Nevanlinna理论在差分多项式中的应用

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具有有限对数增长级亚纯函数的性质与应用

具有有限对数增长级亚纯函数的性质与应用本文研究了具有有限对数增长级亚纯函数的性质与应用,首先研究的是有限对数增长级的亚纯函数的一些函数积与和的对数增长级的性质,接下来利用有限对数增长级亚纯函数的性质和q-差分形式的Wiman-Valiron理论得到了线性q-差分方程亚纯解与系数之间的关系.本文具体安排如下:第二章介绍Nevanlinna 理论,差分Nevanlinna理论.第三章介绍q-差分方程,q-差分Nevanlinna理论.第四章介绍具有有限对数增长级亚纯函数的性质及应用.本文的主要研究结果如下:定理1:设f(z)与g(z)为复平面上的非常数亚纯函数,其对数增长级分别为ρlog(f)与ρlog(g)则:ρlog(fg)≤max{ρlog(f),ρlog(g)},ρlog(f+g)≤max{ρlog(f),ρlog(g)},即两个亚纯函数积与和的对数增长级不大于两个亚纯函数对数增长级中的较大者.定理2:设f(z)与g(z)为复平面上的非常数亚纯函数,f(z)的对数增长级为ρlog(f),g(z)的对数增长级的下级为μlog(g).如果ρlog(f)<μlog(g),则T(r,f =o(T(r,g))(r →∞).定理3:令a0(z),…,an(z)是对数增长级有限的超越整函数,且令q∈C\{0}以至于|q| ≠ 1.对于下面的方程(?)aj(z)f(qjz)=0(1)f(z)是该方程的超越整函数解,并且存在i ∈{0,…,n}使得且则。

有限增长级条件下超越整函数和亚纯函数的一阶差分方程的零点和不动点研究

有限增长级条件下超越整函数和亚纯函数的一阶差分方程的零点和不动点研究章辉梁;高宗升【摘要】对超越整函数和亚纯函数一阶差分方程的零点和不动点的研究,很多的研究结果都是基于函数的增长级σ(f)≤1,而在有限增长级1<σ(f)<∞的情况下,研究结果则相对较少。

利用Nevanlinna的基本理论和方法,探讨了在有限增长级的条件下,超越整函数和亚纯函数一阶差分方程零点和不动点的存在性。

首先,结合Hadmard因子分解定理研究了在一定的条件下超越整函数的一阶差分方程零点和不动点的存在性,证明了其有无穷多个零点和无穷多个不动点。

其次,把对超越整函数的零点和不动点的存在性研究,推广到了亚纯函数,继续探讨了亚纯函数在有限增长级条件下零点和不动点的情况,得出了相应的结论。

【期刊名称】《长江大学学报（自然版）理工卷》【年(卷),期】2016(013)004【总页数】4页(P10-13)【关键词】一阶差分;零点;不动点【作者】章辉梁;高宗升【作者单位】北京航空航天大学数学信息与行为教育部重点实验室，北京100191;北京航空航天大学数学信息与行为教育部重点实验室，北京 100191【正文语种】中文【中图分类】O174.5亚纯函数Nevanlinna理论的一些基本概念和标准记号见文献[1～3]，用T(r,f)表示复平面上亚纯函数f的特征函数，用σ(f)表示f(z)的增长级，用λ(f)表示f(z)的零点收敛指数。

差分算子Δf的具体定义[4]如下：Δf(z)=f(z+1)-f(z)Δn+1f(z)=Δnf(z+1)-Δnf(z)n=0,1,2,…在文献[5]中，Bergweiler 和Langley首先讨论了差分Δf(z)=f(z+c)-f(z)以及差商的零点情况，得到了如下的重要结果。

定理A 设f(z)是超越整函数,若存在，使f(z)的增长级σ(f)、σ满足σ(f)，则有无穷多个零点。

定理B 设f(z)是超越亚纯函数，下级u(f)<1，设c∈C/{0}使得f(z)最多只有有限多个极点zj,zk满足zj-zk=c，则h(z)=f(z+c)-f(z)有无穷多个零点。

vanlee有限差分格式

Van Lee有限差分格式1. 引言Van Lee有限差分格式是一种数值解法，用于求解偏微分方程。

它是由荷兰数学家Van Lee于20世纪50年代提出的，被广泛应用于科学和工程领域。

本文将详细介绍Van Lee有限差分格式的原理、应用和优缺点。

2. 原理Van Lee有限差分格式基于有限差分法，将偏微分方程转化为差分方程，通过有限差分近似求解。

其基本思想是将求解域划分为离散的网格点，然后通过近似替代偏微分方程中的导数项，得到差分方程。

最后，通过迭代求解差分方程，得到数值解。

在Van Lee有限差分格式中，常用的差分近似方法有中心差分、向前差分和向后差分。

其中，中心差分是最常用的方法，它使用当前节点及其相邻节点的函数值来估计导数项。

具体公式如下：其中，f(x)是待求解的函数，h是网格间距，n是当前节点。

3. 应用Van Lee有限差分格式广泛应用于各种偏微分方程的求解，特别是在热传导、流体力学和结构力学等领域。

下面以热传导方程为例，介绍Van Lee有限差分格式的应用。

热传导方程描述了物体内部温度分布随时间的变化。

假设物体的温度分布为u(x, t)，满足以下方程：其中，α是热扩散系数。

通过将空间和时间离散化，可以得到差分方程：其中，i是空间网格索引，j是时间步长索引。

利用Van Lee有限差分格式，可以将差分方程转化为线性方程组，通过求解线性方程组得到温度分布的数值解。

4. 优缺点Van Lee有限差分格式有以下优点：•简单易懂：Van Lee有限差分格式的原理简单，易于理解和实现。

•适用范围广：Van Lee有限差分格式适用于各种偏微分方程的求解，包括热传导、流体力学和结构力学等领域。

•数值稳定性好：Van Lee有限差分格式具有良好的数值稳定性，能够得到准确可靠的数值解。

然而，Van Lee有限差分格式也存在一些缺点：•误差较大：由于使用近似替代导数项，Van Lee有限差分格式的数值解与真实解之间存在误差，且误差随着网格间距的减小而增大。

非线性常微分方程边值问题的有限解析法

非线性常微分方程边值问题的有限解析法非线性常微分方程边值问题（NonlinearOrdinaryDifferentialEquationBoundaryValueProblem，简称BVP）是系统动力学，数学物理，流体动力学及控制等多个学科中的重要问题。

自20世纪60年代以来，BVP的研究得到了迅猛的发展，研究的解析方法从精确解析方法到近似解析方法，再到近似解法及混合解法，主要包括：有限元法，采用多项式进行有限差分法，多项式拟合法，幂级数法，变分法，迭代法等。

