【摘要】：研究復(fù)微分方程有多種方法。局部理論是研究得最多的一種方法。在許多有關(guān)復(fù)微分方程的書中都可以找到局部解的存在和唯一性定理、奇點(diǎn)理論等局部理論的基本結(jié)果。而全局理論也有不同的研究切入點(diǎn)。比如可以從代數(shù)的觀點(diǎn)來研究,可以從微分方程的觀點(diǎn)來研究,也可以從函數(shù)論的觀點(diǎn)來研究。利用Nevanlinna理論研究復(fù)微分方程亞純解的性質(zhì)和結(jié)構(gòu)屬于函數(shù)論的研究范疇。亞純函數(shù)的值分布理論由R.Nevanlinna于1925年創(chuàng)立。首先嘗試將其應(yīng)用于復(fù)微分方程的研究工作包括:F.Nevanlinna在1929年將其與亞純函數(shù)最大虧量和理論結(jié)合,對微分方程f"+A(z)f=0所做的研究[34],其中A(z)為多項(xiàng)式;R.Nevanlinna將其與支點(diǎn)有限的覆蓋曲面理論結(jié)合,對同樣的方程所做的研究[37];以及K.Yosida利用Nevanlinna理論對著名的Malmquist定理給予的證明[47]。1942年,H.Wittich開始了早期對Nevanlinna理論應(yīng)用于復(fù)微分方程的系統(tǒng)研究,而A.Gol'dberg對一般代數(shù)微分方程的研究結(jié)果[11]可能是早期研究工作中最重要的。此后,直到上世紀(jì)60年代末,利用Nevanlinna理論對復(fù)微分方程的全局解進(jìn)行研究才變得流行起來。在隨后的20年里,一些活躍的研究團(tuán)隊(duì)對該領(lǐng)域的研究發(fā)展起到了非常重要的作用。在前人工作的基礎(chǔ)上,I.Laine在1993年前后編寫了一本詳細(xì)介紹Nevanlinna理論如何應(yīng)用于復(fù)微分方程全局解研究的專著《Nevanlinna理論和復(fù)微分方程》[21],內(nèi)容涵蓋經(jīng)典研究結(jié)果以及最新研究趨勢,給有志于該領(lǐng)域研究的后來者提供了一個非常好的學(xué)習(xí)平臺。這之后,大批該領(lǐng)域的研究文章和專著涌現(xiàn),線性和非線性微分方程都得到了充分的研究,獲得了豐碩的成果。作為其中的一個研究方向,如何證明給定的微分方程存在亞純解,能否給出解可能具有的形式,引起了許多學(xué)者的興趣。這篇論文,在前人研究的基礎(chǔ)之上,針對幾種類型的非線性復(fù)微分方程,給出了亞純解的存在條件,并求得了解的結(jié)構(gòu),改進(jìn)和推廣了前人的結(jié)果。全文一共分為五章。第一章主要介紹Nevanlinna理論的經(jīng)典結(jié)果。第二章首先介紹了代數(shù)復(fù)微分方程的基本概念和定義符號;其次介紹了Wiman-Valiron理論,該理論是研究復(fù)微分方程亞純解是否存在的一個有力工具;最后介紹了兩個將Nevanlinna理論應(yīng)用于復(fù)微分方程所獲得的經(jīng)典結(jié)論:Clunie引理和Tumura-Clunie引理。第三章考慮了如下兩類非線性微分方程和超越亞純解的存在情況,給出了亞純解的存在條件和可能具有的形式。在這里,n,k,d均為正整數(shù)。第四章考慮了 一種特殊類型的Briot-Bouquet方程超越亞純解的存在情況,給出了亞純解的存在條件和可能具有的形式。在這里,a1,a2,…,a6均為常數(shù)。第五章研究了如下類型的施瓦茨方程其中Pm(z),Qn(z)分別為m次和n次的不可約多項(xiàng)式,滿足m≤n-2。我們找到了一種新的確定該類方程是否存在亞純解的方法并給出了解的形式。
[Abstract]:There are many ways to study complex differential equations. Local theory is one of the most studied methods. In many books on complex differential equations, the existence and uniqueness theorems of local solutions and the basic results of local theories such as singularity theory can be found. The global theory also has different research entry points. For example, it can be studied from the point of view of algebra, from the point of view of differential equation, or from the point of view of function theory. The properties and structure of meromorphic solutions of complex differential equations are studied by Nevanlinna theory. The value distribution theory of meromorphic functions was founded by R.Nevanlinna in 1925. The first attempt to apply it to complex differential equations involves the combination of the maximum deficiency and the theory of meromorphic functions by: F. Nevanlinna in 1929. A study of the differential equation f "A (z) f ~ 0 [34], where A (z) is a polynomial and R. Nevanlinna combines it with the theory of covering surfaces with finite fulcrum. The study of the same equation [37] and the proof given by K.Yosida to the famous Malmquist theorem by using the Nevanlinna theory [47]. In 1942 H. Wittich began the early systematic study of the application of Nevanlinna theory to complex differential equations. The results of A.Gol'dberg 's study on general algebraic differential equations [11] are probably the most important in the early research work. Thereafter, it became popular to study the global solutions of complex differential equations by using Nevanlinna theory until the late 1960s. Over the next 20 years, a number of active research teams have played an important role in the development of this field. On the basis of previous work, I. Laine wrote a monograph, "Nevanlinna Theory and complex differential equation" [21], about the application of Nevanlinna theory to the global solution of complex differential equations in 1993, which covers the classical research results and the latest research trends. It provides a very good learning platform for the latecomers who wish to study in this field. After this, a large number of research papers and monographs in this field have emerged, and both linear and nonlinear differential equations have been fully studied and fruitful results have been obtained. As one of the research directions, how to prove the existence of meromorphic solutions for a given differential equation and whether to give the possible form of the solution have aroused the interest of many scholars. In this paper, on the basis of previous studies, the existence conditions of meromorphic solutions for several types of nonlinear complex differential equations are given, the structure of solutions is obtained, and the previous results are improved and generalized. The full text is divided into five chapters. The first chapter mainly introduces the classical results of Nevanlinna theory. The second chapter introduces the basic concepts and definitions of algebraic complex differential equations, and then introduces Wiman-Valiron theory, which is a powerful tool to study the existence of meromorphic solutions of complex differential equations. In the end, two classical conclusions: 1. Clunie Lemma and Tumura-Clunie Lemma obtained by applying Nevanlinna theory to complex differential equations are introduced. In chapter 3, we consider the existence of two kinds of nonlinear differential equations and transcendental meromorphic solutions, and give the existence conditions and possible forms of meromorphic solutions. In this case, the number of n / k / d is all positive integers. In chapter 4, we consider the existence of transcendental meromorphic solutions for a special type of Briot-Bouquet equation, and give the conditions and possible forms of meromorphic solutions. Here's a 1, a, a 2,. A6 is constant. In chapter 5, we study the following types of Schwartz equations, where Pm (z) Q n (z) is irreducible polynomials of degree m and degree n, respectively, and satisfies m 鈮，

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