On the stability of a mixed type functional equation in generalized functions
© Lee; licensee Springer. 2012
Received: 18 November 2011
Accepted: 16 February 2012
Published: 16 February 2012
We reformulate the following mixed type quadratic and additive functional equation with n-independent variables
as the equation for the spaces of generalized functions. Using the fundamental solution of the heat equation, we solve the general solution and prove the Hyers-Ulam stability of this equation in the spaces of tempered distributions and Fourier hyperfunctions.
Mathematics Subject Classification 2000: 39B82; 39B52.
In 1940, Ulam  raised a question concerning the stability of group homomorphisms as follows:
Let G1 be a group and let G2 be a metric group with the metric d(·,·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1, then there exists a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x ∈ G1?
for all x ∈ X. For this reason, equation (1.1) is called the mixed type quadratic and additive functional equation. We refer to [9–14] for the stability results of other mixed type functional equations.
Here ○ denotes the pullback of generalized functions and the inequality ||v|| ≤ ε in (1.3) means that for all test functions φ.
where µ is a bounded measurable function such that
In this section, we introduce the spaces of tempered distributions and Fourier hyperfunctions. We first consider the space of rapidly decreasing functions which is a test function space of tempered distributions.
for all nonnegative integers α, β.
In other words, φ(x) as well as its derivatives of all orders vanish at infinity faster than the reciprocal of any polynomial. For that reason, we call the element of as the rapidly decreasing function. It can be easily shown that the function φ(x) = exp(−ax2), a > 0, belongs to , but ψ(x) = (1 + x2)−1 is not a member of . Next we consider the space of tempered distributions which is a dual space of .
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, but not all distributions have one. Imposing the growth condition on || · ||α,βin (2.1) a new space of test functions has emerged as follows.
for some positive constants A, B depending only on φ.
for some constants h, k > 0.
Definition 2.4.  The strong dual space of is called the Fourier hyperfunctions. We denote the Fourier hyperfunctions by .
holds for convolution. It is useful to convert equation (1.2) into the classical functional equation defined on upper-half plane. We also use the following famous result 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 initial values of solutions of the heat equation as a special case of the results as in . In this case, the condition (i) in the above theorem is replaced by the following:
3. General solution in
for all x1, . . . , x n ∈ ℝ, t1, . . . , t n > 0. We here need the following lemma which will be crucial role in the proof of main theorem.
for all x ∈ ℝ, t > 0.
where c = c1 + c2.
Conversely, if f (x, t) = ax2 + bx + c for some a, b, c ∈ ℂ, then it is obvious that f satisfies equation (3.1). □
According to the above lemma, we solve the general solution of (1.2) in the space of (or , resp.) as follows.
for some a, b ∈ ℂ.
for some a, b, c ∈ ℂ. Letting t → 0+ in (3.8), we finally obtain the general solution of (1.2). □
4. Stability in
In this section, we are going to state and prove the Hyers-Ulam stability of (1.3) in the space of (or , resp.).
for all x ∈ ℝ, t > 0.
for all x ∈ ℝ, t > 0, where
for all x ∈ ℝ, t > 0.
for some b, c2 ∈ ℂ.
for all x ∈ ℝ, t > 0, where c = c1 + c2. □
From the above lemma we immediately prove the Hyers-Ulam stability of (1.3) in the space of (or , resp.) as follows.
for all x ∈ ℝ, t > 0. Letting t → 0+ in (4.18) finally we have the stability result (4.17). □
where µ is a bounded measurable function such that .
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