几类线性微分方程解的复振荡性质
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摘要
本文运用复分析的理论和方法，研究了几种类型的线性微分方程解的性质。本文共分五部分：
第一部分：概述了本研究领域的发展历史。
第二部分：引入一些预备知识，主要是后几章要用到的一些概念的定义、记法，以方便读者阅读。
第三部分：研究了一类高阶整函数系数线性微分方程解的复振荡性质，得到了方程解的增长级的精确估计，推广了已有的结果。
第四部分：研究了几种类型的高阶亚纯函数系数线性微分方程解的复振荡性质，得到了方程亚纯解的增长率的精确估计。把已有的结果由整函数系数方程推广到了亚纯函数系数方程。
第五部分：研究了一类亚纯函数系数的二阶线性微分方程的亚纯解及其一阶和二阶导数的不动点及超级问题，得到了有关复域微分方程亚纯解及其一阶和二阶导数的不动点性质，
Content:In this thesis,we investigate the complex oscillation properties of the solutions of some types of linear differential equations by applying the theories and methods of the complex analysis.lt contains the following five parts:
In part 1, we give a brief introduction of history on development of this research field.
In part 2,we introduce some preliminary knowledge,which is mainly definitions and marks of some concepts used in later chapters.
In part 3, we investigate the property of complex oscillation of the solutions of a type of higher-order linear differential equation with entire coefficients. We obtain some precise estimates of the growth of the solutions of the equation and improve the result obtained before.
In part 4,we investigate the property of complex oscillation of the solutions of some types of higher-order linear differential equations with meromorphic coefficients.We obtain some precise estimates of the growth of the meromorphic solutions of the equations,and improve the results of the equations with entire coefficients to the equations with meromorphic coefficients.
In part 5,we investigate the problems on the fixed points and hyper order of solutions and their 1~(st),2~(nd) derivatives of a class of second order linear differential equations with meromorphic coefficients. We obtain the precise properties of fixed points of meromorphic solutions of complex differential equations.
第一部分：概述了本研究领域的发展历史。
第二部分：引入一些预备知识，主要是后几章要用到的一些概念的定义、记法，以方便读者阅读。
第三部分：研究了一类高阶整函数系数线性微分方程解的复振荡性质，得到了方程解的增长级的精确估计，推广了已有的结果。
第四部分：研究了几种类型的高阶亚纯函数系数线性微分方程解的复振荡性质，得到了方程亚纯解的增长率的精确估计。把已有的结果由整函数系数方程推广到了亚纯函数系数方程。
第五部分：研究了一类亚纯函数系数的二阶线性微分方程的亚纯解及其一阶和二阶导数的不动点及超级问题，得到了有关复域微分方程亚纯解及其一阶和二阶导数的不动点性质，
Content:In this thesis,we investigate the complex oscillation properties of the solutions of some types of linear differential equations by applying the theories and methods of the complex analysis.lt contains the following five parts:
In part 1, we give a brief introduction of history on development of this research field.
In part 2,we introduce some preliminary knowledge,which is mainly definitions and marks of some concepts used in later chapters.
In part 3, we investigate the property of complex oscillation of the solutions of a type of higher-order linear differential equation with entire coefficients. We obtain some precise estimates of the growth of the solutions of the equation and improve the result obtained before.
In part 4,we investigate the property of complex oscillation of the solutions of some types of higher-order linear differential equations with meromorphic coefficients.We obtain some precise estimates of the growth of the meromorphic solutions of the equations,and improve the results of the equations with entire coefficients to the equations with meromorphic coefficients.
In part 5,we investigate the problems on the fixed points and hyper order of solutions and their 1~(st),2~(nd) derivatives of a class of second order linear differential equations with meromorphic coefficients. We obtain the precise properties of fixed points of meromorphic solutions of complex differential equations.
引文
[1] 杨乐，值分布及其新研究．北京：科学出版社，1982．
[2] 高仕安，陈宗煊，陈特为，线性微分方程的复振荡理论．武昌：华中理工大学出版社，1997．
[3] 陈宗煊，关于微分方程f″+e~(-z)f′+B(z)f=0的增长性．中国科学(A辑)，2001，9：775-784
[4] Frei M. Uber die subnormaen losungen der differentialgleichung ω~n+e~(-z)ω′+(Kconst.)ω=0[J]. Comment Math Helv, 1962, 36:1-8.
[5] Ozawa M. On a solution of ω~n+e~(-z)ω′+ (az+b) ω=0[J]. Kodai Math J, 1980, 3: 295-309
[6] Amemiya I, Ozawa M. Non-existence of finite order solution of ω~n+e~(-z)ω′+Q (z)ω=0[J]. Hokkaido Math J, 1981,10:1-17
[7] Gundersen G. On the question of whether f″+e~(-z)f′+B(z)f=0 can admit a solution of finite order[J]. Peoc R S E, 1986,102A:9-17
[8] Langley J K. On complex oscillation and a problem of Ozawa[J]. Kodai Math J, 1986, 9:430-439
[9] 李纯红，黄小军，一类高阶线性微分方程解的增长性[J]．数学物理学报，2003，23A(5)：613-618
[10] Chen Zongxuan, Shon KwangHo. On the growth of solutions of a class of higher order differential equations[J]. Acta Mathematica scientia, 2004,24B(1): 52-60
[11] Chen Zongxuan. On the hyper order of solutions of higher order differential equations[J].Chin. Ann. Math2003,24(B):501-508
[12] Gundersen G. Estimates for the logarithmic derivative of a meromorphic function plus similar estimates[J]. J London Math Soc, 1988,37(a):88-104.
