Abstract

We prove the Ulam problem for the cosine addition formula in the spaces of Schwartz distributions and Sato hyperfunctions with respect to bounded distributions and bounded hyperfunctions.

1. Introduction

The Ulam problem for functional equations goes back to 1940 when Ulam proposed the following [1].

Let be a mapping from a group to a metric group with metric such thatThen does there exist a group homomorphism and such thatfor all ?

This problem was solved affirmatively by Hyers under the assumption that is a Banach space (see Hyers [2], Hyers et al. [3]). In 1949–1951, this result was generalized by Aoki [4] and Bourgin [5, 6]. Since then Ulam problems of many other functional equations have been investigated [7–16]. Among the results, Székelyhidi has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with Ulam problem for cosine functional equations [17, 18]. As a direct consequence of the elegant results of Székelyhidi, it was obtained that if satisfy for some , then either there exist , not both zero (either or is not zero), and such that for all , or for all , . Furthermore, the functions and satisfying both (3) and (4) are investigated.

In 1950, Laurent Schwartz introduced the theory of distributions in his monograph Théorie des distributions [19]. In this book Schwartz systematizes the theory of generalized functions, basing it on the theory of linear topological spaces, relates all the earlier approaches, and obtains many important results. After his elegant theory appeared, many important concepts and results on the classical spaces of functions have been generalized to the space of distributions. For example, positive functions and positive-definite functions have been generalized to positive distributions and positive-definite distributions, respectively, and it is shown that every positive distribution is a positive measure [20, page 38] and every positive-definite distribution is the Fourier transform of positive measure such that for some [21, page 157], which is called Bochner-Schwartz theorem and is a natural generalization of the famous Bochner theorem stating that every positive-definite function is the Fourier transform of a positive finite measure. For other examples, the Paley-Wiener theorem has been generalized to the Paley-Wiener-Schwartz theorem which characterizes the distributions with bounded supports [20, page 181]. The main purpose of this paper is to prove a Hyers-Ulam type stability problem for the cosine functional equation in Schwartz distributions and Gelfand hyperfunctions. As a generalization of inequality (3), it is very natural to consider the inequality where and are Lebesgue measurable functions and is the space of all bounded measurable functions defined in . Note that inequality (6) means that inequality (3) holds in almost everywhere sense. In [22–26], some stability problems of several functional equations including inequality (6) were considered in various spaces of generalized functions including Schwartz distributions. In [24–26], for example, replacing and by distributions and in inequality (6) we have considered where , , , and denote the pullback and tensor product of generalized functions. Inequality (7) cannot be considered as a complete formulation in the sense of generalized functions because the differences are assumed to be classical bounded measurable functions and all the previous results in [24–26] have the same formulations as in (7).

Due to Schwartz [19] the space of bounded measurable functions was generalized to the space of bounded distributions. Taking the above generalizations into account, it is very natural and is a complete generalization to consider the stability problem for cosine functional equation in distributions and hyperfunctions with respect to bounded distributions and bounded hyperfunctions, respectively: where and are the spaces of bounded distributions and bounded hyperfunctions, respectively, and , , and are the same as in (7). For some related results in Schwartz distributions, we refer the reader to [19, 20, 22, 23, 27–29].

The main tool of solving (8) and (9) is the heat kernel method initiated by Matsuzawa which represents the generalized functions as the initial values of solutions of the heat equation with some growth conditions [30–33].

2. Spaces of Distributions and Hyperfunctions and Some Preliminary Results

We first introduce the spaces of Schwartz tempered distributions and the space of Gelfand hyperfunctions (see [19–21, 33, 34] for more details of these spaces). We use the notations , , , , and , for , , where is the set of nonnegative integers and .

Definition 1 (see [19]). One denotes by or the Schwartz space of all infinitely differentiable functions in such that for all , , equipped with the topology defined by the seminorms . The elements of are called rapidly decreasing functions and the elements of the dual space are called tempered distributions.

Definition 2 (see [21, 34]). One denotes by or the Gelfand space of all infinitely differentiable functions in such that for some , . One says that as if as for some , and denotes by the strong dual space of and calls its elements Gelfand hyperfunctions.

As a generalization of the space of bounded measurable functions, Schwartz introduced the space of bounded distributions as a subspace of tempered distributions.

Definition 3 (see [19]). One denotes by the space of smooth functions on such that for all equipped with the topology defined by the countable family of seminorms One denotes by the strong dual space of and calls its elements bounded distributions.

Generalizing bounded distributions the space of bounded hyperfunctions has been introduced as a subspace of .

Definition 4 (see [32]). One denotes by the space of smooth functions on satisfying for some constant . One says that in as if there is a positive constant such that One denotes by the strong dual space of .

It is well known that the following topological inclusions hold:

It is known that the space consists of all infinitely differentiable functions on which can be extended to an entire function on satisfying for some , , (see [34]).

Definition 5. Let for . Then the tensor product of and , defined by for , belongs to .