比较近些年，有限解析法受到越来越多的关注，这项研究不仅有助于深入了解BVP的数学本质，还可以指导现实问题的解决。

有限解析法是一种以数学分析的方法求解BVP边界值问题的方法，主要是利用多项式函数近似解，或是采用多项式多项式拟合法进行离散，最后得出精确的解析解。

这种方法被广泛应用于边界值问题的解决，其优势在于不需要迭代求解，即使求解过程复杂，有限解析法仍能得到快速而准确的结果。

二、原理有限解析法的原理是：将BVP边界值问题转换为一个多项式拟合的问题，首先以离散化的方式将非线性常微分方程边值问题转换为一个线性方程组，然后再用多项式函数近似求解有限结点方程组，并通过一组特定的约束条件使多项式函数唯一确定，最终得出有限的解析解。

三、实例下面以一个实例来说明有限解析法的用法。

假设给定一个BVP如下：y + 3y - 2y = x， y(0)=1， y(1)=5此非线性常微分方程边值问题的解析解可以用有限解析法来解决。

首先，以离散化的形式转换为线性方程组，把解区间[0, 1]选择为 N等分，即为xi=i/N，i=0,1,2…N-1，在节点处yi=yi(xi)。

由于边界已知，所以将节点拆分为 N+1个即yi(0)=1，yi(1)=5，那么有限元可以确定y0,y1,y2…yN-1的值，一共N组值。

现在构造N组多项式拟合，即有yi = a0 + a1xi + a2xi2 + +aN-1xiN-1,i=0,1,2…N-1，将构造出的多项式代入原问题，将原问题转移到下面N组线性方程系：(1) a0 + a1(0) + a2(0)2 + +aN-1(0)N-1 = 1；(2) a0 + a1(1/N) + a2(1/N)2 + +aN-1(1/N)N-1 = f(1/N)；(3) a0 + a1(2/N) + a2(2/N)2 + +aN-1(2/N)N-1 = f(2/N)；…………(N) a0 + a1(N-1/N) + a2(N-1/N)2 + +aN-1(N-1/N)N-1 =f(N-1/N)；最后求解上述N组线性方程组的唯一解，即可得出yi的值，从而得出有限的解析解。

关于一类复差分方程组及C~n上复偏差分方程组亚纯解的研究

关于一类复差分方程组及C~n上复偏差分方程组亚纯解的研究
在本文中,主要利用了亚纯函数的Nevanlinna值分布理论以及亚纯函数对数导数引理的差分模拟和差分多项式的值分布性质,研究了一类复差分方程组亚纯解的结构;研究了C~n上的一类偏差分方程组的亚纯解及其性质,进一步推广和改进了之前一些文献的结论.同时例子也表明本文的一些结论更优.全文共分三章:第一章,主要介绍了Nevanlinna值分布理论的基础知识,以及常用的记号和与复差分相关的一些ห้องสมุดไป่ตู้本定理.第二章,研究了一类复差分方程组的亚纯解的结构,对ine等人的结论进行了推广. 例子表明本章定理的条件以及结论是精确的.第三章,利用了 Clunie-Mohon’ko引理的差分模拟,研究了在C~n上一类复偏差分方程组的亚纯解及其性质,得到几个结果.例子也表明定理的

