tailieunhanh - Báo cáo hóa học: " Research Article Fixed Points and Hyers-Ulam-Rassias Stability of Cauchy-Jensen Functional Equations in Banach Algebras"

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 Fixed Points and Hyers-Ulam-Rassias Stability of Cauchy-Jensen Functional Equations in Banach Algebras | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 50175 15 pages doi 2007 50175 Research Article Fixed Points and Hyers-Ulam-Rassias Stability of Cauchy-Jensen Functional Equations in Banach Algebras Choonkil Park Received 16 April 2007 Accepted 25 July 2007 Recommended by Billy E. Rhoades We prove the Hyers-Ulam-Rassias stability of homomorphisms in real Banach algebras and of generalized derivations on real Banach algebras for the following Cauchy-Jensen functional equations f x y 2 z f x y 2 z f x 2 f z 2f x y 2 z f x f y 2f z which were introduced and investigated by Baak 2006 . The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias stability theorem that appeared in his paper 1978 . Copyright 2007 Choonkil Park. 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 The stability problem of functional equations originated from a question of Ulam 2 concerning the stability of group homomorphisms let G1 be a group and let G2 o d be a metric group with the metric d - . Given e 0 does there exist a 8 e 0 such that if a mapping h G1 G2 satisfies the inequality d h x y h x o h y 8 for all x y e G1 then there is a homomorphism H G1 G2 with d h x H x e for all x e G1 If the answer is affirmative we would say that the equation of homomorphism H x y H x o H y is stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that 2 Fixed Point Theory and Applications how do the solutions of the inequality differ from those of the given functional equation Hyers 3 gave a first affirmative answer to the question of Ulam for Banach spaces. Let X and Y be