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 Superstability of Generalized Multiplicative Functionals | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 486375 7 pages doi 2009 486375 Research Article Superstability of Generalized Multiplicative Functionals Takeshi Miura 1 Hiroyuki Takagi 2 Makoto Tsukada 3 and Sin-Ei Takahasi1 1 Department of Applied Mathematics and Physics Graduate School of Science and Engineering Yamagata University Yonezawa 992-8510 Japan 2 Department of Mathematical Sciences Faculty of Science Shinshu University Matsumoto 390-8621 Japan 3 Department of Information Sciences Toho University Funabashi Chiba 274-8510 Japan Correspondence should be addressed to Takeshi Miura miura@ Received 2 March 2009 Accepted 20 May 2009 Recommended by Radu Precup Let X be a set with a binary operation o such that for each x y z e X either x o y o z x o z o y or zo xoy xo zoy . We show the superstability of the functional equation g xoy g x g y . More explicitly if e 0 and f X C satisfies If x o y - f x f y for each x y e X then f x o y f x f y for all x y e X or If x 1 V1 4e 2 for all x e X. In the latter case the constant 1 V1 4e 2 is the best possible. Copyright 2009 Takeshi Miura et al. 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 It seems that the stability problem of functional equations had been first raised by S. M. Ulam cf. 1 Chapter VI . For what metric groups G is it true that an e-automorphism of G is necessarily near to a strict automorphism An e-automorphism of G means a transformation f of G into itself such that pf x ỳ f x f y e for all x y e G. D. H. Hyers 2 gave an affirmative answer to the problem if e 0 and f E1 E2 is a mapping between two real Banach spaces E1 and E2 satisfying f x y - f x - f y e for all x y e E1 then there exists a unique additive mapping T E1 E2 such that f x - T x e for .
