1. Introduction

The Hyers-Ulam stability of functional equations go back to 1940 when Ulam proposed the following problem [1]:

Let be a mapping from a group to a metric group with metric such that

(1.1)

Then does there exist a group homomorphism    and    such that

(1.2)

for all?

This problem was solved affirmatively by Hyers [2] under the assumption that is a Banach space. In 1949-1950, this result was generalized by the authors Bourgin [3, 4] and Aoki [5] and since then stability problems of many other functional equations have been investigated [2, 6–8, 8–19]. In 1990, Székelyhidi [18] has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with stability questions for the sine and cosine functional equations. We refer the reader to [6–8, 12–14, 19] for Hyers-Ulam stability of functional equations of trigonometric type. In this paper, following the method of Székelyhidi [18] we consider a distributional analogue of the Hyers-Ulam stability problem of the trigonometric functional equations

(1.3)
(1.4)

where . Following the formulations as in [6, 20–22], we generalize the classical stability problems of above functional equations to the spaces of generalized functions as

(1.5)
(1.6)

where , and denote the pullback and the tensor product of generalized functions, respectively, and denotes the space of bounded measurable functions on .

We prove as results that if generalized function satisfies (1.5), then satisfies one of the followings

(i) and is arbitrary;

(ii) and are bounded measurable functions;

(iii);

(iv), ,

for some , and a bounded measurable function .

Also if generalized function satisfies (1.6), then satisfies one of the followings:

(i) and are bounded measurable functions,

(ii), , .

2. Generalized Functions

For the spaces of tempered distributions , we refer the reader to [23–25]. Here we briefly introduce the spaces of Gelfand generalized functions and Fourier hyperfunctions. Here we use the following notations: , , , , and , for , , where is the set of nonnegative integers and .

Definition 2.1.

For given , we denote by or the space of all infinitely differentiable functions on such that there exist positive constants and satisfying

(2.1)

The topology on the space is defined by the seminorms in the left-hand side of (2.1) and the elements of the dual space of are called Gelfand-Shilov generalized functions. In particular, we denote by and call its elements Fourier hyperfunctions.

It is known that if and , the space consists of all infinitely differentiable functions on that can be continued to an entire function on satisfying

(2.2)

for some .

It is well known that the following topological inclusions hold:

(2.3)

We refer the reader to [24, chapter V-VI], for tensor products and pullbacks of generalized functions.

3. Stability of the Equations

In view of (2.2), it is easy to see that the -dimensional heat kernel

(3.1)

belongs to the Gelfand-Shilov space for each . Thus the convolution is well defined for all . It is well known that is a smooth solution of the heat equation in and in the sense of generalized functions, that is, for every ,

(3.2)

We call the Gauss transform of . Let be a semigroup and be the field of complex numbers. A function is said to be additive provided , and is said to be exponential provided .

We first discuss the solutions of the corresponding trigonometric functional equations in the space of Gelfand generalized functions. As a consequence of the result [6, 26], we have the following.

Lemma 3.1.

The solutions of the equation

(3.3)

are equal, respectively, to the smooth solution of corresponding classical functional equations (1.3) and (1.4).

Remark 3.2.

We refer the reader to Aczél [27, page 180] and Aczél and Dhombres [28, pages 209–217] for the general solutions and measurable solutions of (1.3) and (1.4).

For the proof of the stability of (1.5), we need the following lemma.

Lemma 3.3.

Let be continuous functions satisfying the inequality; there exists a positive constant such that

(3.4)

for all . Then either there exist , not both zero, and such that

(3.5)

or else

(3.6)

for all .

Also the inequality (3.5) together with (3.4) implies one of the followings;

(i), : arbitrary;

(ii) and are bounded functions;

(iii) and where , , with and is a bounded function.

Proof.

Following the approach as in [29, page 139, Lemma 6.8], we first prove that (3.6) is satisfied if the condition (3.5) fails. Assume that for some implies . Let

(3.7)

Then we can choose and satisfying . It is easy to show that

(3.8)

where , and .

Using (3.8), we have the following two equations.

(3.9)

From (3.9), we have

(3.10)

When are fixed, the right-hand side of the above equation is bounded, so we have the following.

(3.11)

Again considering (3.11) as a function of and for all fixed , we have .

Now we consider the case that the inequality (3.5) holds. If , is arbitrary, which is the case (i). If is a nontrivial bounded function, in view of (3.4) is also bounded, which is the case (ii). If is unbounded, it follows from (3.5) that and

(3.12)

for some and a bounded function . Put (3.12) in (3.4) to get

(3.13)

Replacing by and by and using the triangle inequality, we have

(3.14)

for all . Replacing by , by and using the inequality (3.14), we have for some ,

(3.15)

Using (3.13), (3.14), (3.15), and the triangle inequality, we have

(3.16)

Since is unbounded, it follows from (3.16) that for all . Also, in view of (3.13), for fixed and , is a bounded function of and . Thus it follows from [24, page 104, Theorem 5.2] that is an exponential function. Given the continuity of , we have for some with . Replacing by in (3.13) and using (3.14), we have

(3.17)

Now we consider the stability of (3.17). From (3.17) and the continuity of , it is easy to see that

(3.18)

exists. Putting and letting so that in (3.17) and using the triangle inequality and (3.14), we have

(3.19)

It follows from (3.17), (3.19), and the triangle inequality that

(3.20)

for all . Letting in (3.20), we have

(3.21)

for all . Thus it follows from the Hyer-Ulam stability theorem [2] and the continuity of that there exists such that

(3.22)

for all . Finally, from (3.19) and (3.22), we have

(3.23)

for all . From (3.12) and (3.23), we have (iii). This completes the proof.

