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A Functional Equation of Aczél and Chung in Generalized Functions
Advances in Difference Equations volume 2008, Article number: 147979 (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
In [1], Aczél and Chung introduced the following functional equation:

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

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:

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

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

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

for some .
It is well known that the following topological inclusions hold:

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

for . Let
. Then the multiplication
is defined by

Definition 2.4.
Let ,
. Then, the tensor product
of
and
is defined by

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

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
,

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
,

where ,
,
. Similarly we have for
,

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

where

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

Then,

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

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

where


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

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

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

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

where

By continuing this process, we obtain the following equations:

for all , where
,
,
,

for all , and

By the induction argument, we have for each ,

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

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

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

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

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

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

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

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,

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:

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

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

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

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

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Chung, JY. A Functional Equation of Aczél and Chung in Generalized Functions. Adv Differ Equ 2008, 147979 (2009). https://doi.org/10.1155/2008/147979
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DOI: https://doi.org/10.1155/2008/147979
Keywords
- Smooth Function
- Functional Equation
- Tensor Product
- Differentiable Function
- Heat Kernel