摘要
本文主要研究亚纯函数系数微分方程解的复振荡,分别考虑一类高阶情形和一类二阶情形。全文共分为四个部分。
第一部分(引言)介绍了国内外研究现状,并引入了一些相关定义和记号,及本文所要解决的问题。
第二部分(第二章)考察了Kwon Ki-Ho关于二阶整函数系数微分方程所得的结果,在一定条件下,得到了二阶亚纯函数系数情形的部分结论。
第三部分(第三章)考察Kwon Ki-Ho关于二阶整函数系数微分方程解的增长性研究。对方程的某个系数作适当限制后,得出高阶亚纯函数系数情形的类似结论。
第四部分(第四章)研究了一类高阶亚纯函数解的增长性,在一定程度上改进了陈宗煊,肖丽鹏等的结果,得到了高阶线性微分方程亚纯解级的精确估计。
In this paper,we mainly study the complex oscillation of solutions of differential equations on meromorphic function coefficients,and respectively consider two kinds of equations,including the second order and the higher order.It includes the following four parts.
In part 1(Introduction),we give an introduction of present situation of this field in domestic and foreign,bright in some related definitions and signs,and give the questions that in this paper will be considered.
In part 2(Chapter 2),we study the result made by Kwon Ki-Ho of the second order differential equations on entire function coefficients.In some conditions,we obtain the similar parts of conclusions of the second order differential equations on meromorphic function coefficients.
In part 3(Chapter 3),we investigate the growth of solutions of the second order differential equations on entire function coefficients.We obtain the similar conclusion of higher order differential equations on meromorphic function coefficients,when some coefficients of equations satisfy some special conditions.
In part 4(Chapter 4),we study the growth of solutions of one kind of higher order differential equations on meromorphic function coefficients.To some extent,we improve the result made by Chen ZongXuan and Xiao LiPeng,and we obtain the precise estimations of meromorphic solutions of the equations on higher order linear differential equations.
引文
[1]杨乐.值分布理论及其新研究:科学出版社,1982
[2]K.Kwon,Nonexistence of finite order solutions of certain second order linear differential equations[J],Kodal Math.J.19(1996),378-387.
[3]Z.X.CHEN and K.I-I.SHON,The growth of solutions of higher order differential equtions[J],Southeast Asian Bull.Math.27(2004),995-1004.
[4]BENHARRAT BELAIDI,Some precise estimates of the hyper order of solutions of some complex linear differential equtions[J].Volume 8(2007),issue 4,Article 107,14PP.
[5]G.G.Gundersen,Estimates for the logarithmic derivative of a meromorphic function,plus similar estimates[J].J.Lond0n Math Soc,1988,37(2):88-104.
[6]陈宗煊.二阶亚纯函数系数微分方程解的零点[J].数学物理学报,1996,16(3):276-283.
[7]Barry P.D.Some theorems related to the cos πρ theorem[J].Proc London math Soc,1970,21(2):293-302.
[8]W.K.HAYMAN,Meromorphic Functions[M],Clarendon Press,Oxford,1964.
[9]K.Kwon,On the growth of entire functions satisfying second order linear differential equations [J],Bull.Korean Math.Soc.(2),33(1996),487-496.
[10]陈宗煊.关于高阶线性微分方程亚纯解的增长率[J],数学学报,1999,42(3):551-558.
[11]肖丽鹏,陈宗煊.一类高阶微分方程亚纯解的增长性[J],数学研究,2005,38(3):265-271.
[12]L.Z.Yang,Growth of linear differential equations and their applications[J],Israel,J.Math,147(2005),359-372.
[13]CHEN ZongXuan.On the Hyper-order of Solution of some second-order Linear Differential Equations[J].Acta Mathematica sinica,English Series Jan.2002,Vol.18,No.1,pp.79-88.
[14]I.Lain and Wu.P.C,Growth of solution of second order linear differential equations[J],Proc.Amer.Math.Soc,2000,128(9):2693-2703.
[15]何育赞,肖修治.代数体函数与常微分方程[M],北京:科学出版社,1998.
[16]高仕安,陈宗煊,陈特为.线性微分方程的复振荡理论[M],武汉:华中理工大学出版社,1998.
[17]Frank G and Hellerstein S.On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients[J].Proc London Math.Soc.,1986,53(3):407-428.
[18]CHEN ZongXuan.On the hyper order of solutions of higher order differential equations[J].Chin.Ann.Math.,2003,24,B(4):501-508.
[19]Zong-xuan Chen and Chung-Chun Yang.Some further results on the zeros and growths of entire solutions of second order linear differential equations[J].Kodai Math.,1999,22(2):273-285.
[20]S.B.Bank and I.Laine.On the oscillations theory of f" + Af = 0 where A is entire[J],Trans.Amer.Math.Soc.273(1982),351-363.
[21]G.G.Gundersen,Finite order solutions of second order linear differential equations [J].Trans.Amer.Math.Soc.Volume 305(1988),415-429.