Abstract
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.
Highlights
1 Introduction In, Ulam [ ] raised a question concerning the stability of group homomorphisms as follows: Let G be a group and let G be a metric group with the metric d(·, ·)
The case of approximately additive mappings was solved by Hyers [ ] under the assumption that G is a Banach space
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)
Summary
In , Ulam [ ] raised a question concerning the stability of group homomorphisms as follows: Let G be a group and let G be a metric group with the metric d(·, ·). Stability problems of various functional equations have been extensively studied and generalized by a number of authors (see [ , , , , , , ]). The stability problem of the following quadratic functional equation f (x + y) + f (x – y) = f (x) + f (y). Quadratic functional equations are used to characterize the inner product spaces. F (kx + y) + f (kx – y) = k f (x) + f (y), where k is a fixed positive integer They proved the Hyers-Ulam-Rassias stability of this equation in Banach spaces. ) as the equation for generalized functions and proved that every solution of ) and the related inequality in the spaces of generalized functions as follows:.
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