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Báo cáo hóa học: "Research Article The C1 Solutions of the Series-Like Iterative Equation with Variable Coefficients/;

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 The C1 Solutions of the Series-Like Iterative Equation with Variable Coefficients | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 173028 7 pages doi 10.1155 2009 173028 Research Article The C1 Solutions of the Series-Like Iterative Equation with Variable Coefficients Yuzhen Mi Xiaopei Li and Ling Ma Mathematics and Computational School Zhanjiang Normal University Zhanjiang Guangdong 524048 China Correspondence should be addressed to Xiaopei Li lixp27333@sina.com Received 23 March 2009 Revised 11 June 2009 Accepted 6 July 2009 Recommended by Tomas Dominguez Benavides By constructing a structure operator quite different from that ofZhang and Baker 2000 and using the Schauder fixed point theory the existence and uniqueness of the C1 solutions of the series-like iterative equations with variable coefficients are discussed. Copyright 2009 Yuzhen Mi 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. 1. Introduction An important form of iterative equations is the polynomial-like iterative equation X1f x X2f2 x Xnfn x F x x e I a b 1.1 where F is a given function f is an unknown function Xi e R1 i 1 2 . n and fk k 1 2 . n is the kth iterate of f that is f0 x x fk x f o fk-1 x . The case of all constant Xis was considered in 1-10 . In 2000 W. N. Zhang and J. A. Baker first discussed the continuous solutions of such an iterative equation with variable coefficients Xi Xi x which are all continuous in interval a b . In 2001 J. G. Si and X. P. Wang furthermore gave the continuously differentiable solution of such equation in the same conditions as in 11 . In this paper we continue the works of 11 12 and consider the series-like iterative equation with variable coefficients TO VJXi x fi x F x x e I a b i 1 1.2 2 Fixed Point Theory and Applications where Ằị x I 0 1 are given continuous functions and 2i ikfx 1 A1 x c 0 Vx e Ĩ maxxeITi x ci. We improve the