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Báo cáo hóa học: " Research Article Stability of a Quadratic Functional Equation in the Spaces of Generalized Functions"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Stability of a Quadratic Functional Equation in the Spaces of Generalized Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 210615 12 pages doi 10.1155 2008 210615 Research Article Stability of a Quadratic Functional Equation in the Spaces of Generalized Functions Young-Su Lee Department of Mathematical Sciences Korea Advanced Institute of Science and Technology 373-1 Guseong-dong Yuseong-gu Daejeon 305-701 South Korea Correspondence should be addressed to Young-Su Lee masuri@kaist.ac.kr Received 30 June 2008 Accepted 20 August 2008 Recommended by Laszlo Losonczi Making use of the pullbacks we reformulate the following quadratic functional equation f x y z f x f y f z f x y f y z f z x in the spaces of generalized functions. Also using the fundamental solution of the heat equation we obtain the general solution and prove the Hyers-Ulam stability of this equation in the spaces of generalized functions such as tempered distributions and Fourier hyperfunctions. Copyright 2008 Young-Su Lee. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Functional equations can be solved by reducing them to differential equations. In this case we need to assume differentiability up to a certain order of the unknown functions which is not required in direct methods. From this point of view there have been several works dealing with functional equations based on distribution theory. In the space of distributions one can differentiate freely the underlying unknown functions. This can avoid the question of regularity. Actually using distributional operators it was shown that some functional equations in distributions reduce to the classical ones when the solutions are locally integrable functions 1-4 . Another approach to distributional analogue for functional equations is via the use of the regularization of distributions 5 6 . More exactly