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Báo cáo hóa học: " Research Article On the Stability of Trigonometric Functional Equations"

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 Trigonometric Functional Equations | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 90405 10 pages doi 10.1155 2007 90405 Research Article On the Stability of Trigonometric Functional Equations Gwang Hui Kim Received 17 February 2007 Accepted 5 October 2007 Recommended by Bing Gen Zhang The aim of this paper is to study the superstability related to the d Alembert the Wilson the sine functional equations for the trigonometric functional equations as follows f x y - f x - y 2f x g y f x y - f x - y 2g x f y . Copyright 2007 Gwang Hui Kim. 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 Baker et al. 1 and Bourgin 2 introduced that if f satisfies the stability inequality E1 f - E2 f I e then either f is bounded or E1 f E2 f . This is now frequently referred to as superstability. The superstability of the cosine functional equation also called the d Alembert functional equation f x y f x - y 2f x f y A and the sine functional equation x y 2 x - y 2 f x f y Ạ2 - Ạ 2 S are investigated by Baker 3 and Cholewa 4 respectively. The d Alembert functional equation A is generalized to the following functional equations f x y f x - y 2f x g y f x y f x - y 2g x f y . Afg Agf 2 Advances in Difference Equations Equation Afg raised by Wilson is sometimes called the Wilson equation. We will consider the trigonometric functional equation as follow f x y - f x - y 2f x f y T f x y - f x - y 2f x g y Tfg f x y - f x - y 2g x f y . Tgf The cosine-type functional equations A Afg Agf and sine functional equation have been investigated by Badora Cholewa Ger Kannappan Kim and so forth 3-9 . Given mappings f G C we will denote a difference operator DA G X G C as DA x y f x y f x - y - 2f x f y . 1.1 Badora and Ger 6 proved the superstability under the condition DA x y I ọ x or ọ y for the d Alembert equation A .