Stability of quadratic functional equations in generalized functions
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RESEARCH
Open Access
Stability of quadratic functional equations in generalized functions Young-Su Lee* *
Correspondence: [email protected] Hana Academy Seoul, Seoul, Korea
Abstract In this paper, we consider the following generalized quadratic functional equation with n-independent variables in the spaces of generalized functions:
(f (xi + xj ) + f (xi – xj )) = 2(n – 1)
1≤i , does there exist a δ > such that if a function h : G → G satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G , then there exists a homomorphism H : G → G with d(h(x), H(x)) < for all x ∈ G ? The case of approximately additive mappings was solved by Hyers [] under the assumption that G is a Banach space. In , Rassias [] generalized Hyers’ result to the unbounded Cauchy difference. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [–]). Quadratic functional equations are used to characterize the inner product spaces. Note that a square norm on an inner product space satisfies the parallelogram equality x + y + x – y = x + y © 2013 Lee; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Lee Advances in Difference Equations 2013, 2013:72 http://www.advancesindifferenceequations.com/content/2013/1/72
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for all vectors x, y. By virtue of this equality, the following functional equation is induced: f (x + y) + f (x – y) = f (x) + f (y).
(.)
It is easy to see that the quadratic function f (x) = ax on a real field, where a is an arbitrary constant, is a solution of (.). Thus, it is natural that (.) is called a quadratic functional equation. It is well known that a function f between real vector spaces satisfies (.) if and only if there exists a unique symmetric biadditive function B such that f (x) = B(x, x) (see [–]). The biadditive function B is given by B(x, y) =
f (x + y) – f (x – y) .
The Hyers-Ulam stability for quadratic functional equation (.) was proved by Skof []. Thereafter, many authors studied the stability problems of (.) in various settings (see [– ]). In particular, the Hyers-Ulam-Rassias stability of (.) was proved by Czerwik []. Recently, Eungrasamee et al. [] considered the following functional equation:
n f (xi + xj ) + f (xi – xj ) = (n – ) f (xi ),
≤i . They proved that (.) is equivalent to (.). Also, they proved the Hyers-Ulam-Rassias stability of this equation. In this paper, we solve the general solutions and the stability problems of (.) in the spaces of generalized functions such as S of tempered distributions, F of Fourier hyperfunctions and D of distributions. Using the notions as in [–], we reformulate (.) and the related inequality in the spaces of genera
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