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

(11)

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

(12)

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

(13)

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),

(14)

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

(21)

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

(22)

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

(23)

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

(24)

Adding (2.3) to (2.4) yields

(25)

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

(26)

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

(27)

Interchanging into in (2.7) yields

(28)

Replacing and by and in (1.4) we get

(29)

Substituting by in (2.9) gives

(210)

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

(211)

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

(212)

Setting by in (1.4) and dividing by we obtain

(213)

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

(214)

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

(215)

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

(31)

converges and

(32)

for all . Suppose that a mapping satisfies the inequality

(33)

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

(34)

for all . The mapping is given by

(35)

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

(36)

which is rewritten as

(37)

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

(38)

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

(39)

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

(310)

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

(311)

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

(312)

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

(313)

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

(314)

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

(315)

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

(316)

for all .

Corollary 3.3.

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

(317)

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

(318)

which satisfies (1.4) and the inequality

(319)

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

(41)

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

(42)

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

(43)

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

(44)

for some constants . It is easy to see the following topological inclusions:

(45)

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

(46)

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

(47)

Since for each , belongs to , the convolution

(48)

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

(49)

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

(410)

satisfying

  1. (i)

    There exist positive constants and such that

    (411)
  2. (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

(413)

We note that the Gauss transform

(414)

is well defined and locally uniformly as . Also satisfies semi-homogeneity property

(415)

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

(416)

where .

Proof.

Define . Convolving the tensor product of -dimensional heat kernels in we have

(417)

On the other hand, we figure out

(418)

and similarly we get

(419)

where is the Gauss transform of . Thus, inequality (4.6) is converted into the classical functional inequality

(420)

for all . In view of (4.20), it can be verified that

(421)

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

(422)

which is written in the form

(423)

for all , where . By virtue of the semi-homogeneous property of , substituting by , respectively, in (4.23) and dividing the result by we obtain

(424)

Using induction arguments and triangle inequalities we have

(425)

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

(426)

Letting , we have is a Cauchy sequence. Therefore we may define

(427)

for all . On the other hand, replacing by in (4.20), respectively, and then dividing the result by we get

(428)

Now letting we see by definition of that satisfies

(429)

for all . Letting in (4.25) yields

(430)

To prove the uniqueness of , we assume that is another function satisfying (4.29) and (4.30). Setting and in (4.29) we have

(431)

for all . Then it follows from (4.30) and (4.31) that

(432)

for all . Letting , we have for all . This proves the uniqueness.

It follows from the inequality (4.30) that we get

(433)

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

(434)

exists. Letting in (4.29) we have satisfies quartic functional equation (1.4). Letting we have the inequality

(435)

Now we consider the case . For this case, replacing by in (4.23), respectively, and letting and then multiplying the result by we have

(436)

Using induction argument and triangle inequality we obtain

(437)

for all . Following the similar method in case of , we see that

(438)

is the unique function satisfying (4.29) so that exists. Letting in (4.37) we get

(439)

Now letting in (4.39) we have the inequality

(440)

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

(441)

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

(442)

where .