Remark 3.4.

In particular, if and are solutions of the heat equation the case (iii) of the abovelemma is reduced as

(3.24)

for some and bounded solution of the heat equation.

Theorem 3.5.

Let satisfy (1.5). Then and satisfy one of the followings

(i) and is arbitrary;

(ii) and are bounded measurable functions;

(iii), ;

(iv), ,

for some , and a bounded measurable function .

Proof.

Convolving in (1.5) the tensor product of -dimensional heat kernels, we have

(3.25)

Similarly, we have

(3.26)

where are the Gauss transforms of , respectively. Thus satisfies the inequality (3.4). Now we can apply Lemma 3.3. First we assume that (3.5) holds and consider the cases (i), (ii), (iii) of Lemma 3.3. The case (i) implies (i) of our theorem. For the case (ii); it follows from [30, Page 61, Theorem 1] the initial values are bounded measurable functions, respectively. For the case (iii); using the remark running after Lemma 3.3 and [30, Page 61, Theorem 1], letting the case (iii) of our theorem follows. Finally, if satisfy the (3.6),letting we have

(3.27)

The nontrivial solutions of (3.27) are given by (iv) or which is included in case (iii). This completes the proof.

Now we prove the stability of (1.6).

Lemma 3.6.

Let satisfy the inequality; there exists a positive constant such that

(3.28)

for all . Then either there exist , not both zero, and such that

(3.29)

or else

(3.30)

for all .

Also the inequality (3.29) together with (3.28) implies one of the followings

(i) and are bounded functions;

(ii) is bounded; is an exponential;

(iii) for some bounded exponential ;

(iv), , where , is a bounded exponential and is a bounded function.

Proof.

Suppose that, for , does not hold unless . Note that both and are unbounded. Let

(3.31)

Just for convenience, we consider the following equation which is equivalent to (3.31).

(3.32)

Since is nonconstant, we can choose and satisfying . It is easy to show that

(3.33)

where and .

By the definition of and the use of (3.33), we have the following equations

(3.34)

By equating (3.34), we have

(3.35)

When are fixed, the right-hand side of the above equality is bounded, so we have

(3.36)

Again considering (3.36) as a function of and for all fixed , we have which is equivalent to (3.30). Now we consider the case when (3.29) holds. If is bounded, then is also bounded by the inequality (3.28). It follows from [29, Theorem 5.2] that is bounded or exponential which gives the cases (i) and (ii). If is unbounded, then is also unbounded by the inequality (3.28), hence and . Now let for some bounded function and . Then the inequality (3.28) becomes

(3.37)

Since is bounded, we find that

(3.38)

Thus it follows from [29, page 104, Theorem 5.2] that

(3.39)

for some exponential . Thus if , we have and (iii) follows. If ,

(3.40)

which gives (iv). This completes the proof.

Theorem 3.7.

Let satisfy (1.6). Then and satisfy one of the followings

(i) and are bounded measurable functions;

(ii), ,

where .

Proof.

Similarly as in the proof of Theorem 3.5 convolving in (1.6) the tensor product , we obtain the inequality (3.28) where are the Gauss transforms of , respectively. Now we can apply Lemma 3.6. First assume that (3.29) holds and consider the cases (i), (ii), (iii), and (iv) of Lemma 3.6. For the case (i), it follows from [30, Page 61, Theorem 1] that the initial values of as are bounded measurable functions, respectively. For the case (ii), by the continuity of , we have

(3.41)

for some . Putting (3.41) in (3.28) and letting , and using the triangle inequality, we have

(3.42)

for some . In view of (3.42), we have . Thus is bounded in . Using [25, page 61, Theorem 1], the initial values of as in (3.41) are bounded measurable functions. For the case (iii) putting in the inequality (3.28), we have

(3.43)

for all , where is a bounded exponential. Using the continuity of , it follows from (3.43) that is bounded in and so is , which implies that both and are bounded measurable functions. For the case (iv) since , are unbounded continuous, is unbounded continuous, and . On the other hand, it follows from (3.28) that , which occurs only when . Thus both and are bounded in and are bounded measurable functions.

Secondly, assume that (3.30) holds. Letting in (3.30), we have

(3.44)

By Lemma 3.1, the nonconstant solution of (3.44) is given by , for some . This completes the proof.

Remark 3.8.

Taking the growth of as into account, or only when for some . Thus the Theorems 3.5 and 3.7 are reduced to the followings.

Corollary 3.9.

Let or satisfy (1.5). Then and satisfy one of the followings

(i) and is arbitrary;

(ii) and are bounded measurable functions;

(iii), only ,

for some , , and a bounded measurable function .

Corollary 3.10.

Let or satisfy (1.6). Then and are bounded measurable functions.