摘要
Nevanlinna理论是上个世纪研究亚纯函数性质所取得的最好的成果.这个理论包括了两个基本定理,我们把它们称之为第一基本定理和第二基本定理,这两个定理显著的提高了经典函数理论的研究.至今,以Nevanlinna理论为基础的亚纯函数值分布理论仍是国内外复分析专家研究的热点问题之一
微分方程解的存在性是泛函分析研究的重要内容之一,非线性泛函分析中的拓扑度理论和锥理论是研究微分方程边值问题解的存在性的重要方法.
本文主要包括作者在导师仪洪勋教授指导下得到的一些新结果.论文的结构安排如下
第一章,我们简单介绍了Nevanlinna理论的基本知识；同时介绍了非线性泛函分析的基本理论.
第二章,我们研究了亚纯函数唯一性与Borel方向的关系,得到了亚纯函数IM分担四个不同的值及在包含一条Borel方向的角域内的集的唯一性.我们的结果推广和改进了龙见仁和
伍鹏程[35]近期的一个结果.实际上,我们证明了：定理0.1.若亚纯函数f具有熊庆来无穷级ρ(r),g∈且,(p(r)),argz=θ(0≤θ<2π是f的ρ(r)级Borel方向,若f与9在角域Ω(θ-ε,θ+ε)IM分担aj(j=1,2,3,4)且对任意的ε(0<ε<π),E(S,Ω(θ-ε,θ+ε),f)(?) E(s,Ω(θ-ε,θ+ε),g),这里S={b1,…,bm),m≥1且b1,…,bm).∈c\{01,a2,a3,a4}.则f与gCM分担所有的值,因此f三g或者f是g的Mobius变换.进一步,如果S中元素个数是奇数,则f三g.
第三章,我们研究一类微分差分多项式的值分布问题.实际上,我们得到下面的结果：
定理0.2.设f是有穷级的超越整函数,α(z)(≠0)是f的小函数,cj(j=1,2…d)是有限个不同的复数,n,m,d,k,vj(j=1,2…d)是非负整数.如果n≥k+2,
定理0.3.设f是有穷级的超越整函数,α(z)(≠0)是f的小函数,cj(j=1,2…d)是有限个不同的复数,n,m,d,k,vj(j=1,2…d)是非负整数.如果下面的条件之一成立：(ⅰ)n≥k+2,当m≤k+2,(ⅱ)n≥2k-m+3,当m>k+1.
定理0.4.设f与g是有穷级的超越整函数,α(z)(≠0)是f与9公共的小函数,cj(j=1,2…d)是有限个不同的复数,n,m,d,k,vj(j=1,2…d)是非负整数.
定理0.5.设f与g是有穷级的超越整函数,α(z)(≠0)是f与g公共的小函数,cj(j=1,2…d)是有限个不同的复数,n,m,d,vj(j=1,2...d)是非负整数.
第四章,我们利用凹函数的性质Jensen不等式给出了先验估计,采用不动点指数理论得到本章的结果.本章我们主要研究了带有脉冲效应的四阶p-Laplacian边值问题正解的存在性.这里j=[0,1],f∈C([0,1]×R+,R+),Ik∈C(R+,R+),令0
设p*:=max{1,p),p*:=min{1,p},K1:=2p*-1,K2:=2m(p*-1),K3:=2p*-1,K4:=2m(p*-1),K5:=2p/p*+p*-2,K6:=2(m+1)(p*-1)
下面我们给出本章中用到的假设条件.
(H1)存在ρ>0使得0≤y≤p且0≤t≤1有f(t,y)≤ηpρp,Ik(y)≤ηkρ,
这里η,ηk≥0满足
(H2)存在0 使得f(t,y)≥a1yp,Ik(y)≥a2y,(?)t∈[0,1],0 这里σ:=mint∈[t1,tm]t(1-t)>0.
(H3)存在c>0且a3≥0,a4≥0满足
使得f(t,y)≥a3yp-c,Ik(y)≥a4y-c,(?)t∈[0,1],y≥0.(3)
(H4)存在ρ>0使得σp≤y≤ρ且0≤t≤1有f(t,y)≥ζp ρp,Ik(y)≥ζkp,
这里ζ,ζk≥0满足
(H5)存在O 使得f(t,y)≤b1yp,Ik(y)≤b2y,(?Vt∈[0,1],0 (H6)存在c>0且b3≥0,b4≥0满足
使得f(t,y)≤b3yP+c,Ik(y)≤b4Y+c,(?)t∈[0,1],y≥0.(5)实际上,我们得到了下面的定理：
定理0.6.假设条件(H1)-(H3)满足,则(1)至少有两个正解.
定理0.7.假设条件(H4)-(H6)满足,则(1)至少有两个正解.
将泛函分析与复分析的相关内容有机结合是一个非常重要的研究课题.我们可以继续研究下面的问题：
利用非线性泛函分析中的拓扑度理论(主要是Brouwer度),具体分析亚纯函数族中函数的性质,特别是针对连续函数,通过研究连续函数的零点情况,解决亚纯函数正规族中函数与其导数分担连续函数的问题.目前这方面的结果比较少,1999年,Bargmann,Bonk,Hinkkanen,Martin将亚纯函数族中函数与其导数分担值的问题推广到分担连续函数[2],但仍然有很多问题需要解决.
