关于Fermat型微分差分方程的亚纯解

On Meromorphic Solutions of Fermat Type Complex Differential and Difference Equations

本文研究了Fermat型微分及微分-差分方程亚纯解的存在性问题,证明了如果m,n为正整数,则不存在非常数亚纯函数f(z)满足微分方程f'(z)^m+f(z)^n= 1,但 m=2,n=3或 4和m=1, n=2除外。文中给出例子表明例外情况的方程亚纯解的存在性,并讨论该微分方程整函数解。同时,探讨了复微分-差分方程f'(z)^m+f(z+c)^n=1非常数亚纯解的存在性。

The existence of meromorphic solutions of Fermat type differential equations and differential-difference equations are investigated by value distribution theory and the complex difference theory in this article.We confirm that there don’t exist non-constant meromorp hic function f(z)that satisfy the differential equation f′(z)~m+f(z)~n=1(m,n are positive integers)except when m=2,n=3 or 4 and m=1,n=2.Some examples are given to illustrate the existence of meromorphic solutions of the equation for the particular cases,and we also study the entire function solution of the equation.Meanwhile,we also discuss the existence of non-constant meromorphic solutions of the differential-difference equation f′(z)~m+f(z+c)~n=1.