Stability of Quartic Functional Equations in the Spaces of Generalized Functions
© Y.-S. Lee and S.-Y. Chung. 2009
Received: 6 August 2008
Accepted: 14 January 2009
Published: 10 February 2009
We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.
Lee et al.  obtained the general solution of (1.3) and proved the Hyers-Ulam-Rassias stability of this equation. Also Park  investigated the stability problem of (1.3) in the orthogonality normed space.
for fixed integer with . In the cases of in (1.4), homogeneity property of quartic functional equations does not hold. We dispense with this cases henceforth, and assume that . In Section 2, we show that for each fixed integer with , (1.4) is equivalent to (1.3). Moreover, using the idea of Găvruţa , we prove the Hyers-Ulam-Rassias stability of (1.4) in Section 3. Finally, making use of the pullbacks and the heat kernels, we reformulate and prove the Hyers-Ulam-Rassias stability of (1.4) in the spaces of some generalized functions such as of tempered distributions and of Fourier hyperfunctions in Section 4.
2. General Solution of (1.4)
Stability problems of quadratic functional equations can be found in [16–19]. Similarly, a function satisfies the quartic functional equation (1.3) if and only if there exists a symmetric biquadratic function such that for all (see ). We now present the general solution of (1.4) in the class of functions between real vector spaces.
which proves the validity of (1.4) for . For a negative integer , replacing by one can easily prove the validity of (1.4). Therefore (1.3) implies (1.4) for any fixed integer with .
for each fixed integer . Replacing by in (2.15), and comparing (1.4) with (2.15) we have . Thus (2.14) implies (1.3). This completes the proof.
3. Stability of (1.4)
Now we are going to prove the Hyers-Ulam-Rassias stability for quartic functional equations. Let be a real vector space and let be a Banach space.
for all . Also, if for each fixed the mapping from to is continuous, then for all .
for all . Letting , we must have for all . This completes the proof.
for all .
for all .
4. Stability of (1.4) in Generalized Functions
In this section, we reformulate and prove the stability theorem of the quartic functional equation (1.4) in the spaces of some generalized functions such as of tempered distributions and of Fourier hyperfunctions. We first introduce briefly spaces of some generalized functions. Here we use the multi-index notations, , , and , for , , where is the set of non-negative integers and .
for all . The set of all tempered distributions is denoted by .
Imposing growth conditions on in (4.1) a new space of test functions has emerged as follows.
Definition 4.2 (see ).
for some positive constants depending only on . We say that as if as for some , and denote by the strong dual of and call its elements Fourier hyperfunctions.
where . Here denotes the pullbacks of generalized functions. Also denotes the Euclidean norm and the inequality in (4.6) means that for all test functions defined on . We refer to (see [20, Chapter VI]) for pullbacks and to [21, 23–26] for more details of and .
If , the right side of (4.6) does not define a distribution. Thus, the inequality (4.6) makes no sense in this case. Also, if , it is not known whether Hyers-Ulam-Rassias stability of (1.4) holds even in the classical case. Thus, we consider only the case or .
holds for convolution. Semigroup property will be useful to convert inequality (3.3) into the classical functional inequality defined on upper-half plane. Moreover, the following result called heat kernel method holds .
- (i)There exist positive constants and such that(411)
- (ii)as in the sense that for every ,(412)
Conversely, every -solution of the heat equation satisfying the growth condition (4.11) 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 (see ). In this case, the estimate (4.11) is replaced by the following.
for all .
We are now in a position to state and prove the main result of this paper.
for all . Letting , we have for all . This proves the uniqueness.
This completes the proof.
As an immediate consequence, we have the following corollary.
The first author was supported by the second stage of the Brain Korea 21 Project, The Development Project of Human Resources in Mathematics, KAIST, in 2009. The second author was supported by the Special Grant of Sogang University in 2005.
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