A Functional Equation of Aczél and Chung in Generalized Functions

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

In [1], Aczél and Chung introduced the following functional equation:

(1.1)

where and for . Under the natural assumptions that and are linearly independent, and , for all , , it was shown that the locally integrable solutions of (1.1) are exponential polynomials, that is, the functions of the form

(1.2)

where and 's are polynomials for all .

In this paper, we introduce the following -dimensional version of the functional equation (1.1) in generalized functions:

(1.3)

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.

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

(2.1)

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

Definition 2.2.

For given , one denotes by or the space of all infinitely differentiable functions on such that there exist positive constants and satisfying

(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.

It is known that if and , the space consists of all infinitely differentiable functions on that can be continued to an entire function on satisfying

(2.3)

for some .

It is well known that the following topological inclusions hold:

(2.4)

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

Definition 2.3.

Let . Then, the th partial derivative of is defined by

(2.5)

for . Let . Then the multiplication is defined by

(2.6)

Definition 2.4.

Let , . Then, the tensor product of and is defined by

(2.7)

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

We employ a function such that

(3.1)

Let and , . Then, for each , is well defined. We call a regularizing function of the distribution , since is a smooth function of satisfying in the sense of distributions, that is, for every ,

(3.2)

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.

By convolving the tensor product in each side of (1.3), we have, for ,

(3.3)

where , , . Similarly we have for ,

(3.4)

Thus (1.3) is converted to the following functional equation:

(3.5)

where

(3.6)

for , . We first prove that are smooth functions and equal to for all . Let

(3.7)

Then,

(3.8)

is a smooth function of for each , , and is linearly independent. We may choose , such that . Then, it follows from (3.5) that

(3.9)

where , . Putting (3.9) in (3.5), we have

(3.10)

where

(3.11)

(3.12)

Since is a smooth function of for each , , it follows from (3.11) that

(3.13)

is a smooth function of for each , . Also, since is linearly independent, it follows from (3.12) that

(3.14)

is linearly independent. Thus we can choose , such that . Then, it follows from (3.10) that

(3.15)

where , . Putting (3.15) in (3.10), we have

(3.16)

where

(3.17)

By continuing this process, we obtain the following equations:

(3.18)

for all , where , , ,

(3.19)

for all , and

(3.20)

By the induction argument, we have for each ,

(3.21)

is a smooth function of for each , . Thus, in view of (3.20),

(3.22)

is a smooth function. Furthermore, converges to locally uniformly, which implies that in the sense of distributions, that is, for every ,

(3.23)

In view of (3.19) and the induction argument, for each , we have

(3.24)

is a smooth function and for all . Changing the roles of and for , we obtain, for each ,

(3.25)

is a smooth function and . Finally, we show that for each , is equal to a smooth function. Letting in (3.5), we have

(3.26)

For each fixed , , replacing by , multiplying and integrating with respect to , we have

(3.27)

where for all , . Letting in (3.27), we have

(3.28)

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.

The result of Theorem 3.1 holds for , , . Using the following -dimensional heat kernel,

(3.29)

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

The well-known Cauchy equation, Pexider equation, Jensen equation, quadratic functional equation, and d'Alembert functional equation are typical examples of the form (1.1). For the distributional version of these equations and their stabilities, we refer the reader to [7, 8]. In this section, as well-known examples of (1.1), we introduce the following trigonometric differences:

(4.1)

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].

By generalizing the differences (4.1), we consider the differences

(4.2)

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 .

Similarly as in the proof of Theorem 3.1, convolving the tensor product of heat kernels and using the semigroup property

(4.4)

of the heat kernels, we can convert the inequalities , to the classical Hyers-Ulam stability problems, respectively,

(4.5)

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.

The referee of the paper has recommended the author to consider the Hyers-Ulam stability of the equations, which will be one of the most interesting problems in this field. However, the author has no idea of solving this question yet. Instead, Baker [25] proved the Hyers-Ulam stability of the equation

(4.6)