tailieunhanh - Báo cáo hóa học: " Research Article On the Stability of a General Mixed Additive-Cubic Functional Equation in Random Normed Spaces"

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 On the Stability of a General Mixed Additive-Cubic Functional Equation in Random Normed Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 328473 16 pages doi 2010 328473 Research Article On the Stability of a General Mixed Additive-Cubic Functional Equation in Random Normed Spaces Tian Zhou Xu 1 John Michael Rassias 2 and Wan Xin Xu3 1 Department of Mathematics School of Science Beijing Institute of Technology Beijing 100081 China 2 Pedagogical Department . Section of Mathematics and Informatics National and Kapodistrian University of Athens 4 Agamemnonos Str. Aghia Paraskevi 15342 Athens Greece 3 School of Communication and Information Engineering University of Electronic Science and Technology of China Chengdu 611731 China Correspondence should be addressed to Tian Zhou Xu xutianzhou@ Received 6 June 2010 Revised 1 August 2010 Accepted 23 August 2010 Academic Editor Radu Precup Copyright 2010 Tian Zhou Xu 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. We prove the generalized Hyers-Ulam stability of the following additive-cubic equation f kx y f fkx - y kf x y kf x - y 2f fkx - 2kf x in the setting of random normed spaces. 1. Introduction A basic question in the theory of functional equations is as follows when is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation If the problem accepts a unique solution we say the equation is stable see 1 . The first stability problem concerning group homomorphisms was raised by Ulam 2 in 1940 and affirmatively solved by Hyers 3 . The result of Hyers was generalized by Rassias 4 for approximate linear mappings by allowing the Cauchy difference operator CDf x y f x y - f x f y to be controlled by e x p llynp . In 1994 a generalization of Rassias theorem was obtained by Gavruta 5 who replaced e x p yhp by a .