一类二阶迭代泛函微分方程在共振点附近的解析解的存在性

The existence of analytic solutions of a second-order iterative functional differential equation near resonance

本文在复域C内研究了二阶迭代微分方程x″(x^([r])(z))=(x^([m])(z))~2,r,m≥2;r,m∈N解析解的存在性.通过Schr(o|¨)der变换,即x(z)=y(α^(-1)(z)),作者把这类方程转化为一种不含未知函数迭代的泛函微分方程α~2y″(α^(r+1)z)y′(α~rz)=αy′(α^(r+1)z)y″(α~rz)+(y′(α~rz))~3(y(α~mz))~2,并给出它的局部可逆解析解.本文不仅讨论了双曲型情形|α|>1,0<|α|<1和共振的情形(α是一个单位根),而且还在Brjuno条件下讨论了近共振点情形(即单位根附近).

In this paper, the second-order iterate differential equation x＂（x^[r]（z））=（x^[m]（z））^2,r,m≥2;r,m∈N is investigated in the complex field C for the existence of analytic solutions. By reducing the equation with the Schrosder transformation, x（z）=y（ay^-1 （z））, to the another functional differential equation without iteration of the unknown function a62 y^n （a^r＋1 z） y＇ （a＇ z） =ay＇ （a^r＋1 z） y＂ （a^r z） ＋ （ y＇ （a^r z ） ）3 （y（a^m z））^2 ,the author obtains existence of its local invertible analytic solutions. Then, the author discusses not only these a given in Schr6der transformation at the hyperbolic case ｜a｜〉1,0〈｜a｜〈1 and resonance（i, e. , at a root of the unity）, but also those a near resonance （i. e. , near a root of the unity） under Brjuno condition.