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
A function f : R Âź R is called an additive function if and only if it satisfies the Cauchy functional equation f (x + y) = f (x) + f (y) (1:1)for all x, y Ă R
The above stability theorem was motivated by Ulam [2]
Recall that a distribution u is a linear functional on Câ c (Ăm) of infinitely differentiable functions on Rm with compact supports such that for every compact set K â Rm there exist constants C > 0 and N Ă N0 satisfying
Summary
(1.2) and the related inequality in the spaces of generalized functions as follows: n n u ⊠A = u ⊠Pi + We prove that every solution u in S (Ăm) or F (Ăm) of the inequality (1.4) can be written uniquely in the form u = a · x + ÎŒ(x), where a â m and ÎŒ is a bounded measurable function such that ÎŒ We denote by S(Ăm) the Schwartz space of all infinitely differentiable functions in Rm satisfying Ï Î±,ÎČ = sup | xαâÎČ Ï(x) |< â
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