差分和q-差分多项式的Clunie定理

J.London Math.Soc.(2)76(2007)556–566Ce2007London Mathematical Societydoi:10.1112/jlms/jdm073 CLUNIE THEOREMS FOR DIFFERENCE ANDq-DIFFERENCE POLYNOMIALSILPO LAINE and CHUNG-CHUN YANGAbstractThe main purpose of this paper is to prove diﬀerence and q-diﬀerence counterparts of the Clunie and Mohon’ko lemmas from the Nevanlinna theory of diﬀerential polynomials.We also give applications to the value distribution theory of meromorphic solutions of some complex diﬀerence equations.1.IntroductionMeromorphic solutions of complex diﬀerence equations have become a subject of great interest recently,due to the apparent role of the existence of such solutions ofﬁnite order for the integrability of discrete diﬀerence equations.These types of considerations were initiated by Ablowitz,Halburd and Herbst[1]a few years ago,although the idea of relating growth of solutions with the integrability goes back to Veselov[17]at least.See also Ramani et al.[16]. Recently,Halburd and Korhonen[9]were able to show that whenever a complex diﬀerence equation of typew(z+1)+w(z−1)=R(z,w),(1.1) where R is rational in both of its arguments,has a meromorphic solution ofﬁnite order,either w satisﬁes a diﬀerence Riccati equation or(1.1)may be transformed into a diﬀerence Painlev´e or a linear diﬀerence equation.For a precise statement of this result,see[9,Theorem1.1].A natural tool for such considerations is the Nevanlinna theory of meromorphic functions and, more precisely,the counterparts of its basic notions and results in the framework of diﬀerence expressions.In this paper,we assume that the reader is familiar with the standard notation and basic results of Nevanlinna theory;see,for example,[10].With respect to Nevanlinna theory and diﬀerence expressions,there are two recent papers [4,8]containing very similar results.However,their presentations are slightly diﬀerent in detail.A typical example of these presentations is the diﬀerence analogue of the logarithmic derivative lemma.In the presentation of[8],this key lemma reads as follows.Lemma1.1.Let f be a non-constant meromorphic function,c∈C,δ<1andε>0.Thenmr,f(z+c)f(z)=oT(r+|c|,f)1+εrδ(1.2)for all r outside of a possible exceptional set E withﬁnite logarithmic measure.On the other hand,the same result will be formulated in[4]in the following form.Received9June2006;revised19December2006;published online24October2007.2000Mathematics Subject Classiﬁcation30D35(primary),39B32(secondary).The authors were supported by the RGC Grant of Hong Kong605205.Theﬁrst author has also been supported by the Academy of Finland grant210245.DIFFERENCE AND q -DIFFERENCE POLYNOMIALS 557Lemma 1.2.Given two distinct complex numbers η1,η2,let f be a meromorphic function of ﬁnite order ρ.Then for each ε>0,m r,f (z +η1)f (z +η2)=O(r ρ−1+ε).(1.3)In what follows,we mostly refer to [4]while considering the Nevanlinna theory of diﬀerence expressions.Note that in such considerations,the estimates derived are valid outside of a possible exceptional set of ﬁnite logarithmic (or,respectively,linear)measure at most.As we are only considering ﬁnitely many of such exceptional sets,we may remove the union of such sets from our consideration.In [8],Halburd and Korhonen proved diﬀerence counterparts (see [8,Theorem 3.1and Theorem 3.2],for the well-known Clunie and Mohon’ko lemmas in Nevanlinna theory);see[5,15]for the original versions of these results,as well as [13,Lemma 2.4.2,Proposition 9.2.3].A slightly more general version of the Clunie lemma can be found in [11,pp.218–220];see also[13,Lemma 2.4.5.].Recently,the additional assumptions in the He–Xiao variant of the Clunie lemma have been removed by Yang and Ye in [21,Theorem 1].Closely related to diﬀerence expressions are q -diﬀerence expressions,where the usual shift f (z +c )of a meromorphic function will be replaced by the q -shift f (qz ),q ∈C \{0}.The Nevanlinna theory of q -diﬀerence expressions and its applications to q -diﬀerence equations have recently been considered by Barnett et al.;see [3].Similarly,as the ﬁnite order solutions play a key role in complex diﬀerence equations,solutions of order zero are in focus for complex q -diﬀerence equations;see the introduction in [3].Moreover,typical results in the q -diﬀerence Nevanlinna theory are valid on sets of logarithmic density 1.This means that we may have to consider the intersection of several such sets to ensure that a ﬁnite number of related observations remain valid simultaneously.This will not be mentioned explicitly in what follows.The main purpose of this paper is to prove diﬀerence and q -diﬀerence counterparts (Theorem 2.3and Theorem 2.5below)to the Yang–Ye theorem.We also state the diﬀerence counterpart (Theorem 2.4)of the Mohon’ko lemma in the Chiang–Feng setting.Finally,we add some applications of these diﬀerence counterparts to obtain observations on the value distribution of meromorphic solutions of some complex diﬀerence equations.2.Main resultsIn what follows,we shall consider diﬀerence products and diﬀerence polynomials.By a diﬀerence product ,we mean a diﬀerence monomial,that is,an expression of typesj =1f (z +γj )νj ,(2.1)where γ1,...,γs are complex numbers and ν1,...,νs are natural numbers.A diﬀerence polynomial is a ﬁnite sum of diﬀerence products,that is,an expression of the form U (z,f )= {J }αJ (z )⎛⎝ j ∈Jf (z +c j )⎞⎠,(2.2)where c j ,j ∈J ,is a set of distinct complex numbers.In what follows,we assume that the coeﬃcients of diﬀerence polynomials are,unless otherwise stated,small functions as understood in the usual Nevanlinna theory;that is,their characteristic is of type S (r,f ).Combining[14,Lemma 2.1](based on an observation by Ramani et al.in [16])with [4,Theorem 2.1],we obtain the following lemma.558ILPO LAINE AND CHUNG-CHUN YANG Lemma 2.1.The characteristic function of a diﬀerence polynomial U (z,f )in (2.2)satisﬁesT (r,U (z,f )) nT (r,f )+O(r ρ−1+ε)+S (r,f ),(2.3)provided that f is a meromorphic function of ﬁnite order ρ,and the index set J consists of n elements.We ﬁrst state a slight generalization of the diﬀerence version of the Clunie lemma (see[8,Theorem 3.1]),expressed in the terminology of [4].However,we omit the proof,as this version is just a special case of the next theorem.Theorem 2.2.Let f be a transcendental meromorphic solution of ﬁnite order ρof a diﬀerence equation of the formH (z,f )P (z,f )=Q (z,f ),(2.4)where H (z,f )is a diﬀerence product of total degree n in f (z )and its shifts,and where P (z,f ),Q (z,f )are diﬀerence polynomials such that the total degree of Q (z,f )is at most n .Then for each ε>0,m (r,P (z,f ))=O(r ρ−1+ε)+S (r,f ),(2.5)possibly outside of an exceptional set of ﬁnite logarithmic measure.Remark.For a transcendental meromorphic solution of inﬁnite order of (2.4),the proximity function m (r,P (z,f ))is not necessarily small with respect to T (r,f )as in the Nevanlinna theory.To give an example,see [12,pp.101–102].The entire function f (z )=e A cos(πz/3+α)with parameters A,α∈C satisﬁes the diﬀerence equation f (z )f (z +2)=f (z +1).Another example (see [6,pp.102–103])follows fromf (z )2f (z +2)=f (z +1)2,which has the two-parameter family f (z )=exp e (1/2)z log 2 c 1cos πz 4 +c 2sin πz 4of solutions.Finally,the assumption that f should be transcendental is needed as well.In fact,if f is rational,then the coeﬃcients in (2.4)should be constants.Suppose then f ∼z −k ,k 1,close to inﬁnity.Also assume that Q (z,f )in (2.4)is a diﬀerence product,in addition to H (z,f ).Then H (z,f )∼z −kn ,whereas Q (z,f )∼z −kp for some p n .If now p <n ,then m (r,P (z,f ))∼k (n −p )log r .On the other hand,T (r,f )∼s log r for some ﬁnite s ∈N .Hence,m (r,P (z,f ))is not small with respect to f .We next prove a more general version of the preceding theorem by making use of the reasoning applied by Yang and Ye in [21].Theorem 2.3.Let f be a transcendental meromorphic solution of ﬁnite order ρof a diﬀerence equation of the formU (z,f )P (z,f )=Q (z,f ),(2.6)where U (z,f ),P (z,f )and Q (z,f )are diﬀerence polynomials such that the total degree deg U (z,f )=n in f (z )and its shifts,and deg Q (z,f ) n .Moreover,we assume that U (z,f )contains just one term of maximal total degree in f (z )and its shifts.Then for each ε>0,m (r,P (z,f ))=O(r ρ−1+ε)+S (r,f ),(2.7)possibly outside of an exceptional set of ﬁnite logarithmic measure.DIFFERENCE AND q -DIFFERENCE POLYNOMIALS 559Remark.It remains open whether the assumption of just one term of maximal total degree could be removed.For an example,see the remark after the proof of this theorem.The reasoning applied by Yang and Ye in [21](see the proof of Theorem 2)may also be used to obtain a diﬀerence counterpart of the Mohon’ko lemma in the Chiang–Feng setting.See [8,Theorem 3.2]for a previous diﬀerence variant of this lemma.We omit the proof,as this is just an easy modiﬁcation of the corresponding proof in [8].Theorem 2.4.Let f be a transcendental meromorphic solution of ﬁnite order ρof the diﬀerence equationP (z,f )=0,where P (z,f )is a diﬀerence polynomial in f (z )and its shifts.If P (z,a )≡0for a slowly moving target function a ,that is,T (r,a )=S (r,f ),then m r,1f −a=O(r ρ−1+ε)+S (r,f )outside of a possible exceptional set of ﬁnite logarithmic measure.Replacing diﬀerence polynomials by q -diﬀerence polynomials,we ﬁnd an analogue of Theorem 2.3,by generalizing the basic q -diﬀerence Clunie lemma,recently proved by Barnett et al.;see [3,Theorem 2.1].Here a q -diﬀerence polynomial of f for q ∈C \{0,1}is a polynomial in f (z )and ﬁnitely many of its q -shifts f (qz ),...,f (q n z )with meromorphic coeﬃcients in the sense that their Nevanlinna characteristic functions are o(T (r,f ))on a set of logarithmic density 1.Theorem 2.5.Let f be a transcendental meromorphic solution of order zero of a q -diﬀerence equation of the formU q (z,f )P q (z,f )=Q q (z,f ),(2.8)where U q (z,f ),P q (z,f )and Q q (z,f )are q -diﬀerence polynomials such that the total degree deg U q (z,f )=n in f (z )and its q -shifts,whereas deg Q q (z,f ) n .Moreover,we assume that U q (z,f )contains just one term of maximal total degree in f (z )and its q -shifts.Thenm (r,P q (z,f ))=o(T (r,f ))(2.9)on a set of logarithmic density 1.3.Proof of Theorem 2.3We ﬁrst ﬁx some notation for the proof as follows:P (z,f )=λ∈I a λ(z )σλ j =1f (z +αλ,j )l λ,j ,(3.1)Q (z,f )=μ∈J b μ(z )τμ j =1f (z +βμ,j )m μ,j ,(3.2)U (z,f )= ν∈K c ν(z )U ν(z,f )=ν∈K c ν(z )υν j =1f (z +γν,j )n ν,j .(3.3)560ILPO LAINE AND CHUNG-CHUN YANGBy an assumption concerning the total degree of U(z,f)and Q(z,f),we havemax μ∈Jτμj=1mμ,j n=maxν∈Kυνj=1nν,j.(3.4)Moreover,we use the notation gλ,j(z):=f(z+αλ,j)/f(z),lλ:= σλj=1lλ,j and hμ,j(z):=f(z+βμ,j)/f(z).We next rearrange the expression for the diﬀerence polynomial U(z,f)by collecting together all terms having the same total degree and then writing U(z,f)in the formU(z,f)=nj=0c j(z)f(z)j.(3.5)Now each of the coeﬃcients c j is aﬁnite sum of products of functions of the form f(z+δ)/f(z),δ∈C,each such product being multiplied by one of the original coeﬃcients cν(z),ν∈K. However, c n(z)contains just one product of the described form,multiplied by one of the original coeﬃcients.By[4,Corollary8.3],we have,for all j=0,...,n and for allε>0small enough,the estimate| c j(z)|ν∈K|cν(z)|exp(rρ−1+ε)outside of a possible exceptional set ofﬁnite logarithmic measure.Taking logarithms and integrating,we getm(r, c j)=O(rρ−1+ε)+S(r,f),(3.6)for all j=0,...,n,as well asm(r,1/ c n)=O(rρ−1+ε)+S(r,f);(3.7)here(3.7)follows from the assumption that we have just one term of maximal total degree in U(z,f).Making use of the reasoning in[21],weﬁrst deﬁnec(z):=max1 j n1,2c n−jc n1/j.(3.8)Although c(z)is not meromorphic,we may however compute its proximity function to obtainm(r,c)nj=0m(r, c j)+m(r,1/ c n)+O(1)=O(rρ−1+ε)+S(r,f).(3.9)We next divide the circle|z|=r into two parts:F1:={θ∈[0,2π):|f(re iθ)| c(re iθ)},F2:=[0,2π)\F1.(3.10)It is routine to see that whenever0<α<1and x k 0,we have(x k)αxαk.Applyingnow the notation introduced at the beginning of the proof,we may drop,for brevity,the argument‘z’in what follows.We may alsoﬁxαto satisfy0<α<1.DIFFERENCE AND q-DIFFERENCE POLYNOMIALS561 For z=re iθwithθ∈F1,we now get|P(z,f)|αλ∈I|aλ||f|lλσλj=1|gλ,j|lλ,jα=λ∈I|aλf lλ|ασλj=1|gλ,j|αlλ,jλ∈I|aλf lλ|2α1/2λ∈Iσλj=1|gλ,j|2αlλ,j1/2λ∈I|aλc lλ|2α1/2λ∈Iσλj=1|gλ,j|2αlλ,j1/2.(3.11)In the second case to be considered,θ∈F2,we have|f(z)|>c(z) 2c n−jc n1/j,and hencec n−jc n|f(z)|j2jfor all j=1,...,n.This means that we have|U(z,f)| | c n||f|n1−nj=1| c n−j|| c n||f|j| c n||f|n2n.Moreover,recall that we also have|f(z)|>c(z) 1in the present case.Similarly,as in the preceding case ofθ∈F1,and recalling the assumption on the total degrees of Q(z,f)and U(z,f),we may now estimate|P(z,f)|α=Q(z,f)α2n| c n||f|μ∈J|bμ(z)|τμj=1|f(z+βμ,j)|mμ,jα2n| c n|αμ∈J|bμ|τμj=1|hμ,j|mμ,jα2n| c n|αμ∈J|bμ|2α1/2μ∈Jτμj=1|hμ,j|2αmμ,j1/2.