劉瑞,高凌云

(暨南大學(xué)數(shù)學(xué)系,廣東 廣州 510632)

一類復(fù)微分方程的亞純允許解的值分布

劉瑞,高凌云

(暨南大學(xué)數(shù)學(xué)系,廣東 廣州 510632)

利用亞純函數(shù)的Nevanlinna值分布理論,研究了一類復(fù)高階微分方程的亞純允許解的存在性問題.證明了在適當(dāng)條件的假設(shè)下,該類復(fù)微分方程的亞純解不是允許解的結(jié)果,推廣了以前一些文獻(xiàn)的結(jié)論,并且文中有例子表明結(jié)果是精確的.

值分布;高階微分方程;亞純允許解

1 引言與主要結(jié)果

全文采用Nevanlinna值分布理論的通常記號(見文獻(xiàn)[1-2]).

關(guān)于涉及亞純函數(shù)的微分多項(xiàng)式和微分方程亞純解的值分布問題,一些作者已做了討論[35],并得到了一些結(jié)果.2002年,文獻(xiàn)[4]中研究了高階代數(shù)微分方程:

2 一些引理

3 定理1.2的證明

4 兩個(gè)例子

[1]何育贊,肖修治.代數(shù)微分方程與常微分方程[M].北京:科學(xué)出版社,1988.

[2]儀洪勛,楊重駿.亞純函數(shù)的唯一性理論[M].北京:科學(xué)出版社,1995.

[4]高凌云.高階微分方程允許解的存在性[J].數(shù)學(xué)雜志,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.

國家自然科學(xué)基金(10471065);廣東省自然科學(xué)基金(04010474).

劉瑞(1986-),碩士生,研究方向：復(fù)分析與微分方程.

2010 MSC:30D05,30D35,34M05