一类复微分方程的亚纯允许解的值分布

刘瑞,高凌云

(暨南大学数学系,广东 广州 510632)

一类复微分方程的亚纯允许解的值分布

刘瑞,高凌云

(暨南大学数学系,广东 广州 510632)

利用亚纯函数的Nevanlinna值分布理论,研究了一类复高阶微分方程的亚纯允许解的存在性问题.证明了在适当条件的假设下,该类复微分方程的亚纯解不是允许解的结果,推广了以前一些文献的结论,并且文中有例子表明结果是精确的.

值分布;高阶微分方程;亚纯允许解

1 引言与主要结果

全文采用Nevanlinna值分布理论的通常记号(见文献[1-2]).

关于涉及亚纯函数的微分多项式和微分方程亚纯解的值分布问题,一些作者已做了讨论[35],并得到了一些结果.2002年,文献[4]中研究了高阶代数微分方程:

2 一些引理

3 定理1.2的证明

4 两个例子

[1]何育赞,肖修治.代数微分方程与常微分方程[M].北京:科学出版社,1988.

[2]仪洪勋,杨重骏.亚纯函数的唯一性理论[M].北京:科学出版社,1995.

[4]高凌云.高阶微分方程允许解的存在性[J].数学杂志,2003,23(3):381-384.

[5]Gao Lingyun.The analytic form of solutions of a type of algebraic di ff erential equations[J].Pure and Applied Mathematics,2005,21(4):305-309.

[6]Toda N.On the growth of meromorphic solutions of some higher-order di ff erential equations[J].J.Math. Soc.Japan,1986,38(3):439-451.

[7]Toda N.On the conjecture of Gackstatter and Laine concerning the di ff erential equation[J].Kodai Math.J.,1983,6(2):238-249.

The value distribution of meromorphic admissible solutions of higher-order di ff erential equations

Liu Rui,Gao Lingyun
(Department of Mathematics,Ji′nan University,Guangzhou 510632,China)

Using Nevanlinna theory of the value distribution of meromorphic functions,the problem of the existence of solutions of complex higher-order di ff erential equation is investigated.And under the assumption of certain proper condition,we prove a result which the meromorphic solutions of the di ff erential equation must be non-admissible solutions and improve some results in previous references.At the end of this paper,the examples are given to show that the result is precise.

value distribution,higher-order di ff erential equations,admissible solutions

O174.52

A

1008-5513(2012)01-0025-04

2011-06-08.

国家自然科学基金(10471065);广东省自然科学基金(04010474).

刘瑞(1986-),硕士生,研究方向：复分析与微分方程.

2010 MSC:30D05,30D35,34M05