一阶迭代泛函微分方程的局部可逆解析解

本文研究迭代泛函微分方程x′(z)=1/(x(az+b/(x′(z)))),z∈C的解析解,其中a,b均为复常数.首先利用Schr(o|¨)der变换,把迭代泛函微分方程转化为不含迭代的泛函微分方程.针对Schr(o|¨)der变换中的常数α在单位圆上,不是单位根但满足Brjuno条件;α不但在单位圆上,而且是单位根;α在单位圆内三种情况,讨论了辅助方程的解析解.在此基础上,我们证明原方程局部可逆解析解存在,并且计算出解析解表达式.最后举例说明定理的应用.

Local Invertible Analytic Solutions of a First Order Iterative Functional Differential Equation

This paper concerns the analytic solutions of iterative functional differential equation x'(z)=1/(x(az+(b/(x'(z))))), z∈C where a,b are complex numbers. First of all, we reduce the equation with the Schröder transformation to another functional differential equation without iteration. Then analytic solutions of auxiliary equation are discussed under the following conditions: the constant α given in Schröder transformation is on the unit circle, but not a root of the unity, the Brjuno condition is satisfied; α is not only on the unit circle, but also a root of the unity; α is inside of the unit circle. Following this, we prove the existence of local invertible analytic solution of primary function, and calculate its explicit form. Finally, our main result is illustrated with an example.