for some , and a bounded measurable function .

Theorem 4.4.

Let satisfy . Then, and satisfy one of the following items:

- (i)
and are bounded measurable functions,

- (ii)
, , .

For the proof of the theorems, we employ the -dimensional heat kernel(4.3)

In view of (2.3), it is easy to see that for each , belongs to the Gelfand-Shilov space . Thus the convolution is well defined and is a smooth solution of the heat equation in and in the sense of generalized functions for all .

for the smooth functions , . Proving the Hyers-Ulam stability problems for the inequalities (4.5) and taking the initial values of and as , we get the results. For the complete proofs of the result, we refer the reader to [24].

Remark 4.5.

## Authors’ Affiliations

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