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Advances in Difference Equations

, 2009:838347 | Cite as

Stability of Quartic Functional Equations in the Spaces of Generalized Functions

  • Young-Su LeeEmail author
  • Soon-Yeong Chung
Open Access
Research Article

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.

Keywords

Functional Equation Heat Kernel Stability Problem Cauchy Sequence Induction Argument 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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, 5, 6, 7, 8, 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 Open image in new window 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 Open image in new window is a solution of (1.1) if and only if Open image in new window , where the function Open image in new window 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 Open image in new window with Open image in new window . In the cases of Open image in new window in (1.4), homogeneity property of quartic functional equations does not hold. We dispense with this cases henceforth, and assume that Open image in new window . In Section 2, we show that for each fixed integer Open image in new window with Open image in new window , (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 Open image in new window of tempered distributions and Open image in new window of Fourier hyperfunctions in Section 4.

2. General Solution of (1.4)

Throughout this section, we denote Open image in new window and Open image in new window by real vector spaces. It is well known [15] that a function Open image in new window satisfies the quadratic functional equation
if and only if there exists a unique symmetric biadditive function Open image in new window such that Open image in new window for all Open image in new window . The biadditive function Open image in new window is given by

Stability problems of quadratic functional equations can be found in [16, 17, 18, 19]. Similarly, a function Open image in new window satisfies the quartic functional equation (1.3) if and only if there exists a symmetric biquadratic function Open image in new window such that Open image in new window for all Open image in new window (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 Open image in new window satisfies the functional equation (1.3) if and only if for each fixed integer Open image in new window with Open image in new window , a mapping Open image in new window satisfies the functional equation (1.4).

Proof.

Suppose that Open image in new window satisfies (1.3). Putting Open image in new window in (1.3) we have Open image in new window . Also letting Open image in new window in (1.3) we get Open image in new window . Using an induction argument we may assume that (1.4) is true for all Open image in new window with Open image in new window . Replacing Open image in new window by Open image in new window and Open image in new window by Open image in new window in (1.4) we have
Substituting Open image in new window by Open image in new window in (2.3) and using the evenness of Open image in new window we get
Adding (2.3) to (2.4) yields
According to the inductive assumption for Open image in new window , (2.5) can be rewritten as

which proves the validity of (1.4) for Open image in new window . For a negative integer Open image in new window , replacing Open image in new window by Open image in new window one can easily prove the validity of (1.4). Therefore (1.3) implies (1.4) for any fixed integer Open image in new window with Open image in new window .

We now prove the converse. For each fixed integer Open image in new window with Open image in new window , we assume that Open image in new window satisfies (1.4). Putting Open image in new window in (1.4) we have Open image in new window . Also letting Open image in new window in (1.4) we get Open image in new window for all Open image in new window . Setting Open image in new window in (1.4) we obtain the homogeneity property Open image in new window for all Open image in new window . Replacing Open image in new window by Open image in new window in (1.4) we have
Plugging (2.7) into (2.8), and using (2.9) and (2.10) we have
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 Open image in new window satisfies the following functional equation

for each fixed integer Open image in new window . Replacing Open image in new window by Open image in new window in (2.15), and comparing (1.4) with (2.15) we have Open image in new window . 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 Open image in new window be a real vector space and let Open image in new window be a Banach space.

Theorem 3.1.

Let Open image in new window be a mapping such that
converges and
for all Open image in new window . Suppose that a mapping Open image in new window satisfies the inequality
for all Open image in new window . Then there exists a unique quartic mapping Open image in new window which satisfies quartic functional equation (1.4) and the inequality

for all Open image in new window . Also, if for each fixed Open image in new window the mapping Open image in new window from Open image in new window to Open image in new window is continuous, then Open image in new window for all Open image in new window .

Proof.

