# Stability of Quartic Functional Equations in the Spaces of Generalized Functions

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## 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## 1. Introduction

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.

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)

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.

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 .

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.

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.

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.

for all Open image in new window .

Corollary 3.3.

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]).

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]).

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.

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 .

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].

- (i)There exist positive constants Open image in new window and Open image in new window such that(411)
- (ii)Open image in new window as Open image in new window in the sense that for every Open image in new window ,(412)

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 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.

where Open image in new window .

Proof.

exists.

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.

This completes the proof.

As an immediate consequence, we have the following corollary.

Corollary 4.4.

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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