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 Functional Inequalities Associated with Jordan-von Neumann-Type Additive Functional | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 41820 13 pages doi 2007 41820 Research Article Functional Inequalities Associated with Jordan-von Neumann-Type Additive Functional Equations Choonkil Park Young Sun Cho and Mi-Hyen Han Received 27 September 2006 Accepted 1 November 2006 Recommended by Sever S. Dragomir We prove the generalized Hyers-Ulam stability of the following functional inequalities llf x f y f z 2f x y z 2 f x f y f z f x y z H I If x f y 2 f z 11 I12 f x y 2 z I in the spirit of the Rassias stability approach for approximately homomorphisms. Copyright 2007 Choonkil Park 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 and preliminaries Ulam 1 gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms. We are given a group G and a metric group G with metric p - - . Given e 0 does there exist a 8 0 such that if f G G satisfies p f xy f x f y 8 for all x W-G then a homomorphism h G G exists with p f x h x e for all G G Hyers 2 considered the case of approximately additive mappings f E E where E and E are Banach spaces and f satisfies Hyers inequality f x y - f x - f y e for all x y G E. It was shown that the limit L x lim n- -oo f 2nx 2n 2 Journal of Inequalities and Applications exists for all XE E and that L E E is the unique additive mapping satisfying f X -L x e. Rassias 3 provided a generalization of Hyers theorem which allows the Cauchy difference to be unbounded. Theorem Rassias . Let f E E be a mapping from a normed vector space E into a Banach space E subject to the inequality f X y - f x - f y e II x p 11 y p for all X y eE where e and p .
