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Báo cáo hoa học: "Research Article On the Superstability Related with the Trigonometric Functional Equation"

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 Superstability Related with the Trigonometric Functional Equation | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 503724 11 pages doi 10.1155 2009 503724 Research Article On the Superstability Related with the Trigonometric Functional Equation Gwang Hui Kim Department of Mathematics Kangnam University Youngin Gyeonggi 446-702 South Korea Correspondence should be addressed to Gwang Hui Kim ghkim@kangnam.ac.kr Received 22 August 2009 Accepted 6 November 2009 Recommended by Patricia J. Y. Wong We will investigate the superstability of the hyperbolic trigonometric functional equation from the following functional equations f x y g x -y Xf x g y f x y g x-y Xg x f y f x y g x - y Xf x f y f x y g x - y Xg x g y which can be considered the mixed functional equations of the sine function and cosine function of the hyperbolic sine function and hyperbolic cosine function and of the exponential functions respectively. Copyright 2009 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. in 1 introduced the following if f satisfies the inequality E1 f - E2 f e then either f is bounded or E1 f E2 f . This is frequently referred to as superstability. The superstability of the cosine functional equation also called the d Alembert equation f x y f x - y 2f tofty C and the sine functional equation 2 2 S 2 Advances in Difference Equations were investigated by Baker 2 and Cholewa 3 respectively. Their results were improved by Badora 4 Badora and Ger 5 Forti 6 and Gavruta 7 as well as by Kim 8 9 and Kim and Dragomir 10 . The superstability of the Wilson equation f x y f x - y 2f x g y Cfg was investigated by Kannappan and Kim 11 . The superstability of the trigonometric functional equation with the sine and the cosine equation f x y - f x - y 2f WHy T f x y - f x - y 2f x g y Tfg was investigated by Kim 12 . The .