Nevanlinna theory can be seen the most important achievements in the preceding century to understand the properties of meromorphic functions. This theory is composed of two main theorems, which are called Nevanlinna's first and second main theorems that had been significant breakthroughs in the development of the classical function theory.Today,many mathematics scholars still pay great attentions to this field which would always be based on Nevanlinna theory.
The existence of Solutions of differential equations is an important part in functional analysis, and the topological degree theory and cone theory are very important methods in nonlinear function analysis.
The present thesis involves some results of the author under the guidance of my supervisor Hongxun Yi. It consists of four parts and the matters are explained as below.
In Chapter1, We introduce the general background of Nevanlinna Theo-ry. Meanwhile, we give some preliminary definitions and properties of nonlin-ear functional analysis, and several lemmas on the existence of fixed point.
In Chapter2, we investigate the relationship between Borel directions and uniqueness of meromorphic functions and obtain some results of mero-morphic functions sharing four distinct values IM and one set in an angular domain containing a Borel line. Our result is an improvement of recent the-orem which given by Long and Wu [35]. In fact, we obtained the following result:
Theorem0.1. Let.f be a meromorphic function of infinite order p(r), g∈M(p(r)),argz=θ(0<θ<2π)be one Borel direction of p(r)order of meromorphic function f,we assume that f and g share four distinct values aj(j=1,2,3,4)IM in Ω(θ-ε,θ+ε)and E(S,Ω(θ-ε,θ+ε),f)(?) E(S,Ω(θ ε,θ+ε)E,g),,for anyε(0<ε<π),where S={b1,…,bm},m>1and b1,…,bm∈C\{a1,a2,a3,a4).Then f and g share all values CM,thus it follows that either f=g or f is a Mobius transformation of g.Furthermore, if the number of the values in S is odd,then f=9.
In Chapter3,we investigate the distribution and uniqueness of a class of differential-difference polynomials.In fact,we obtained the following result:
Theorem0.2.Let f be transcendental entire function of finite order,a(z)(≠0)be a small function with respect to f,cj(j=1,2…d)be distinct finite complex numbers,n,m,d,k,vj(j=1,2…d)are non-negative integers. If n>k+2,then the differential-difference polynomial(fn(fm(z)-1)(?) f(z+cj)vj)(k)一a(z)has infinitely many zeros.
Theorem0.3.Let f be transcendental entire function of finite order,a(z)(≠0)be a small function with respect to,f,cj(j=1,2...d)be distinct finite complex numbers,n,m,d,k,vj(J=1,2...d)are non-negative integers.If one of the following conditions holds:(i)n>k+2when m 2k-m+3when M>k+1.Then the differential-difference polynomial(fn(f(z)一1)m(?)f(z+cj)vj)(k)-a(z)has infinitely many zeros.
Theorem0.4.Let,and g be transcendental entire functions of finite order, a(z)(≠0)be a common small function with respect to,andg,cj(j=1,2...d) be distinct finite complex numbers,n,m,d,k,vj(j=1,2…d)are non-negative integers. If n>2k+m+σ+5,and the differential-difference polynomial a(z)CM,then f=tg,where tm=tn+σ=1.
Theorem0.5.Let f and g be transcendental entire functions of finite order, a(Z)(≠0)be a common small function with respect to f and g,cj(j=1,2…d) be distinct finite complex numbers,n,m,d,k,vj(j=1,2...d)are non-negative integers. If n>4k-m+σ+9,and the differential-difference polynomial
In Chapter4,we devoted to study the existence and multiplicity of positive solutions for the fourth order p-Laplacian boundary value problem involving impulsive effects where J=[0,1],f∈C([0,1]×R+,R+),Ik∈C(R+,R+)(R+:=[0,∞)). Based on a priori estimates achieved by uti-lizing properties of concave func-tions and Jensen's inequality,we adopt fixed point index theory to establish our main results.
Let p*:=max{1,p},p.:=min{1,p},K1:=2p*-1,k2:=m(p*-1), K3:=2p*-1,K4:=2m(p*-1),K5:=2p/p*p-2,k6:=2(m+1)(p*-1). We now list our hypotheses.
(H1)There is a p>0such that0 whereη,ηk>0satisfy
(H2)There exist00,a2>0satisfying
such that f(t, y)> a1yp, Ik(y)> a2y,(?)t∈[0,1],0 where σ:=min t€[t1,tm] t(1-t)>0.
(H3) There exist c>0and a3>0, a4>0satisfying
such that f(t,y)>a3yp-c, Ik(y)> a4y-c,(?)t∈[0,1], y>0.(8)
(H4) There is a p>0such that σpζPPP, Ik(y)>-ζkA
whereζ,ζk>0satisfy
(H5) There exist00, b2>0satisfying
such that f(t, y) (H6) There exist c>0and b3>0, b4>0satisfying
such that f(t, y)0.(10)
In fact, we obtained the following result:
Theorem0.6.Suppose that (H1)-(H3) are satisfied, then (6) has at least two positive solutions.
Theorem0.7. Suppose that (H4)-(H6) are satisfied, then (6) has at least two positive solutions.
Combining functional analysis with complex analysis is a very important research topic, we can continue to study the following question:
Using the topological degree theory (mainly Brouwer degrees) in Non-linear functional analysis, we can study the normality of function family, especially for a continuous function. By studying the case of zeros of con-tinuous functions, we will resolve the problem of meromorphic functions and its derivative sharing continuous Functions. Currently there are very few results in this area, in1999, Bargmann, Bonk, Hinkkanen, Martin focus on this problem [2]. At present, this area still has a lot of problems to be solved.
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