Stability of an additive functional equation in the spaces of generalized functions
© Lee; licensee Springer. 2011
Received: 24 July 2011
Accepted: 2 November 2011
Published: 2 November 2011
We reformulate the following additive functional equation with n-independent variables
as the equation for the spaces of generalized functions. Making use of the fundamental solution of the heat equation we solve the general solutions and the stability problems of this equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the regularizing functions, we extend these results to the space of distributions.
2000 MSC: 39B82; 46F05.
KeywordsCauchy functional equation stability distribution heat kernel Gauss transform
for all x, y ∈ ℝ. 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 of (1.1) as follows:
for all x ∈ E 1.
The above stability theorem was motivated by Ulam . Forti  noticed that the theorem of Hyers is still true if E 1 is replaced by an arbitrary semigroup. In 1978, Rassias  generalized Hyers' result to the unbounded Cauchy difference. Thereafter, many authors studied the stability problems of (1.1) in various settings (see [5–7]).
was proposed by Nakmahachalasint , where n is a positive integer with n > 1. He proved that (1.2) is equivalent to (1.1). For that reason, we say that (1.2) is a generalization of the Cauchy functional equation. The stability theorem of (1.2) was also proved.
Here, ○ denotes the pullback of generalized functions and the inequality ||v|| ≤ ε in (1.4) means that for all test functions φ.
where and μ is a bounded measurable function such that . Subsequently, in Section 3, these results are extended to the space .
2. Stability in
We first introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the m-dimensional notations, |α| = α 1 + ⋯ + α m , α! = α 1! ⋯ α m !, and , for ζ = (ζ 1, ..., ζ m ) ∈ ℝ m , , where ℕ0 is the set of non-negative integers and .
for all . The set of all tempered distributions is denoted by .
Note that tempered distributions are generalizations of L p -functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform. Imposing the growth condition on ||·|| α,β in (2.1) a new space of test functions has emerged as follows:
for some positive constants A, B depending only on φ. The strong dual of , denoted by , is called the Fourier hyperfunction.
of the heat kernel is very useful to convert Equation (1.3) 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:
Conversely, every C ∞ -solution U(x, t) of the heat equation satisfying the growth condition (2.4) can be uniquely expressed as for some .
Similarly, we can represent Fourier hyperfunctions as a special case of the results as in . In this case, the estimate (2.4) is replaced by the following:
We are now going to solve the general solutions and the stability problems of (1.2) in the spaces of and . Here, we need the following lemma.
where , .
where , . □
From the above lemma, we can find the general solutions of (1.2) in the spaces of and . Taking the inclusions of (2.3) into account, it suffices to consider the space .
This completes the proof. □
From the above theorem, we have the general solution of (1.1) in the spaces of and immediately.
We are going to prove the stability theorem of (1.2) in the spaces of and as follows:
Now inequality (2.18) implies that u - a · x belongs to (L 1)' = L ∞. Thus, all the solution u in can be written uniquely in the form u = a · x + h(x), where . □
From the above theorem, we shall prove the stability theorem of (1.1) in the spaces of and as follows:
3. Stability in
Making use of the regularizing functions we can find the general solution of (1.2) in the space as follows:
This completes the proof. □
In a similar manner, we have the following corollary immediately.
Using the regularizing functions, Chung  extended the stability theorem of the Cauchy functional equation (1.1) to the space . Similarly, we shall extend the stability theorem of (1.2) mentioned in the previous section to the space .
Inequality (3.13) implies that h(x) : = u - g(x) belongs to (L 1)' = L ∞. Thus, we conclude that . □
From the above theorem, we have the following corollary.
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