[13] CHEN Zong-xuan, YANG Chung-chun. Quantitative estimations on the zeros and growths of entire solutions of linear differential equations[J].Complex Variables, 2000, (42): 119-133.
[14] 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．
[15] 陈宗煊．二阶亚纯系数微分方程亚纯解的零点[J]．数学物理学 报，1996，16(30)：276-283．
[16] CHEN Zong-xuan. The growth of solutions of second order linear differential equations with meromorphic coefficients [J].Kodai Math J, 1999,22(2):208-221.
[17] G. Frank and S. Hellerstein. On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients[J].Proc. London Math. Soc. 1986,53:407-428.
[18] 易才凤，邱玲．二阶亚纯函数系数的非齐次微分方程解的增长性[J]．江西师范大学学报(自然科学版)，2002，26(4)：324-328．
[19] 郑秀敏，陈宗煊，等．一类亚纯函数系数的高阶非齐次线性微分方程解的增长性[J]．江西师范大学学报(自然科学版)，2004，28(5)：431-435
[20] 肖丽鹂．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．
[21] 何育赞，萧修治．代数体函数与常微分方程[M]．北京：科学出版社，1988．
[22] 仪洪勋，杨重骏．亚纯函数唯一性理论[M]．北京：科学出版社，1995．
[23] 陈宗煊．二阶复域微分方程解的不动点与超级[J]．数学物理学报，2000，20(3)：425-432．
[24] 王珺，吕巍然．二阶线性微分方程亚纯解的不动点与超级[J]应用数学学报，2004，27(1)：72-79．
[25] Chen, Z.X. zeros of meromorphic solutions of higher order linear differential equations[J]. Analysis, 1994,14:425-438
[26] Hayman W. Meromorphic Functions[M].Oxford:Clarendon
[2] 高仕安，陈宗煊，陈特为，线性微分方程的复振荡理论．武昌：华中理工大学出版社，1997．
[3] 陈宗煊，关于微分方程f″+e~(-z)f′+B(z)f=0的增长性．中国科学(A辑)，2001，9：775-784
[4] Frei M. Uber die subnormaen losungen der differentialgleichung ω~n+e~(-z)ω′+(Kconst.)ω=0[J]. Comment Math Helv, 1962, 36:1-8.
[5] Ozawa M. On a solution of ω~n+e~(-z)ω′+ (az+b) ω=0[J]. Kodai Math J, 1980, 3: 295-309
[6] Amemiya I, Ozawa M. Non-existence of finite order solution of ω~n+e~(-z)ω′+Q (z)ω=0[J]. Hokkaido Math J, 1981,10:1-17
[7] Gundersen G. On the question of whether f″+e~(-z)f′+B(z)f=0 can admit a solution of finite order[J]. Peoc R S E, 1986,102A:9-17
[8] Langley J K. On complex oscillation and a problem of Ozawa[J]. Kodai Math J, 1986, 9:430-439
[9] 李纯红，黄小军，一类高阶线性微分方程解的增长性[J]．数学物理学报，2003，23A(5)：613-618
[10] Chen Zongxuan, Shon KwangHo. On the growth of solutions of a class of higher order differential equations[J]. Acta Mathematica scientia, 2004,24B(1): 52-60
[11] Chen Zongxuan. On the hyper order of solutions of higher order differential equations[J].Chin. Ann. Math2003,24(B):501-508
[12] Gundersen G. Estimates for the logarithmic derivative of a meromorphic function plus similar estimates[J]. J London Math Soc, 1988,37(a):88-104.
[13] CHEN Zong-xuan, YANG Chung-chun. Quantitative estimations on the zeros and growths of entire solutions of linear differential equations[J].Complex Variables, 2000, (42): 119-133.
[14] 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．
[15] 陈宗煊．二阶亚纯系数微分方程亚纯解的零点[J]．数学物理学 报，1996，16(30)：276-283．
[16] CHEN Zong-xuan. The growth of solutions of second order linear differential equations with meromorphic coefficients [J].Kodai Math J, 1999,22(2):208-221.
[17] G. Frank and S. Hellerstein. On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients[J].Proc. London Math. Soc. 1986,53:407-428.
[18] 易才凤，邱玲．二阶亚纯函数系数的非齐次微分方程解的增长性[J]．江西师范大学学报(自然科学版)，2002，26(4)：324-328．
[19] 郑秀敏，陈宗煊，等．一类亚纯函数系数的高阶非齐次线性微分方程解的增长性[J]．江西师范大学学报(自然科学版)，2004，28(5)：431-435
[20] 肖丽鹂．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．
[21] 何育赞，萧修治．代数体函数与常微分方程[M]．北京：科学出版社，1988．
[22] 仪洪勋，杨重骏．亚纯函数唯一性理论[M]．北京：科学出版社，1995．
[23] 陈宗煊．二阶复域微分方程解的不动点与超级[J]．数学物理学报，2000，20(3)：425-432．
[24] 王珺，吕巍然．二阶线性微分方程亚纯解的不动点与超级[J]应用数学学报，2004，27(1)：72-79．
[25] Chen, Z.X. zeros of meromorphic solutions of higher order linear differential equations[J]. Analysis, 1994,14:425-438
[26] Hayman W. Meromorphic Functions[M].Oxford:Clarendon
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