- Open Access
Stability of quadratic functional equations in tempered distributions
© Lee; licensee Springer 2012
- Received: 12 September 2011
- Accepted: 6 August 2012
- Published: 16 August 2012
We reformulate the following quadratic functional equation:
as the equation for generalized functions. Using the fundamental solution of the heat equation, we solve the general solution of this equation and prove the Hyers-Ulam stability in the spaces of tempered distributions and Fourier hyperfunctions.
- quadratic functional equation
- tempered distribution
- heat kernel
- Gauss transform
In 1940, Ulam  raised a question concerning the stability of group homomorphisms as follows: The case of approximately additive mappings was solved by Hyers  under the assumption that is a Banach space. In 1978, Rassias  generalized Hyers’ result to the unbounded Cauchy difference.
Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?
where k is a fixed positive integer. They proved the Hyers-Ulam-Rassias stability of this equation in Banach spaces. Wang  considered the intuitionistic fuzzy stability of (1.2) by using the fixed-point alternative. Saadati and Park  proved the Hyers-Ulam-Rassias stability of (1.2) in non-Archimedean -fuzzy normed spaces.
Here, ∘ denotes the pullback of generalized functions and the inequality in (1.4) means that for all test functions φ. We refer to  for pullbacks and to [2, 7–11] for more details of the spaces of generalized functions.
In this section, we introduce the spaces of tempered distributions and Fourier hyperfunctions. Here, we use the n-dimensional notations. If , where is the set of nonnegative integers, then , . For , we denote and .
2.1 Tempered distributions
We present a very useful space of test functions for the tempered distributions as follows.
for all . The set of all tempered distributions is denoted by .
for all , where is the Fourier transform of φ.
Note that tempered distributions are generalizations of -functions. These are very useful for the study of Fourier transforms in generality, since all tempered distributions have a Fourier transform.
2.2 Fourier hyperfunctions
Imposing the growth condition on in (2.1) Sato and Kawai introduced the new space of test functions for the Fourier hyperfunctions as follows.
Definition 2.2 ()
for some positive constants A, B depending only on φ. The strong dual of , denoted by , is called the Fourier hyperfunction.
for some .
of the heat kernel is very useful to convert Eq. (1.3) into the classical functional equation defined on upper-half plane. We also use the following famous result, the so-called heat kernel method, which is stated as follows.
Theorem 3.1 ()
- (i)There exist positive constants C, M, and N such that(3.1)
- (ii)as in the sense that for every ,
Conversely, every -solution of the heat equation satisfying the growth condition (3.1) 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 (3.1) is replaced by the following:
Here, we need the following lemma to solve the general solution of (1.3).
for some .
for all , . □
As a direct consequence of the above lemma, we present the general solution of the quadratic functional equation (1.3) in the spaces of generalized functions.
for some .
as in (3.8). □
In this section, we are going to solve the stability problem of (1.4). For the case of in (1.4), the result is known as follows.
We here need the following lemma to solve the stability problem of (1.4).
for all , , . Taking the limit as , we conclude that for all , . □
We now state and prove the main theorem of this paper.
This completes the proof. □
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