tailieunhanh - Báo cáo hóa học: " Superstability of approximate d’Alembert harmonic functions"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Superstability of approximate d’Alembert harmonic functions | Kim et al. Journal of Inequalities and Applications 2011 2011 118 http content 2011 1 118 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Superstability of approximate d Alembert harmonic functions Hark-Mahn Kim 1 Gwang Hui Kim2 and Mi Hyun Han1 Correspondence good1014@cnu. department of Mathematics Chungnam National University 79 Daehangno Yuseong-gu Daejeon 305-764 Korea Full list of author information is available at the end of the article Springer Abstract In this article we study the superstability problem for the complex-valued functional equation f x y z f x y z f y z x f z x y 4f x f y f z on an abelian group and on a commutative semisimple Banach algebra. As a result we obtain application to harmonic functions satisfying the equation approximately. 1 Introduction The problem of stability of functional equations was originally stated by Ulam 1 . For Banach spaces Hyers 2 gave the first partial solution to Ulam s question in 1941 which states that if Ỏ 0 and f X Y is a mapping where X Y are Banach spaces such that f x y f x f y 8 for all x y e X then there exists a unique additive mapping T X e Y such that Ilf x T x 8 for all x e X. And then Aoki 3 and Bourgin 4 generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences for approximately additive mappings. Rassias 5 succeeded in extending the result of Hyers for approximate linear mappings by weakening the condition for the Cauchy difference to be unbounded. The stability phenomenon that was presented by Rassias may be called the generalized Hyers-Ulam stability. This terminology may also be applied to the cases of other functional equations see 6 . The stability problem for functional equations has been extensively investigated by a number of mathematicians 7-15 . On the other hand there is a strong stability phenomenon which is known as a superstability. An equation of a .