(3.12)Estimating nowm(r,P(z,f))=12πα2πlog+|P(z,f)|αdθby the preceding inequality(3.12)and the corresponding inequality(3.11)from the caseθ∈F1, we immediately see that the last factor on the right-hand side of both of these inequalities yields,after taking logarithms and integrating,O(rρ−1+ε)+O(1).The factors formed by the coeﬃcients aλand bμin(3.11)and(3.12),respectively,are of type S(r,f),by assumption. Finally,the factor(2n/| c n|)αin the caseθ∈F2results in O(rρ−1+ε)+S(r,f)by(3.7).Adding all the components,we get the required estimate for m(r,P(z,f)).562ILPO LAINE AND CHUNG-CHUN YANGRemark.Theorem 2.3fails if U (z,f )contains more than one term of maximal total degree.Indeed,the diﬀerence equation f (z )2+f z +π22 f (z )=−f (z +π)is solved by f (z )=sin z .Looking at the proof above,we would get U (z,f )= c 2(z )f (z )2= 1+ f (z +π2)f (z )2 f (z )2.But then we getm (r,1/ c 2)=m (r,sin 2(z ))=O(r ),contradicting what we need in (3.7).4.Proof of Theorem 2.5As this is completely parallel to the proof of Theorem 2.3in the preceding section,a few remarks are suﬃcient.First,we again rearrange U q (z,f )in the formU q (z,f )=nj =0 γj (z )f (z )j ,(4.1)where each of the coeﬃcients γj is a ﬁnite sum of products of functions of the form f (q k z )/f (z ),each such product being multiplied by one of the original coeﬃcients of U q (z,f ).Again, γn (z )consists of just one such product.Making use of [3,Theorem 1.1]instead of [4,Corollary 8.5],we infer thatm (r, γj )=o(T (r,f ))for j =0,...,n andm (r,1/ γn )=o(T (r,f )),both these estimates being valid on a set of logarithmic density 1.Deﬁning now,as before,γ(z ):=max 1 j n 1,2 γn −j γn1/j ,we ﬁnd thatm (r,γ)=o(T (r,f )),again on a set of logarithmic density 1.To compute now m (r,P q (z,f )),we divide the circle |z |=r into two:F 1:={0∈[0,2π):|f (re iθ)| γ(re iθ)},F 2:=[0,2π)\F 1as before.We may now repeat the corresponding reasoning of the proof of Theorem 2.3to complete the proof.5.Applications to complex diﬀerence equationsAn immediate consequence of Theorem 2.2is the following proposition.Proposition 5.1.Given three non-zero complex numbers α,β,γ,not all equal,let f be a meromorphic solution of ﬁnite order ρof a Riccati diﬀerence equationf (z +α)f (z +β)+a (z )f (z +γ)+b (z )f (z )+c (z )=0(5.1)DIFFERENCE AND q-DIFFERENCE POLYNOMIALS563 with small coeﬃcients in the Nevanlinna theory sense.Thenm(r,f)=O(rρ−1+ε)+S(r,f)(5.2) outside of a possible exceptional set ofﬁnite logarithmic measure.Moreover,for any meromorphic functionω(z)small with respect to f,we obtainm(r,1/(f−ω))=O(rρ−1+ε)+S(r,f),(5.3) provided thatω2+(a(z)+b(z))ω+c(z)≡0.Proof.Theﬁrst assertion is an immediate consequence of Theorem2.2and[4,Corollary 2.6],whereas the second follows from Theorem2.4.Remark.We remark that in this case of Riccati diﬀerence equations,ﬁnite-order mero-morphic solutions may well exist,as shown by the concrete examples given below.See also[2], pp.122–123and[18],Theorem1.Example[6,pp.99–100].The Pielou logistic equation(which is a special case of a Riccati diﬀerence equation)f(z+1)=αf(z)1+βf(z),β=0,has the following non-constant solutions(ofﬁnite order),where c=β/(α−1)is a complex parameter:f(z)=1c+βz,in the case ofα=1,andf(z)=(α−1)e z logα−β+c(α−1)+βe,ifα=1.Looking at the last case only and writing the equation in the formf(z)f(z+1)+1βf(z+1)−αβf(z)=0,we immediately conclude that m(r,f)=O(rε)+S(r,f)by Theorem 2.2,and that m(r,1/(f−ω))=O(rε)+S(r,f),provided thatω=(α−1)/β.The complex constant(α−1)/βis in fact a Picard value for f(z),as can be immediately veriﬁed.Example[12,pp.103–106].The following diﬀerence equationf(z+1)=αf(z)(1−f(z)),α=0,comes out of a well-known discrete logistic model in biology.Although it looks somewhat similar to the previous example,the behaviour of its solutions is completely diﬀerent.Indeed, apart from the constant solutions f(z)≡0,f(z)≡(α−1)/α,all other meromorphic solutions are of inﬁnite order of growth.For a possible entire solution,an immediate contradiction follows by the Clunie theorem above(Theorem2.2).If f(z)is meromorphic non-entire,then we may start at an arbitrary pole to construct an arithmetic progression of poles so that their multiplicities grow exponentially.This readily implies a growth of inﬁnite order;see the proof of Proposition5.4.Examples of one-parameter families of entire solutions of inﬁnite order aref(z)=12(1−exp(Ae z log2)),A∈C\{0}forα=2andf(z)=sin2(Be z log2),B∈C\{0} forα=4.564ILPO LAINE AND CHUNG-CHUN YANGExample[19,p.124].Similarly,the diﬀerence equationf(z+1)=f(z)p,p∈N,admits an entire solutionf(z)=exp(e z log p)of inﬁnite order.Again,as in the previous example,we may show that the only meromorphic solutions ofﬁnite order are constants,provided that p 2.Concerning linear diﬀerence equations,weﬁrst complete[4,Theorem9.2].Theorem 5.2.Let A0,...,A n be entire functions ofﬁnite order such that among those having the maximal orderρ:=max0 k nρ(A k),exactly one has its type strictly greater than the others.Then for any meromorphic solution ofA n(z)f(z+ωn)+...+A1(z)f(z+ω1)+A0(z)f(z)=0,(5.4) we haveρ(f) ρ+1.Proof.First recall that for an entire function of orderρand of typeτ<∞,we have M(r,f)=O(exp((τ+ε)rρ)).For simplicity of notation,we rewrite equation(5.4)in the form a0(z)f(z+α0)+...+a k(z)f(z+αk)+b0(z)f(z+β0)+...+b l(z)f(z+βl)=0,(5.5) where k+l=n,ρ(a0)=...=ρ(a k)=ρand max0 j lρ(b j)=:μ<ρ.Moreover,we may assume thatτ=max1 j kτ(a j)<τ(a0).Suppose now that f is a meromorphic solution of (5.4),and hence of(5.5)as well,of orderρ(f)<ρ+1.Now takeε>0small enough to satisfy μ+ε<ρandρ(f)+2ε<ρ+1.Fix alsoσso thatτ<σ<σ+3ε<τ(a0).Dividing now(5.5) by f(z+α0)and recalling[4,Corollary8.3],we obtain,outside of a possible exceptional set ofﬁnite logarithmic measure,M(r,a0) e rρ(f)−1+ε(O(e(τ+ε)rρ)+O(e rμ+ε))=O(e(σ+2ε)rρ).By[7,Lemma5],the exceptional set may be removed by slightly increasingε,if needed.Hence ) σ+2ε,a contradiction.