The Research Of Some Problems On Value Distribution Of Meromorphic Function And Differential Difference Equations

This thesis study the uniqueness of entire functions that share a small function with their linear difference polynomials,and the problem that entire solutions for several Fermat type differential difference equations in value distribution theory.It contains the following parts:In Chapter 1,we give an introduction about the research background,the significance and the development of the value distribution theory of meromorphic function and differential difference value distribution theory.In Chapter 2,we give a brief summarization of the basic knowledge of value distribution theory and some important concepts and theorems about differential difference value distribution theory.In Chapter 3,we study the problem that the uniqueness of entire functions that share a small function with their linear difference polynomials,and proveLet f(z)be a transcendental entire function with ρ2(f)<1,let L(z,f)=b1(z)f(z+c1)+b2(z)f(z+c2)+…+bn(z)f(z+cn)be a linear difference polynomial with bi(z)((?)0,i=1,2,…,n)are small functions of f(z)and ci(i=1,2,…,n)are distinct finite values,and let a(z)be a small function of f(z)satisfying a(z)(?)L(z,a)and L(z,f)(?)L(z,a).Ifδ(a,f)=1,then f(z)and L(z,f)can not share either a(z)or L(z,a)IM.Let f(z)be a transcendental entire function with ρ2(f)<1,let a1(z),a2(z)be two small functions of f(z)satisfying a1(z)(?)a2(z),and let L(z,f)=b1(z)f(z+c1)+b2(z)f(z+c2)+…+bn(z)f(z+cn)be a linear difference polynomial with bi(z)((?)0,i=1,2,…,n)are small functions of f(z)and ci(i=1,2,…,n)are distinct finite values,and a1(z)(?)L(z,a2(z)).If δ(a2,f)+δ(a2,L(z,f))>1,and f(z)and L(z,f)share a1(z)CM,then f(z)≡L(z,f).In Chapter 4,we focus on the problem of entire solutions for several Fermat type differential difference equations,and proveLet g(z)be a nonconstant polynomial,and let f(z)be a transcendental entire solution with finite order of the difference equation f(z)2+2αf(z)Δcf(z)+Δcf(z)2=eg(z).Then g(z)must be of the form g(z)=az+b,and f(z)=Ae1/2az,where a(≠0),b,A(≠0)are constants satisfying A2[eac+ 2(α-1)e1/2ac-2(α-1)]=eb.Let g(z)be a nonconstant polynomial,and let f(z)be a transcendental entire solution with finite order of the difference equation f(z+c)2+ 2af(z+c)Δcf(z)+ Δcf(z)2=eg(z).Then g(z)must be of the form g(z)=az+b,and f(z)=Ae1/2az,where a(≠0),b,A(≠0)are constants satisfying A2[2(1+α)eac-2(1+α)e1/2ac+1]=eb.Let g(z)be a nonconstant polynomial,and let f(z)be a transcendental entire solution with finite order of the differential difference equation f’(z)2+2αf’(z)Δcf(z)+Δcf(z)2=eg(z).Then g(z)must be of the form g(z)=az+b,where a(≠0),b are constants,and f(z)must satisfy one of the following cases:(1)f(z)=Ae1/2az+c1,where A(≠0),c1 are constants satisfying A2[eac+(αa-2)e1/2ac+1/4a2-αa+1]=eb;(2)f(z)=B1z+ B2eaz+c1,where Bi(#0,i=1,2),c1 are constants satisfying(3)f(z)=B1ea1z+B2e(a-a1)z+c1,where a1(≠0),Bi(≠0,i=1,2),c1 are constants satisfyingIn Chapter 5,we give a conclusion about this paper and point out the further conjecture on the basis of this paper.