Abstract
We consider the Hyers-Ulam stability of a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions, Fourier hyperfunctions, and Gelfand generalized functions.
Highlights
The Hyers-Ulam stability of functional equations go back to 1940 when Ulam proposed the following problem 1 : Let f be a mapping from a group G1 to a metric group G2 with metric d ·, · such that d f xy, f x f y ≤
Journal of Inequalities and Applications following the method of Szekelyhidi 18 we consider a distributional analogue of the HyersUlam stability problem of the trigonometric functional equations f x−yfxgy −g x f y, 1.3 g x−ygxgyfxfy, 1.4 where f, g : Rn → C
Following the formulations as in 6, 20–22, we generalize the classical stability problems of above functional equations to the spaces of generalized functions u, v as u ◦ S − u ⊗ v v ⊗ u ∈ L∞ R2n, 1.5 v ◦ S − v ⊗ v − u ⊗ u ∈ L∞ R2n, 1.6 where S x, y x − y, x, y ∈ Rn, ◦ and ⊗ denote the pullback and the tensor product of generalized functions, respectively, and L∞ R2n denotes the space of bounded measurable functions on R2n
Summary
The Hyers-Ulam stability of functional equations go back to 1940 when Ulam proposed the following problem 1 : Let f be a mapping from a group G1 to a metric group G2 with metric d ·, · such that d f xy , f x f y ≤. In 1949-1950, this result was generalized by the authors Bourgin 3, 4 and Aoki 5 and since stability problems of many other functional equations have been investigated 2, 6–8, 8–19. Journal of Inequalities and Applications following the method of Szekelyhidi 18 we consider a distributional analogue of the HyersUlam stability problem of the trigonometric functional equations f x−yfxgy −g x f y , 1.3 g x−ygxgyfxfy , 1.4 where f, g : Rn → C. We prove as results that if generalized function u, v satisfies 1.5 , u, v satisfies one of the followings:. If generalized function u, v satisfies 1.6 , u, v satisfies one of the followings:. I u and v are bounded measurable functions, ii u cos c · x , v sin c · x , c ∈ Cn
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