τ(aRemark.It is natural to ask whether all meromorphic solutions f of equation(5.4)satisfy ρ(f) 1+max0 j nρ(A j),even if there is no dominating coeﬃcient.We next apply the previous considerations to complex diﬀerence equations loosely related to discrete Painlev´e equations.However,as our interest is purely mathematical,we permit more general coeﬃcients than is customary in questions directly dealing with Painlev´e equations. On the other hand,the search for what could be called the‘discrete Painlev´e property’has prompted us to determine whether a given complex diﬀerence equation permits meromorphic solutions ofﬁnite order;see[1,16]for a detailed argument.As for the next proposition from this point of view;see[16,p.1007].The existence of meromorphic solutions has been proved in the case of constant coeﬃcients andα=1,β=2;see[20].Yanagihara also proved that meromorphic solutions of(5.6)below are of inﬁnite order in the case of constant coeﬃcients. Before proceeding,we recall some notation and a lemma from[9].Where f has more than S(r,f)poles of a certain type,we mean that the integrated counting function of these poles is not of type S(r,f).We next use∞k to denote a pole of f of multiplicity k.Similarly,we may denote0k and a+0k for zeros and a-points of f,respectively.Moreover,we may use notation such as f(z0)=a+0k as short-hand notation for the corresponding expansions of f around z0.We now recall the following lemma from[9,Lemma3.1].DIFFERENCE AND q-DIFFERENCE POLYNOMIALS565 Lemma5.3.Let w be a meromorphic function having more than S(r,w)poles,and let a s, s=1,...,n,be small meromorphic functions with respect to f.Denote by m j the maximum order of zeros and poles of the functions a s at z j.Then for anyε>0,there are at most S(r,w) points z j such thatw(z j)=∞k j,where m j εk j.Proposition 5.4.Given two distinct non-zero complex numbersα,β,let f be a transcendental meromorphic solution off(z+α)+f(z+β)=a(z)f(z)2+b(z)f(z)+c(z)(5.6) with meromorphic coeﬃcients satisfying T(r,a)=S(r,f),T(r,b)=S(r,f)and T(r,c)= S(r,f).Then f is of inﬁnite order of growth.Proof.Let us assume that f is a meromorphic solution of(5.6)ofﬁnite orderρ.For simplicity,letF L(z):=f(z+α)+f(z+β)andF R(z):=a(z)f(z)2+b(z)f(z)+c(z).By Theorem2.1,we readily conclude thatm(r,f)=O(rρ−1+ε)+S(r,f).Therefore,f has more than S(r,f)poles,counting multiplicity.Denoting points in the pole-sequence by z j,we may invoke the notation introduced above to denote f(z j)=∞k j.By Lemma5.3,f has more than S(r,f)poles so that we have m j<εk j at z j.Here m j refers to the coeﬃcients a,b,c of(5.6).Denote the sequence of such poles by z1,j,and take this sequence as our starting point.Supposing,as we may,thatε<1/2,we see thatF R(z1,j)=∞k 2,j,k 2,j (2−ε)k1,j.Comparing this with F L,we conclude that at least one of the points z1,j+α,z1,j+βis a pole of f of multiplicity k2,j k 2,j.Weﬁrst apply Lemma5.3to obtain more than S(r,f)such points z2,j with f(z2,j)=∞k 2,j and m2,j<εk2,j.We then pick only one of these points,denoting it by z2,j.Continuing to the next phase,we observe that F R(z2,j)=∞k 3,j,and weﬁx,for each permitted z2,j,a pole z3,j of the next phase so that f(z3,j)=∞k3,j,wherek3,j k 3,j (2−ε)k2,j (2−ε)2k1,j.Continuing inductively,we mayﬁnally choose a sequence z n of poles of f which satisfy the conditions f(z n)=∞k n and k n (2−ε)n−1k1 (2−ε)n−1.To estimate now the counting function N(r,f),let M:=max(|α|,|β|).Then it is geometri-cally obvious thatz n∈B(z1,(n−1)M)⊂B(0,|z1|+(n−1)M)=B(0,r n).For n large enough,we have r n 2(n−1)M,which means thatn(r n,f) (2−ε)n−1>(3/2)n−1.Hence,N(2r n,f) (log2)32n−1(log2)32rn/3M.This obviously contradicts our hypothesis that f is ofﬁnite order,proving theassertion.566DIFFERENCE AND q-DIFFERENCE POLYNOMIALS Acknowledgement.We thank the referees for pointing out several inaccuracies in the original version of this paper.References1.M.J.Ablowitz,R.Halburd and B.Herbst,‘On the extension of the Painlev´e property to diﬀerenceequations’,Nonlinearity13(2000)889–905.2.R.P.Agarwal,Diﬀerence equations and inequalities.Theory,methods and applications,5th edn(MarcelDekker,Inc.,New York,2000).3. D.C.Barnett,R.G.Halburd,R.J.Korhonen and W.Morgan,‘Nevanlinna theory for the q-diﬀerenceoperator and meromorphic solutions of q-diﬀerence equations’,Proc.Roy.Soc.Edinburgh Sect.A.137 (2007)457–474.4.Y.M.Chiang and S.Feng,‘On the Nevanlinna characteristic of f(z+η)and diﬀerence equations in thecomplex plane’,Ramanujan J.,to appear,arXiv:math.CV/0609324.5.J.Clunie,‘On integral and meromorphic functions’,J.London Math.Soc.37(1962)17–27.6.S.Elaydi,An introduction to diﬀerence equations,3rd edn(Springer,New York,2005).7.G.Gundersen,‘Finite order solutions of second order linear diﬀerential equations’,Trans.Amer.Math.Soc.305(1988)415–429.8.R.G.Halburd and R.J.Korhonen,‘Diﬀerence analogue of the lemma on the logarithmic derivativewith applications to diﬀerence equations’,J.Math.Anal.Appl.314(2006)477–487.9.R.G.Halburd and R.J.Korhonen,‘Finite order solutions and the discrete Painlev´e equations’,Proc.London Math.Soc.94(2007)443–474.10.W.K.Hayman,Meromorphic functions(Clarendon Press,Oxford,1964).11.Y.He and X.Xiao,Algebroid functions and ordinary diﬀerential equations(Science Press,Beijing,1988)(Chinese).12.W.Kelley,and A.Peterson,Diﬀerence equations.An introduction with applications(Academic Press,Inc.,Boston,1991).ine,Nevanlinna theory and complex diﬀerential equations(Walter de Gruyter,Berlin,1993).ine,J.Rieppo and H.Silvennoinen,‘Remarks on complex diﬀerence equations’,Comput.MethodsFunct.Theory5(2005)77–88.15. A.Z.Mohon’ko and V.D.Mohon’ko,‘Estimates of the Nevanlinna characteristics of certain classesof meromorphic function,and their applications to diﬀerential equations’,Sibirsk.Mat.Zh.15(1974) 1305–1322(Russian).16. A.Ramani,B.Grammaticos,T.Tamizhmani and K.M.Tamizhmani,‘The road to the discrete analogueof the Painlev´e property:Nevanlinna meets singularity conﬁnement’,Comput.Math.Appl.45(2003) 1001–1012.17. A.P.Veselov,‘Growth and integrability in the dynamics of mappings’,Comm.Math.Phys.145(1992)181–193.18.N.Yanagihara,‘Meromorphic solutions of some diﬀerence equations’,Funkcial.Ekvac.23(1980)309–326.19.N.Yanagihara,‘Meromorphic solutions of some diﬀerence equations.II’,Funkcial.Ekvac.24(1981)113–124.20.N.Yanagihara,‘Meromorphic solutions of some diﬀerence equations of n th order’,Arch.Ration.Mech.Math.91(1985)169–192.21. C.-C.Yang and Z.Ye,‘Estimates of the proximate function of diﬀerential polynomials’,Proc.JapanAcad.Ser.A.Math.Sci.83(2007)50–55.Ilpo LaineUniversity of Joensuu Department of Mathematics P.O.Box111FI-80101JoensuuFinlandilpo·laine@joensuu·ﬁChung-Chun YangHong Kong University of Science and TechnologyDepartment of Mathematics Clear WaterKowloonHong Kongmayang@ust·hk。

