Analytic Solutions of an Iterative Differential Equation under Brjuno Condition

作     者：Jian LIU Jian Guo SI 

作者机构：School of Science University of Ji'nan Ji'nan 250022 P.R. China School of Mathematics Shandong University Ji'nan 250100 P.R. China 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2009年第25卷第9期

页      面：1469-1482页

核心收录：

学科分类：080701[工学-工程热物理] 07[理学] 08[工学] 0807[工学-动力工程及工程热物理] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by Natural Science Foundation of University of Ji'nan (Grant No. XKY0704) the second author is partially supported by National Natural Science Foundation of China (Grant No. 10871117) NSFSP (Grant No. Y2006A07) 

主　　题：iterative differential equation analytic solution Banach fixed point theorem resonance Diophantine condition Brjuno condition 

摘      要：In this paper, the differential equation involving iterates of the unknown function, x＇（z）=[a^2-x^2（z）]x^[m]（z） with a complex parameter a, is investigated in the complex field C for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the SchrSder transformation x（z） = y（αy^-1（z）） to the following another functional differential equation without iteration of the unknown function αy＇（αz）=[a^2-y^2（αz）]y＇（z）y（α^mz）, which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y（αy^-1 （z）） for the original iterative differential equation are obtained. We discuss not only these α given in SchrSder transformation in the hyperbolic case 0 〈 ｜α｜ 〈 1 and resonance, i.e., at a root of the unity, but also those α near resonance （i.e., near a root of the unity） under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.