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Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài :Stability of an additive functional equation in the spaces of generalized functions | Lee Advances in Difference Equations 2011 2011 50 http www.advancesindifferenceequations.eom content 2011 1 50 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Stability of an additive functional equation in the spaces of generalized functions Young-Su Lee Correspondence masuri@sogang. ac.kr Department of mathematics Sogang university Seoul 121-741 Republic of Korea Springer Abstract We reformulate the following additive functional equation with n-independent variables nf t xj tfrt E fx xj i 1 i 1 1 i j n 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. Keywords Cauchy functional equation stability distribution heat kernel Gauss transform 1. Introduction A function f R 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 e R. It is well-known that every measurable solution of 1.1 is of the form f x ax for some constant a. In 1941 Hyers proved the stability theorem of 1.1 as follows Theorem 1.1 1 . Let E1 be a normed vector space E2 a Banach space. Suppose thatf E1 E2 satisfies the inequality II f x y - f x - f y II s for all x y e E1. Then there exists the unique additive mapping g E1 E2 such that II f x - g x II s for all x e E1. The above stability theorem was motivated by Ulam 2 . Forti 3 noticed that the theorem of Hyers is still true if E1 is replaced by an arbitrary semigroup. In 1978 2011 Lee licensee Springer.This is an Open Access article distributed under the terms ofthe Creative Commons Attribution License http creativecommons.org licenses by 2.0 which permits unrestricted use distribution and reproduction in any medium provided the
