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 ).
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 .
Let . Then its Gauss transform is a -solution of the heat equation
There exist positive constants and such that
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 ). 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.
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
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
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.
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