在共振点附近的一类二阶泛函微分方程的解析解

在复域C内研究一类包含未知函数迭代的二阶微分方程x″(z)=G(z,x(z),x~2(z),…,x~m(z))解析解的存在性.通过Schr(?)der变换,即x(z)=y(αy~(-1)(z)),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″(αz)y″(z)-αy′(αz)y″(z)= (y′(z))~3G(y(z),y(αz),…,y(α~mz)),并给出它的局部可逆解析解.本文不仅讨论了双曲型情形0<|α|<1和共振的情形(α是一个单位根),而且还在Brjuno条件下讨论了共振点附近的情形(即单位根附近).

Analytic Solutions for a Second-Order Iterative Functional Differential Equation near Resonance

In this paper,the second-order differential equation involving iterates of the unknown function x″(z)=G(z,x(z),x~2(z),...,x~m(z))is investigated in the com- plex field C for the existence of analytic solutions.By reducing the equation with the Schr(?)der transformation,x(z)=y(αy~(-1)(z)),to another functional differential equation without iteration of the unknown functionα~2y″(αz)y′(z)-αy′(αz)y″(z)= (y′(z))~3G(y(z),y(αz),...,y(α~mz)),we give the existence of its local invertible analytic solutions.We discuss not only thoseαgiven in Schr(?)der transformation in the hyper- bolic case 0<|α|<1 and resonance,i.e.,at a root of the unity,but also thoseαnear resonance(i.e.,near a root of the unity)under Brjuno condition.