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Stability of Quartic Functional Equations in the Spaces of Generalized Functions
Advances in Difference Equations volume 2009, Article number: 838347 (2009)
Abstract
We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.
1. Introduction
One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam [1]. The case of approximately additive mappings was solved by Hyers [2]. In 1978, Rassias [3] generalized Hyers' result to the unbounded Cauchy difference. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [4–9]). The terminology Hyers-Ulam-Rassias stability originates from these historical backgrounds and this terminology is also applied to the cases of other functional equations. For instance, Rassias [10] investigated stability properties of the following functional equation

It is easy to see that is a solution of (1.1) by virtue of the identity

For this reason, (1.1) is called a quartic functional equation. Also Chung and Sahoo [11] determined the general solution of (1.1) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (1.1) if and only if
, where the function
is symmetric and additive in each variable. Since the solution of (1.1) is even, we can rewrite (1.1) as

Lee et al. [12] obtained the general solution of (1.3) and proved the Hyers-Ulam-Rassias stability of this equation. Also Park [13] investigated the stability problem of (1.3) in the orthogonality normed space.
In this paper we consider the following quartic functional equation, which is a generalization of (1.3),

for fixed integer with
. In the cases of
in (1.4), homogeneity property of quartic functional equations does not hold. We dispense with this cases henceforth, and assume that
. In Section 2, we show that for each fixed integer
with
, (1.4) is equivalent to (1.3). Moreover, using the idea of Găvruţa [14], we prove the Hyers-Ulam-Rassias stability of (1.4) in Section 3. Finally, making use of the pullbacks and the heat kernels, we reformulate and prove the Hyers-Ulam-Rassias stability of (1.4) in the spaces of some generalized functions such as
of tempered distributions and
of Fourier hyperfunctions in Section 4.
2. General Solution of (1.4)
Throughout this section, we denote and
by real vector spaces. It is well known [15] that a function
satisfies the quadratic functional equation

if and only if there exists a unique symmetric biadditive function such that
for all
. The biadditive function
is given by

Stability problems of quadratic functional equations can be found in [16–19]. Similarly, a function satisfies the quartic functional equation (1.3) if and only if there exists a symmetric biquadratic function
such that
for all
(see [12]). We now present the general solution of (1.4) in the class of functions between real vector spaces.
Theorem 2.1.
A mapping satisfies the functional equation (1.3) if and only if for each fixed integer
with
, a mapping
satisfies the functional equation (1.4).
Proof.
Suppose that satisfies (1.3). Putting
in (1.3) we have
. Also letting
in (1.3) we get
. Using an induction argument we may assume that (1.4) is true for all
with
. Replacing
by
and
by
in (1.4) we have

Substituting by
in (2.3) and using the evenness of
we get

Adding (2.3) to (2.4) yields

According to the inductive assumption for , (2.5) can be rewritten as

which proves the validity of (1.4) for . For a negative integer
, replacing
by
one can easily prove the validity of (1.4). Therefore (1.3) implies (1.4) for any fixed integer
with
.
We now prove the converse. For each fixed integer with
, we assume that
satisfies (1.4). Putting
in (1.4) we have
. Also letting
in (1.4) we get
for all
. Setting
in (1.4) we obtain the homogeneity property
for all
. Replacing
by
in (1.4) we have

Interchanging into
in (2.7) yields

Replacing and
by
and
in (1.4) we get

Substituting by
in (2.9) gives

Plugging (2.7) into (2.8), and using (2.9) and (2.10) we have

Replacing and
by
and
in (1.4), respectively, we get

Setting by
in (1.4) and dividing by
we obtain

It follows from (2.12) and (2.13) that (2.11) can be rewritten in the form

Using an induction argument in (2.14), it is easy to see that satisfies the following functional equation

for each fixed integer . Replacing
by
in (2.15), and comparing (1.4) with (2.15) we have
. Thus (2.14) implies (1.3). This completes the proof.
3. Stability of (1.4)
Now we are going to prove the Hyers-Ulam-Rassias stability for quartic functional equations. Let be a real vector space and let
be a Banach space.
Theorem 3.1.
Let be a mapping such that

converges and

for all . Suppose that a mapping
satisfies the inequality

for all . Then there exists a unique quartic mapping
which satisfies quartic functional equation (1.4) and the inequality

for all . The mapping
is given by

for all . Also, if for each fixed
the mapping
from
to
is continuous, then
for all
.
Proof.
Putting in (3.3) and then dividing the result by
we have

which is rewritten as

for all , where
. Making use of induction arguments and triangle inequalities we have

for all . Now we prove the sequence
is a Cauchy sequence. Replacing
by
in (3.8) and then dividing by
we see that for
,

Since the right-hand side of (3.9) tends to as
, the sequence
is a Cauchy sequence. Therefore we may define

for all . Replacing
by
, respectively, in (3.3) and then dividing by
we have

Taking the limit as , we verify that
satisfies (1.4) for all
. Now letting
in (3.8) we have

for all . To prove the uniqueness, let us assume that there exists another quartic mapping
which satisfies (1.4) and the inequality (3.12). Obviously, we have
and
for all
. Thus, we have

for all . Letting
, we must have
for all
. This completes the proof.
Corollary 3.2.
Let be fixed integer with
and let
be real numbers such that
and either
or
. Suppose that a mapping
satisfies the inequality

for all . Then there exists a unique quartic mapping
which satisfies (1.4) and the inequality

for all and for all
if
. The mapping
is given by

for all .
Corollary 3.3.
Let be fixed integer with
and
be a real number. Suppose that a mapping
satisfies the inequality

for all . Then there exists a unique quartic mapping
defined by

which satisfies (1.4) and the inequality

for all .
4. Stability of (1.4) in Generalized Functions
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
(411) -
(ii)
as
in the sense that for every
,
(412)
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 .
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Acknowledgments
The first author was supported by the second stage of the Brain Korea 21 Project, The Development Project of Human Resources in Mathematics, KAIST, in 2009. The second author was supported by the Special Grant of Sogang University in 2005.
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Lee, YS., Chung, SY. Stability of Quartic Functional Equations in the Spaces of Generalized Functions. Adv Differ Equ 2009, 838347 (2009). https://doi.org/10.1155/2009/838347
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DOI: https://doi.org/10.1155/2009/838347
Keywords
- Functional Equation
- Heat Kernel
- Stability Problem
- Cauchy Sequence
- Induction Argument