有限元中, 是怎样处理分布载荷的。并用圣维南定理解释

有限元中, 是怎样处理分布载荷的。

并用圣维南定理解释有限元中是怎样处理分布载荷的有限元分析是一种工程数值分析方法，它用于评估结构在受力情况下的行为和性能。

在实际工程中，结构通常会受到分布载荷的作用，如风荷载、自重、地震力等。

因此，有限元分析需要能够准确地处理这些分布载荷，以便对结构的行为进行精确评估。

在本文中，我们将详细介绍有限元中处理分布载荷的方法，并且通过圣维南定理来解释这些方法的原理。

1. 划分载荷为有限元模型所能处理的载荷有限元模型通常是用离散的有限元单元来描述结构，在分析中需要将分布载荷离散化为有限元模型所能处理的载荷。

这通常可以通过将分布载荷按照一定的规则分布到有限元节点上来实现。

例如，将分布载荷按照线性或者二次分布规则离散化到有限元节点上。

然后，在节点上建立载荷的插值函数，将其传递给单元，从而得到整个有限元模型受力情况的离散表述。

2. 在有限元模型中引入等效节点载荷在有限元模型中，有时会将分布载荷的作用效果近似为等效节点载荷。

这通常可以通过对分布载荷进行积分得到等效节点载荷，并将其施加到有限元模型的节点上。

这样一来，整个有限元模型就可以通过节点载荷的叠加来模拟分布载荷的作用效果。

3. 使用圣维南定理来解释处理分布载荷的原理为了更加深入地理解有限元处理分布载荷的方法，我们可以借助圣维南定理来解释其原理。

圣维南定理是结构力学中的一个基本定理，它描述了受力结构的力平衡条件。

在有限元分析中，我们可以将分布载荷在有限元模型中抽象成为等效节点载荷，然后利用圣维南定理来解释这些等效节点载荷如何在有限元模型中引起结构的应力和变形。

圣维南定理可以描述为：对于一个受到外载荷作用的结构，在平衡状态下，结构内部的每个点，都能受到平衡的力和力矩。

在有限元模型中，由于分布载荷的作用，结构内部会受到一定的内力和内力矩。

因此，通过引入等效节点载荷，并且利用有限元模型的离散单元表述，我们可以用圣维南定理来解释结构内部的力平衡条件，从而理解分布载荷如何在有限元模型中引起结构的响应。

关于亚纯函数分担值及一些微分差分方程的值分布

关于亚纯函数分担值及一些微分差分方程的值分布上世纪二十年代,芬兰数学家R. Ncvanlinna引入了亚纯函数的特征函数,并建立了两个基本定理,从而创立了Nevanlinna值分布理论.他所创立的这一理论被认为是二十世纪最重大的数学成就之一,不仅奠定了亚纯函数理论研究的基础,而且对数学其它分支的发展产生了重大而深远的影响.尽管现在亚纯函数值分布理论已经趋于完善,但是对于其中一些经典问题的研究仍在继续,并且随着Ncvanlinna理论自身的不断发展,这一理论也广泛地应用到了其它数学领域,如位势理论,正规族,多复变量理论,复微分方程以及复差分方程等.我们知道,多项式除了一个常数因子外,由其零点而定.但对于超越整函数及亚纯函数来说,仅仅考虑零点是不够的,因此如何确定一个亚纯函数的讨论就显得复杂而有趣了.亚纯函数唯一性理论就是探讨在什么情况下只存在一个函数满足给定的条件,以及满足给定条件的函数之间有什么样的关系R. Ncvanlinna给出了唯一性理论上的经典的结果,即五值定理和四值定理.许多国内外著名的学者在这一方面做了大量的工作,取得了一系列引人注目的成果.这一理论也出现了越来越多的分支,比如亚纯函数与其导数的唯一性问题,这方面的研究成果可见[1].对应于亚纯函数与其导数的唯一性问题,Hcittokangas [2]等人最近开始考虑亚纯函数与其平移的分担值问题,这方面的研究基于复差分的Ncvanlinna理论,其中最关键的结果就是差分上的对数导数引理Halburd和Korhoncn [3,4], Chiang和Feng [5]分别独立给出了这个引理的两种表达形式.本文主要讨论了亚纯函数分担两个值的唯一性问题,以及一些微分差分方程的值分布问题.论文的结构安排如下.第一章,作为背景知识,我们首先简单介绍了Ncvanlinna理论的一些经典结果,其次介绍了差分中的对数导数引理,该引理是差分中Nevanlinna理论的奠基石,最后介绍了本文中用到的一个重要定理:Wiman-Valiron定理.第二章,我们研究了关于两个亚纯函数的非线性微分多项式分担值的唯一性问题,得到了几个唯一性的定理.它们改进了Fang和Hua[6],Yang和Hua[7]以及Fang和Qiu[8]的结果.实际上,我们证明了以下定理.定理0.1.设f和g是两个超越亚纯函数,它们零点的重级至少是k,其中k是正整数.令n>max{2k-1,k+4/k+4}是一个正整数.如果fnf(k)和gng(k)分担z CM,f和g分担∞IM,那么f和9满足下列情形之一?(i)fnf(k)=gng(k).(ii)f(z)=c1ecz2,g(z)=c2e-cz2,其中c1,c2和c是常数,且满足4(c1c2)n+1c2=-1.定理0.2.设f和g是两个非常数亚纯函数,它们零点的重级至少是k,其中k是正整数.令n>max{2k一1,k+4/k+4}是一个正整数.如果fnf(k)和gng(k)分担1CM,f和g分担∞IM那么f和9满足下列情形之一.(i)fnf(k)=gng(k).(ii)f(z)=c3edz,g(z)=c4e-dz,其中c3,c4和d是常数,且满足(-1)k(c3c4)n+1d2k=1.第三章,我们首先讨论了一类非线性差分方程和多项式的解的存在和增长性问题,得到以下一些结果.同时我们给出几个例子,证明我们结果中的有些条件是必须的.定理0.3.设p(z), r(z)和s(z)是非零多项式,c是非零复数.如果n>m+1(或m>n+1)是两个正整数,那么非线性差分方程f(z)n+p(z)f(z+c)m=r(z)es(z),没有有穷级超越整函数解.定理0.4.考虑差分方程f(z)n+p(z)f(z+c)m=g(z),其中p和q是非零有穷级整函数,m和n是正整数,c是非零复数.如果整函数f满足A(f)<(σ(f)=∞且σ2(f)<∞,那么f 不是该方程的解.定理0.5.假定方程f(z)n+p(z)f(z+c)m=q(z)具有有穷级整函数解f,其中q是有穷级非零整函数,p是f的小函数,m≠n是正整数,c是非零复数.那么σ(f)=σ(q).对应于fnf’值分布问题的研究,我们也探讨了整函数差分乘积f(z)n(f(z)-1)f(z+c)的值分布情况,解决了Zhang[9]和Qi[10]没有解决的n=1时的情形,得到以下主要结果.定理0.6.设f是有穷级超越整函数,且有Borel例外值a.c是一个非零复数.则有λ(f(f-1)f(z+c)-b)=σ(f),其中b≠a3-a.定理0.7.设f是有穷级超越整函数,c是一个非零复数.如果f(z)或f(z)-1有无穷多个重级零点,那么f(z)(f(z)-1)f(z+c)取每一个a∈C无穷多次.第四章,我们研究了与Bruck猜想有关的一些微分差分方程的值分布问题,改进了Chen,Shon[11],Wang[12]和Liu,Chen[13]的结果,得到以下主要定理.定理0.8.设f是非常数整函数,σ2(f)<∞且不是正整数.令L1(f)=ak(z)f(k)+ak-1(z)f(k-1)+…+a2(z)f(k+1)+f,(k≥2)其中aj(z)(2≤j≤k)是级小于1的整函数,且ak(z)≠0.如果f和L1(f)分担z IM,并且那么L1(f)-z=h(z)(f-z),其中b是级不大于s的亚纯函数.定理0.9.设f是非常数整函数,级σ(f)<1/2,a是f的一个非零小函数.令A(f)=ak(z)△kf+…+a1(z)△f+a0(z)f,其中aj(z)(j=0,1,…,k)是多项式且ak(z)≠0.如果f-a(z)=0→A(f)-a(z)=0,那么其中B(z)是一个非零多项式.。

