Stability in Generalized Functions

and Applied Analysis 3 where a ∈ C. Also, we shall figure out that every solution u in S′ R or F′ R of the inequality 1.6 can be written uniquely in the form


Introduction
The most famous functional equation is the Cauchy equation any solution of which is called additive.It is well known that every measurable solution of 1.1 is of the form f x ax for some constant a.In 1941, Hyers proved the stability theorem for 1.1 as follows.
Theorem 1.1 see 1 .Let E 1 be a normed vector space, E 2 a Banach space.Suppose that f : E 1 → E 2 satisfies the inequality for all x, y ∈ E 1 , then there exists the unique additive mapping g : The above stability theorem was motivated by Ulam 2 .As noted in the above theorem, the stability problem of the functional equations means how the solution of the inequality differs from the solution of the original equation.Forti 3 noticed that the theorem of Hyers is still true if E 1 is replaced by an arbitrary semigroup.In 1950 Aoki 4 and in 1978 Rassias 5 generalized Hyers' result to the unbounded Cauchy difference.Thereafter, many authors studied the stability problems for 1.1 in various settings see 6, 7 .During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors see 8-17 .Among them, the following additive functional equation with n-independent variables: was proposed by Nakmahachalasint 18 , where n is a positive integer with n > 1 and x 0 ≡ x n .He solved the general solutions and the stability problems for the above equation.Actually, he proved that 1.4 is equivalent to 1.1 .
In this paper, in a similar manner as in 19-23 , we solve the general solutions and the stability problems for 1.4 in the spaces of generalized functions such as the space S R m of tempered distributions, the space F R m of Fourier hyperfunctions, and the space D R m of distributions.Making use of the pullbacks, we first reformulate 1.4 and the related inequality in the spaces of generalized functions as follows: where A, P i , and B i are the functions defined by

1.7
Here • denotes the pullback of generalized functions, and the inequality v ≤ in 1.6 means that | v, ϕ | ≤ ϕ L 1 for all test functions ϕ.
In Section 2, we will prove that every solution u in S R m or F R m of 1.5 has the form where a ∈ C m .Also, we shall figure out that every solution u in S R m or F R m of the inequality 1.6 can be written uniquely in the form where μ is a bounded measurable function such that μ L ∞ ≤ 10n − 3 / 2 2n − 1 .Subsequently, in Section 3, these results are extended to the space D R m .

Stability in F R m
We first introduce the spaces of tempered distributions and Fourier hyperfunctions.Here, we use the m-dimensional notations, |α| for some positive constants A, B depending only on ϕ.The strong dual of F R m , denoted by F R m , is called the Fourier hyperfunction.
It can be verified that the seminorm 2.3 is equivalent to for some constants h, k > 0. It is easy to see the following topological inclusions: Taking the inclusions 2.5 into account, it suffices to consider the space F R m .In order to solve the general solutions and the stability problems for 1.4 in the spaces F R m and S R m , we employ the m-dimensional heat kernel, fundamental solution of the heat equation,

