Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Approximate Cauchy functional inequality in quasi-Banach spaces | Kim and Son Journal of Inequalities and Applications 2011 2011 102 http content 2011 1 102 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Approximate Cauchy functional inequality in quasi-Banach spaces Hark-Mahn Kim and Eunyoung Son Correspondence sey8405@ Department of Mathematics Chungnam National University 79 Daehangno Yuseong-gu Daejeon 305-764 Korea Springer Abstract In this article we prove the generalized Hyers-Ulam stability of the following Cauchy functional inequality I If x f y nf z II nf IKx y x I n in the class of mappings from n-divisible abelian groups to p-Banach spaces for any fixed positive integer n 2. 1 Introduction The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. We are given a group G1 and a metric group G2 with metric p . Given e 0 does there exist a Ỗ 0 such that iff G1 G2 satisfies p f xy f x f y ỏ for all x y e G1 then a homomorphism h G1 G2 exists with p fx h x e for all x e G1 In other words we are looking for situations when the homomorphisms are stable i. e. if a mapping is almost a homomorphism then there exists a true homomorphism near it. In 1941 Hyers 2 considered the case of approximately additive mappings between Banach spaces and proved the following result. Suppose that E1 and E2 are Banach spaces and f E1 E2 satisfies the following condition there is a constant e 0 such that If x y - f x - y II s . f 2nx for all x y e E1. Then the limit h x lirn TO v 7 exists for all x e E1 and it is a n 2n unique additive mapping h E1 E2 such that fx - h x e. The method which was provided by Hyers and which produces the additive mapping h was called a direct method. This method is the most important and most powerful tool for studying the stability of various functional equations. Hyers theorem was generalized by Aoki 3 and Bourgin 4 for additive mappings by .