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Nevanlinna理论在差分多项式中的应用
在1922年至1925年,芬兰数学家R.Nevanlinna在做了一些简短的注记之后,发表了他关于亚纯函数理论的文章,也就是后来的重要的数学理论Nevanlinna
理论,即复平面C上的亚纯函数值分布理论,10余年后L.Ahlfors建立了此理论
的几何形式.Nevanlinna理论,与后来的一些推广是函数论的重要组成部分,是
研究亚纯函数性质方面最重要的理论。

该理论不断自我完善和发展,同时广泛的应用到其他的复分析领域,如势理论,复微分及差分方程理论,多复变量理论,极
小曲面理论等。

复差分方程的基础建立于20世纪的早期,Batchelder[2],N（?）rlund[52]和Whittaker[57]在这个方面做了重要的贡献。

后来,Shimomura[55]和Yanagihara[59,60,61]利用Nevanlinna理论来研究了非线性的复差分方程的解。

由于亚纯函数有穷级解的存在性是考察差分方程可解性的一个好的性质,所以最近在这个方面的领域得到了广范的研究兴趣。

从这个角度出发,Nevanlinna 理论在处理复差分方程方面是一个很有用的工具。

复差分方面的Nevanlinna理论是最近才确立的。

其中,最关键的结果是差分对数导数引理,Halburd-Korhonen[20]和Chiang-Feng[8]给出了这个引理的两
种表达形式。

Halburd和Korhonen[21]在差分算子的基础上建立了Nevanlinna理论。

Ishizaki和Yanagihara[33]研究了差分方程慢增长的解的性质,并且给出了在
微分方程中著名的Wiman-Valiron理论的差分定理.Bergweiler和Langley[4,38]研究了慢增长的亚纯函数的差分算子的值分布论。

本论文利用Nevanlinna理论去研究差分多项式的值分布。

论文的结构安排
如下:第一章,我们简单介绍Nevanlinna唯一性理论和一些经常用的符号,还介绍了唯一性理论的一些经典的结果。

第二章,我们简单的回忆差分的对数导数引理,差分Clunie引理,还有差分的第二基本定理及其应用的结果。

另外差分方程解的存在性及增长性的一些重要结果也包含在本章节中。

第三章,我们介绍了差分乘积的值分布论,我们得到一些重要结果,可以看作关于Hayman经典微分多项式结果的差分推广,也就是关于微分多项式
fnf’的推广.实际上我们得到了下面的结果。

定理0.1.假设f（z）是超越的有穷级的整函数,令c是非零常数,并且n≥2是整数,则f（z）
nf（z+c）-p（z）和f（z）n△cf-p（z）都有无穷多个零点,其中p（z）（?）0是z的多项式。

对于更一般的差分乘积,也就是f是超越的亚纯函数,我们考察具有下面形式的差分乘积的值分布论Πj=1nf（z+cj）vj,其中cj∈C是一些不同的复常数。

我们不仅改进了定理0.1,而且我们得到了一个量化的估计:定理0.2.假设f为超越的有穷级的亚纯函数,级为ρ（f）,S（z）=R（z）eQ（z）,其中R（z）是非零的有理函数,Q（z）是多项式满足degQ（z）＜ρ（f）和λ（?）＜ρ（f）.如果（?）≥3,至少一个vj≥2,则Πj=1nf （z+cj）vj-S（z）有无穷多个零点。

如果有其中一个指数满足则另外我们研究了具有某种特定形式的亚纯函数的差分算子的值分布论,我们的目的是去寻找某些和微分算子类似的性质.也就是,我们证明了fkΔcf-a（z）的零点情况,k∈N∪{0}。

这个结果可以看作是fkf’-a（z）的差分的版本,可参见Hayman[27]。

我们的结果可以表述成:定理0.3.假设f有穷级的亚纯函数,1≤ρ（f）＜∞,且令a,c∈C\{0}满足△cf（?）0,f有无穷个零点并且λ（f）＜1.如果f有无穷多个极点,则△cf-a有无穷多个零点。

定理0.4.假设f超越的有穷级的亚纯函数,ρ（f）＜1,c是一个非零的常
数,B={bj}包含所有的f的极点,满足bj+kc（?）B
（k=1,…,m）至多有限个例外值,则f（z）n△cf-a有无穷多个零点。

在这一部分,也包含很多的例子说明我们的结果中的限制条件是必不可少的。

第四章,我们介绍亚纯函数的差分多项式的值分布论的结果。

我们首先回忆Hay-man[25,Theorem 8&9]经典的结果,我们的结果可以表述成:定理0.5.设f超越的有穷级的亚纯函数,并且ρ（f）=ρ,不是以c为周期的函数,λ（?）＜ρ（f）,s是有理的和a是非零的常数。

如果n≥3或者s=0,n
≥2,则差分多项式f（z）n+aΔcf-s（z）在复平面有
无穷多个零点。

定理0.6.设f超越的有穷级的亚纯函数,ρ（f）=ρ,不是以c为周期的函数,a是非零的常数,如果n≥8,则差分多项式f（z）n+aΔ
cf-s（z）有无穷多个零点。

在最后的第五章,我们得到了整函数f （z）与其平移f（z+c）或者差分算子△cf分担公共集合的唯一性的结果。

我们的结果可以看作是函数及其导数分担公共值[39]的差分版本。

其中一个重要的结果:定理0.7.假设f（z）为超越的有穷级的整函数,c∈C\{0},令a（z）
∈S（f）为非零的周期的整函数,周期为c.如果f（z）和f（z+c）分担集合{a （z）,-a（z）}CM,则f（z）满足下面结论中的其中之一:（ⅰ）f（z）（?）f（z+c）,（ⅱ）f（z）+f（z+c）（?）0,（ⅲ）f（z）=（?）（h1（z）+h2（z））,这里（?）,（?）,h1（z）h2（z）=a（z）
2(1-e-2γ),γ是一个多项式。

如果f（z）和△cf分担集合{a,-a}CM,其中a∈C,我们可以得到类似上面定理的结果。

作为上面定理的应用,我们研究了非线性的差分方程的解的情况,给出了f（z）2+f（z+c）2=a（z）2的解的形式,同时得到f（z）2+（△cf）
2=a2不存在非常数的有穷级的整函数解。

另外,我们给出了C.C.Yang在University of Joensuu的一次讨论课中给出的一个问题的部分答案。

他猜测下面的方程不存在无穷级的整函数解其中n≥2,b∈C\{0}和h（z）是一个有穷级的整函数。

我们也进一步改善[32,Theorem 8]中的分担值的条件,我们的结果说明f（z）仅是一个周期为c的函数。