一类二阶迭代泛函微分方程的解析解

在复域C内研究了一类含有未知函数迭代的二阶微分方程λ2x″(z)+λ1x′(z)+λ0x(z)=f(∑mj=0cjxj(z))+G(z)的解析解的存在性。通过Schrder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λ2［α2y″(αz)y′(z)-αy′(αz)y″(z)］+λ1αy′(αz)(y′(z))2+λ0y(αz)(y′(z))3=(y′(z))3［f(∑mj=0cjy(αjz))+G(y(z))］,并给出了它的局部可逆解析解。讨论了双曲型情形0<|α|<1和共振的情形,还在Brjuno条件下讨论了在共振点附近的情形。

Analytic solutions of a second-order iterative functional differential equation

The second-order differential equation involving iterates of the ilnknown function λ_2x″(z)+λ_1x′(z)+λ_0x(z)=f(Σ_(j=0)~mc_jx~j(z))+G(z) is investigated in the complex field C for the existence of analytic solutions.By reducing the equation with the Schr(o)der transformtion,x(z)=y(αy~(-1)(z)),to another functional differential equation without iteration of the unknown function λ_2α~2y″(αz)y′(z)-αy′(αz)y″(z)]+λ_1αy'(αz)(y′(z))~2+λ_0y(az)(y′(z))~3=(y′(z))~3f(Σ_(j=0)~mc_jy(α~jz))+G(y(z))],we sive the existence of its local invertible analytic solutions.We discuss not only those α given in the Schr(o)der transformation in the hyperbolic case 0＜|α|＜1 and resonance,but also those α near resonance under Brjuno condifion.