In this section, we reformulate and prove the stability theorem of the quartic functional equation (1.4) in the spaces of some generalized functions such as
of tempered distributions and
of Fourier hyperfunctions. We first introduce briefly spaces of some generalized functions. Here we use the multi-index notations,
,
,
and
, for
,
, where
is the set of non-negative integers and
.
Definition 4.1 (see [20, 21]).
We denote by
the Schwartz space of all infinitely differentiable functions
in
satisfying
for all
,
, equipped with the topology defined by the seminorms
. A linear form
on
is said to be tempered distribution if there is a constant
and a nonnegative integer
such that
for all
. The set of all tempered distributions is denoted by
.
Imposing growth conditions on
in (4.1) a new space of test functions has emerged as follows.
Definition 4.2 (see [22]).
We denote by
the Sato space of all infinitely differentiable functions
in
such that
for some positive constants
depending only on
. We say that
as
if
as
for some
, and denote by
the strong dual of
and call its elements Fourier hyperfunctions.
It can be verified that the seminorms (4.3) are equivalent to
for some constants
. It is easy to see the following topological inclusions:
From the above inclusions it suffices to say that we consider (1.4) in the space
. Note that (3.14) itself makes no sense in the spaces of generalized functions. Following the notions as in [23–25], we reformulate the inequality (3.14) as
where
. Here
denotes the pullbacks of generalized functions. Also
denotes the Euclidean norm and the inequality
in (4.6) means that
for all test functions
defined on
. We refer to (see [20, Chapter VI]) for pullbacks and to [21, 23–26] for more details of
and
.
If
, the right side of (4.6) does not define a distribution. Thus, the inequality (4.6) makes no sense in this case. Also, if
, it is not known whether Hyers-Ulam-Rassias stability of (1.4) holds even in the classical case. Thus, we consider only the case
or
.
In order to prove the stability problems of quartic functional equations in the space of
we employ the
-dimensional heat kernel, that is, the fundamental solution
of the heat operator
in
given by
Since for each
,
belongs to
, the convolution
is well defined for each
, which is called the Gauss transform of
. In connection with the Gauss transform it is well known that the semigroup property of the heat kernel
holds for convolution. Semigroup property will be useful to convert inequality (3.3) into the classical functional inequality defined on upper-half plane. Moreover, the following result called heat kernel method holds [27].
Let
. Then its Gauss transform
is a
-solution of the heat equation
satisfying
-
(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 (4.11) can be uniquely expressed as
for some
. Similarly, we can represent Fourier hyperfunctions as initial values of solutions of the heat equation as a special case of the results (see [28]). In this case, the estimate (4.11) is replaced by the following.
For every
there exists a positive constant
such that
We note that the Gauss transform
is well defined and
locally uniformly as
. Also
satisfies semi-homogeneity property
for all
.
We are now in a position to state and prove the main result of this paper.
Theorem 4.3.
Let
be fixed integer with
and let
be real numbers such that
and either
or
. Suppose that
in
or
satisfies the inequality (4.6). Then there exists a unique quartic mapping
which satisfies (1.4) and the inequality
where
.
Proof.
Define
. Convolving the tensor product
of
-dimensional heat kernels in
we have
On the other hand, we figure out
and similarly we get
where
is the Gauss transform of
. Thus, inequality (4.6) is converted into the classical functional inequality
for all
. In view of (4.20), it can be verified that
exists.
We first prove the case
. Choose a sequence
of positive numbers which tends to
as
such that
as
. Letting
,
in (4.20) and dividing the result by
we get
which is written in the form
for all
, where
. By virtue of the semi-homogeneous property of
, substituting
by
, respectively, in (4.23) and dividing the result by
we obtain
Using induction arguments and triangle inequalities we have
for all
. Let us prove the sequence
is convergent for all
. Replacing
by
, respectively, in (4.25) and dividing the result by
we see that
Letting
, we have
is a Cauchy sequence. Therefore we may define
for all
. On the other hand, replacing
by
in (4.20), respectively, and then dividing the result by
we get
Now letting
we see by definition of
that
satisfies
for all
. Letting
in (4.25) yields
To prove the uniqueness of
, we assume that
is another function satisfying (4.29) and (4.30). Setting
and
in (4.29) we have
for all
. Then it follows from (4.30) and (4.31) that
for all
. Letting
, we have
for all
. This proves the uniqueness.
It follows from the inequality (4.30) that we get
for all test functions
. Since
is given by the uniform limit of the sequence
,
is also continuous on
. In view of (4.29), it follows from the continuity of
that for each 
exists. Letting
in (4.29) we have
satisfies quartic functional equation (1.4). Letting
we have the inequality
Now we consider the case
. For this case, replacing
by
in (4.23), respectively, and letting
and then multiplying the result by
we have
Using induction argument and triangle inequality we obtain
for all
. Following the similar method in case of
, we see that
is the unique function satisfying (4.29) so that
exists. Letting
in (4.37) we get
Now letting
in (4.39) we have the inequality
This completes the proof.
As an immediate consequence, we have the following corollary.
Corollary 4.4.
Let
be fixed integer with
and
be a real number. Suppose that
in
or
satisfies the inequality
Then there exists a unique quartic mapping
which satisfies (1.4) and the inequality
where
.