Abstract
We reformulate the following additive functional equation with n-independent variables as the equation for the spaces of generalized functions. Making use of the fundamental solution of the heat equation we solve the general solutions and the stability problems of this equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the regularizing functions, we extend these results to the space of distributions. 2000 MSC: 39B82; 46F05.
Highlights
In 1940, Ulam [1] raised a question concerning the stability of group homomorphisms: “Let f be a mapping from a group G1 to a metric group G2 with metric d(·, ·) such that d f, f (x) f (y) ≤ ε
Does there exist a group homomorphism L : G1 → G2 and δ > 0 such that d f (x),L(x) ≤ δ for all x ∈ G1?” The case of approximately additive mappings was solved by Hyers [2] under the assumption that G1 and G2 are Banach spaces
The stability problems of several functional equations have been extensively investigated by a number of authors
Summary
In 1940, Ulam [1] raised a question concerning the stability of group homomorphisms: “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) ≤ ε. For all x ∈ G1?” The case of approximately additive mappings was solved by Hyers [2] under the assumption that G1 and G2 are Banach spaces. The stability problems of several functional equations have been extensively investigated by a number of authors (see [4,5,6,7,8,9,10,11,12]). The terminology Hyers-Ulam-Rassias stability originates from these historical backgrounds and this terminology is applied to the case of other functional equations
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