2.6
Since for each t > 0, E •, t belongs to the space F R m , the convolution is well defined for all u in F R m , which is called the Gauss transform of u.Subsequently, the semigroup property of the heat kernel is very useful to convert 1.5 into the classical functional equation defined on upper-half plane.We also use the following famous result, so-called heat kernel method, which states as follows.
satisfying the following: i there exist positive constants C, M, and N such that Conversely, every C ∞ -solution U x, t of the heat equation satisfying the growth condition 2.10 can be uniquely expressed as U x, t u x, t for some u ∈ S R m .
Similarly, we can represent Fourier hyperfunctions as a special case of the results as in 28 .In this case, the estimate 2.10 is replaced by the following.
For every > 0, there exists a positive constant C such that We need the following lemma in order to solve the general solutions for the additive functional equation in the spaces of F R m and S R m .In what follows, we denote x 0 ≡ x n and t 0 ≡ t n .Lemma 2.4.Suppose that f : R m × 0, ∞ → C is a continuous function satisfying for all x 1 , . . ., x n ∈ R m , t 1 , . . ., t n > 0, then the solution f has the form for some a ∈ C m .
From the above lemma, we can solve the general solutions for the additive functional equation in the spaces of F R m and S R m .Theorem 2.5.Every solution u in F R m (or S R m , resp.) of 1.5 has the form for some a ∈ C m .
Proof.Convolving the tensor product E t 1 x 1 • • • E t n x n of the heat kernels on both sides of 1.5 , we have where u is the Gauss transform of u.Thus, 1.5 is converted into the following classical functional equation: for some a ∈ C m .Letting t → 0 in 2.26 , we finally obtain the general solution for 1.5 .
We are going to solve the stability problems for the additive functional equation in the spaces of F R m and S R m .Lemma 2.6.Suppose that f : R m × 0, ∞ → C is a continuous function satisfying for all x 1 , . . ., x n ∈ R m , t 1 , . . ., t n > 0, then there exists a unique a ∈ C m such that for all x ∈ R m , t > 0.
Proof.Putting x 1 , . . ., x n 0, . . ., 0 in 2.27 yields for all k ∈ N, x ∈ R m , t > 0. Replacing x, t by 2 l x, 2 l t in 2.37 , respectively, and dividing the result by 2 l , we see that for k ≥ l > 0,

2.38
Since the right-hand side of 2.38 tends to 0 as l → ∞, the sequence {2 −k f 2 k x, 2 k t } is a Cauchy sequence which converges uniformly.Thus, we may define for all x ∈ R m , t > 0. Now, we verify from 2.27 that the function A satisfies From the above lemma, we have the following stability theorem for the additive functional equation in the spaces of F R m and S R m .Theorem 2.7.Suppose that u in F R m (or S R m , resp.) satisfies the inequality 1.6 , then there exists a unique a ∈ C m such that

43
Proof.Convolving the tensor product x n of the heat kernels on both sides of 1.6 , we have for all x 1 , . . ., x n ∈ R m , t 1 , . . ., t n > 0, where u is the Gauss transform of u.By Lemma 2.6, we have for all x ∈ R m , t > 0. Letting t → 0 in 2.45 , we obtain the conclusion.

Stability in D R m
In this section, we shall extend the previous results to the space of distributions.Recall that a distribution u is a linear functional on C ∞ c R m of infinitely differentiable functions on R m with compact supports such that for every compact set K ⊂ R m , there exist constants C > 0 and N ∈ N 0 satisfying The set of all distributions is denoted by D R m .It is well known that the following topological inclusions hold: As we see in 19,20,23 , by the semigroup property of the heat kernel, 1.5 can be controlled easily in the spaces F R m and S R m .But we cannot employ the heat kernel in the space D R m .For that reason, instead of the heat kernel, we use the regularizing functions.We denote by ψ the function on R m satisfying where

3.4
It is easy to see that ψ is an infinitely differentiable function supported in the set {x : |x| ≤ 1} with ψ x dx 1.For each t > 0, we define ψ t x : t −m ψ x/t , then ψ t has all the properties of ψ except that the support of ψ t is contained in the ball of radius t with center at 0. If u ∈ D R m , then for each t > 0, u * ψ t x u y , ψ t x − y is a smooth function in R m and u * ψ t x → u as t → 0 in the sense of distributions, that is, for every For each t > 0, the function u * ψ t is called a regularization of u, and the transform which maps u to u * ψ t is called a mollifier.Making use of the mollifiers, we can solve the general solution for the additive functional equation in the space D R m as follows.for some a ∈ C m .
Proof.Convolving the tensor product ψ t 1 x 1 • • • ψ t n x n of the regularizing functions on both sides of 1.5 , we have

3.7
Thus, 1.5 is converted into the following functional equation:   Inequality 3.30 implies that h x : u − g x belongs to L 1 L ∞ .Thus, we conclude that u g x h x ∈ S R m .