Putting Open image in new window in (3.3) and then dividing the result by Open image in new window we have
which is rewritten as
for all Open image in new window , where Open image in new window . Making use of induction arguments and triangle inequalities we have
for all Open image in new window . Now we prove the sequence Open image in new window is a Cauchy sequence. Replacing Open image in new window by Open image in new window in (3.8) and then dividing by Open image in new window we see that for Open image in new window ,
Since the right-hand side of (3.9) tends to Open image in new window as Open image in new window , the sequence Open image in new window is a Cauchy sequence. Therefore we may define
for all Open image in new window . Replacing Open image in new window by Open image in new window , respectively, in (3.3) and then dividing by Open image in new window we have
Taking the limit as Open image in new window , we verify that Open image in new window satisfies (1.4) for all Open image in new window . Now letting Open image in new window in (3.8) we have
for all Open image in new window . To prove the uniqueness, let us assume that there exists another quartic mapping Open image in new window which satisfies (1.4) and the inequality (3.12). Obviously, we have Open image in new window and Open image in new window for all Open image in new window . Thus, we have

for all Open image in new window . Letting Open image in new window , we must have Open image in new window for all Open image in new window . This completes the proof.

Corollary 3.2.

Let Open image in new window be fixed integer with Open image in new window and let Open image in new window be real numbers such that Open image in new window and either Open image in new window or Open image in new window . Suppose that a mapping Open image in new window satisfies the inequality
for all Open image in new window . Then there exists a unique quartic mapping Open image in new window which satisfies (1.4) and the inequality

for all Open image in new window .

Corollary 3.3.

Let Open image in new window be fixed integer with Open image in new window and Open image in new window be a real number. Suppose that a mapping Open image in new window satisfies the inequality
for all Open image in new window . Then there exists a unique quartic mapping Open image in new window defined by
which satisfies (1.4) and the inequality

for all Open image in new window .

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 Open image in new window of tempered distributions and Open image in new window of Fourier hyperfunctions. We first introduce briefly spaces of some generalized functions. Here we use the multi-index notations, Open image in new window , Open image in new window , Open image in new window and Open image in new window , for Open image in new window , Open image in new window , where Open image in new window is the set of non-negative integers and Open image in new window .

Definition 4.1 (see [20, 21]).

We denote by Open image in new window the Schwartz space of all infinitely differentiable functions Open image in new window in Open image in new window satisfying
for all Open image in new window , Open image in new window , equipped with the topology defined by the seminorms Open image in new window . A linear form Open image in new window on Open image in new window is said to be tempered distribution if there is a constant Open image in new window and a nonnegative integer Open image in new window such that

for all Open image in new window . The set of all tempered distributions is denoted by Open image in new window .

Imposing growth conditions on Open image in new window in (4.1) a new space of test functions has emerged as follows.

Definition 4.2 (see [22]).

We denote by Open image in new window the Sato space of all infinitely differentiable functions Open image in new window in Open image in new window such that

for some positive constants Open image in new window depending only on Open image in new window . We say that Open image in new window as Open image in new window if Open image in new window as Open image in new window for some Open image in new window , and denote by Open image in new window the strong dual of Open image in new window and call its elements Fourier hyperfunctions.

It can be verified that the seminorms (4.3) are equivalent to
for some constants Open image in new window . 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 Open image in new window . Note that (3.14) itself makes no sense in the spaces of generalized functions. Following the notions as in [23, 24, 25], we reformulate the inequality (3.14) as

where Open image in new window . Here Open image in new window denotes the pullbacks of generalized functions. Also Open image in new window denotes the Euclidean norm and the inequality Open image in new window in (4.6) means that Open image in new window for all test functions Open image in new window defined on Open image in new window . We refer to (see [20, Chapter VI]) for pullbacks and to [21, 23, 24, 25, 26] for more details of Open image in new window and Open image in new window .

If Open image in new window , the right side of (4.6) does not define a distribution. Thus, the inequality (4.6) makes no sense in this case. Also, if Open image in new window , it is not known whether Hyers-Ulam-Rassias stability of (1.4) holds even in the classical case. Thus, we consider only the case Open image in new window or Open image in new window .