For the proof of our theorems we employ the -dimensional heat kernel given by In view of (16), we can see that the heat kernel belongs to the Gelfand space for each . Thus, for each , the convolution is well defined. We call the Gauss transform of . From now on we denote by the Gauss transform of . It is well known that the Gauss transform is a smooth solution of the heat equation such that in weak star topology as ; that is, for all .

For the proof of our main result the following results are very useful. The proof of Theorem 2.3 of [35] works even when ; that is, we obtain the following.

Lemma 6 (see [35]). The Gauss transform of is a smooth solution of the heat equation satisfying the following.(i)There exist constants , such that (ii) as in the sense that, for every , Conversely, every smooth solution of the heat equation satisfying the estimate (20) can be uniquely expressed as for some .

The following lemma is a special case of Theorem 3.5 of [32] when where the space is denoted by .

Lemma 7 (see [32]). The Gauss transform of is a smooth solution of the heat equation satisfying the following.(i)For every there exists a constant such that (ii) as in the sense that, for every , Conversely, every smooth solution of the heat equation satisfying the estimate (22) can be uniquely expressed as for some .

The following structure theorem for bounded distributions is well known.

Lemma 8 (see [19]). Every can be expressed as for some , where are bounded measurable functions on . Equality (24) implies that for all .

As a special case of Theorem 3.4 of [32] when where the space is denoted by we obtain the following.

Lemma 9 (see [32]). Every can be expressed by where denotes the Laplacian, , are bounded continuous functions on , and , , satisfy the following estimates; for every there exists such that for all .

The following properties of the heat kernel will be useful which can be found in [33].

Proposition 10 (see [33]). For each , is an entire function and the following estimate holds; there exists such that Also for each , one has

3. Stability Problems with Time Variable

The main tools of the proof of our main result are based on the heat kernel method initiated by Matsuzawa [33] (see Lemmas 6 and 7 in Section 2) and the structure theorems (see Lemmas 8 and 9). Making use of the heat kernel and the structure theorems we convert (8) and (9) to the following classical stability problems for the Gauss transforms , of , , respectively:

In this section we prove the stability problems of inequality (30) with more general settings: let , with an Abelian group and let be a nonnegative real number. We consider the following stability problems, respectively:

From now on, a function from a semigroup to the field of complex numbers is said to be an additive function provided for all , and is said to be an exponential function provided for all . We introduce the following conditions on and :

For the proof of the following lemma we refer to [36, Lemma 2.2].

Lemma 11. Let , satisfy the inequality; for each and there exist positive constants and (resp., for each , , and there exists a positive constant ) such that for all , . Then either satisfies (33) (resp., (34)) or is an exponential function.

For the proof of the following lemma we refer to [36, Lemma 2.3].

Lemma 12. Let be a nonzero exponential function satisfying (33) (resp., (34)). Then can be written in the form where is an exponential function on satisfying for all and is an exponential function on satisfying for all .

For the proof of the following lemma we refer to [36, Lemma 2.4].

Lemma 13. Let be a nonzero exponential function satisfying (33) (resp., (34)). Suppose that satisfies the inequality; there exist and (resp., for every there exists ) such that for all , , , . Then one has where and are exponential functions on and , respectively, such that for all and for all , and is an additive function on and is a function satisfying (33) (resp., (34)).

Theorem 14. Let , satisfy (31) (resp., (32)). Then satisfies one of the following:(i)both and satisfy (33) (resp., (34));(ii) is an exponential function and satisfies (33) (resp., (34));(iii), , where , is a nonzero exponential function, and is a function satisfying (33) (resp., (34));(iv), , where is an additive function, and are exponential functions on and , respectively, such that for all and for all , and is a function satisfying (33) (resp., (34));(v) for all , , , .

Proof. We first prove that either there exist , , not both zero, and , (resp., for every , there exists ) such that or for all , , .
Suppose that inequality (39) holds only when . Then we can choose and satisfying . Let Then we can write where and . Using (41) and (42) we have
On the other hand, we can write By equating the right hand sides of (43) and (44) we have
If we fix , , , in (45), the right hand side of (45) satisfies (33) (resp., (34)). Thus, our assumption implies Now if we fix , , , in (46), the left hand side of (46) satisfies (33) (resp., (34)) as a function of and . Thus our assumption implies . Thus we have case (v).
From now on, we assume that inequality (39) holds: if satisfies (33) (resp., (34)), then cases (i) and (ii) follow immediately from Lemma 11. It is easy to see that if does not satisfy (33) (resp., (34)), then neither does . For this case, inequality (39) implies and we can write for some and a function satisfying (33) (resp., (34)).
Putting (47) in (31) (resp., (32)) using the triangle inequality and fixing , we have for all , and for some positive constants and (resp., for every there exists a positive constant ). Using Lemma 11, we have for all , , where is an exponential function on .
Now if , case (iii) follows. If , then we have for all , . Viewing (50) and using Lemma 12 we have where , for all , . Putting (51) in (31) (resp., (32)) and using the triangle inequality we have for all , . By Lemma 13 we have where is an additive function and is a function satisfying (33) (resp., (34)). Thus it follows from (51) and (54) that for all , . Replacing by we obtain (iv). This completes the proof.