Theorem 3 . 1 .
Every solution u in D R m of 1.5 has the form u a • x, 3.6

− 2 u 25 which is equivalent to 2f x y 2 −
* ψ t x − u * ψ s y − u * ψ s −y − 2n − 5 f 0 ≤ , 3.20 for all x, y ∈ R m , t, s > 0. It follows from 3.19 that the inequality 3.20 can be rewritten asu * ψ t * ψ s x y − u * ψ t * ψ s y − x − 2 u * ψ t x 2f 0 ≤ 2 , 3.21 for all x, y ∈ R m , t, s > 0. Letting t → 0 in 3.21 yields u * ψ s x y − u * ψ s y − x − 2f x 2f 0 ≤ 2 , 3.22for all x, y ∈ R m , s > 0. Using 3.19 we may write the inequality 3.22 asu * ψ s x y u * ψ s x − y − 2f x 2n − 1 f 0 ≤ 3 , 3.23 for all x, y ∈ R m , t, s > 0. Putting y 0 in 3.23 and dividing the result by 2 give u * ψ s x − f x 2n − x ∈ R m , s > 0. From 3.23 and 3.24 , we have f x y f x − y − 2f x ≤ 6 , 3.f x − f y ≤ 6 , 3.26 for all x, y ∈ R m .Thus, by virtue of the result as in 29 , there exists a unique function g : R m → C satisfying x − g x ≤ 6 g 0 , 3.28 for all x ∈ R m .It follows from 3.18 , 3.24 , and 3.28 that u * ψ s x − g x ≤ 8 g 0 , 3.29 for all x ∈ R m , s > 0. Letting s → 0 in 3.29 , we obtain u − g x ≤ 8 g 0 .3.30 where N 0 is the set of nonnegative integers and ∂ j ∂/∂ζ j .Definition 2.2 see 26 .We denote by F R m the set of all infinitely differentiable functions ∈ R m , t > 0. Substituting x 1 , x 2 , x 3 , . . ., x n with x, y, 0, . . ., 0 and letting t 1 t, t 2 s, t 3 • • • t n → 0 in 2.13 , we obtain from 2.17 that which shows that f x, t is independent with respect to t > 0. For that reason, we see from 2.18 that F x : f x, t satisfiesF x y F x − y 2F x , 2.20for all x, y ∈ R m .Replacing x by x y /2 and y by x − y /2 in 2.20 , we haveF x y F x F y , 2.21for all x, y ∈ R m .Given the continuity, we obtain , . . ., x n ∈ R m , t 1 , . . ., t n > 0. In view of 3.8 , it is easy to see that, for each fixed x ∈ R m , ∈ R m , t > 0. Substituting x 1 , x 2 , x 3 , . . ., x n with x, y, 0, . . ., 0 and letting t 1 t, t 2 s, t 3 • • • t n → 0 in 3.8 , we obtain from 3.10 that → 0 in 3.13 , we finally obtain the general solution for 1.5 .Now, we shall extend the stability theorem for the additive equation mentioned in the previous section to the space D R m .Suppose that u in D R m satisfies the inequality 1.6 , then there exists a unique a ∈ C m such that It suffices to show that every distribution satisfying 1.6 belongs to the space S R m .Convolving the tensor product ψ t 1 x 1 • • • ψ t n x n on both sides of 1.6 , we have which is equivalent to the Cauchy equation 1.1 for all x, y ∈ R m .Since f is a smooth function in view of 3.13 , it follows that f x a • x for some a ∈ C m .Letting s t 1 , . . ., t n > 0. In view of 3.16 , it is easy to see that for each fixed x, , x 2 , . . ., x n x, 0, . . ., 0 and letting t 1 t, t 2 • • • t n → 0 in 3.16 , we have u * ψ t x u * ψ t −x 2n − 3 f 0 ≤ , 3.19 for all x ∈ R m , t > 0. Substituting x 1 , x 2 , x 3 , . . ., x n with x, y, 0, . . ., 0 and letting t 1 t, t 2 s, t 3 • • • t n → 0 in 3.16 , we have u * ψ t * ψ s x y − u * ψ t * ψ s y − x