In order to prove the stability problems of quartic functional equations in the space of Open image in new window we employ the Open image in new window -dimensional heat kernel, that is, the fundamental solution Open image in new window of the heat operator Open image in new window in Open image in new window given by
is well defined for each Open image in new window , which is called the Gauss transform of Open image in new window . 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 Open image in new window . Then its Gauss transform Open image in new window is a Open image in new window -solution of the heat equation
satisfying

Conversely, every Open image in new window -solution Open image in new window of the heat equation satisfying the growth condition (4.11) can be uniquely expressed as Open image in new window for some Open image in new window . 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 Open image in new window there exists a positive constant Open image in new window such that
We note that the Gauss transform
is well defined and Open image in new window locally uniformly as Open image in new window . Also Open image in new window satisfies semi-homogeneity property

for all Open image in new window .

We are now in a position to state and prove the main result of this paper.

Theorem 4.3.

Let Open image in new window be fixed integer with Open image in new window and let Open image in new window be real numbers such that Open image in new window and either Open image in new window or Open image in new window . Suppose that Open image in new window in Open image in new window or Open image in new window satisfies the inequality (4.6). Then there exists a unique quartic mapping Open image in new window which satisfies (1.4) and the inequality

where Open image in new window .

Proof.

Define Open image in new window . Convolving the tensor product Open image in new window of Open image in new window -dimensional heat kernels in Open image in new window we have
On the other hand, we figure out
and similarly we get
where Open image in new window is the Gauss transform of Open image in new window . Thus, inequality (4.6) is converted into the classical functional inequality
for all Open image in new window . In view of (4.20), it can be verified that

exists.

We first prove the case Open image in new window . Choose a sequence Open image in new window of positive numbers which tends to Open image in new window as Open image in new window such that Open image in new window as Open image in new window . Letting Open image in new window , Open image in new window in (4.20) and dividing the result by Open image in new window we get
which is written in the form
for all Open image in new window , where Open image in new window . By virtue of the semi-homogeneous property of Open image in new window , substituting Open image in new window by Open image in new window , respectively, in (4.23) and dividing the result by Open image in new window we obtain
Using induction arguments and triangle inequalities we have
for all Open image in new window . Let us prove the sequence Open image in new window is convergent for all Open image in new window . Replacing Open image in new window by Open image in new window , respectively, in (4.25) and dividing the result by Open image in new window we see that
Letting Open image in new window , we have Open image in new window is a Cauchy sequence. Therefore we may define
for all Open image in new window . On the other hand, replacing Open image in new window by Open image in new window in (4.20), respectively, and then dividing the result by Open image in new window we get
To prove the uniqueness of Open image in new window , we assume that Open image in new window is another function satisfying (4.29) and (4.30). Setting Open image in new window and Open image in new window in (4.29) we have
for all Open image in new window . Then it follows from (4.30) and (4.31) that

for all Open image in new window . Letting Open image in new window , we have Open image in new window for all Open image in new window . This proves the uniqueness.

It follows from the inequality (4.30) that we get
for all test functions Open image in new window . Since Open image in new window is given by the uniform limit of the sequence Open image in new window , Open image in new window is also continuous on Open image in new window . In view of (4.29), it follows from the continuity of Open image in new window that for each Open image in new window
exists. Letting Open image in new window in (4.29) we have Open image in new window satisfies quartic functional equation (1.4). Letting Open image in new window we have the inequality
Now we consider the case Open image in new window . For this case, replacing Open image in new window by Open image in new window in (4.23), respectively, and letting Open image in new window and then multiplying the result by Open image in new window we have
Using induction argument and triangle inequality we obtain
for all Open image in new window . Following the similar method in case of Open image in new window , we see that
is the unique function satisfying (4.29) so that Open image in new window exists. Letting Open image in new window in (4.37) we get
Now letting Open image in new window in (4.39) we have the inequality

This completes the proof.

As an immediate consequence, we have the following corollary.

Corollary 4.4.

Then there exists a unique quartic mapping Open image in new window which satisfies (1.4) and the inequality

where Open image in new window .

Notes

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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Copyright information

© Y.-S. Lee and S.-Y. Chung. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKorea Advanced Institute of Science and TechnologyDaejeonSouth Korea
  2. 2.Department of Mathematics and Program of Integrated BiotechnologySogang UniversitySeoulSouth Korea

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