Properties Of Solutions Of Some Complex Differential And Differential-Difference Equations

In this thesis,we use the theory of function spaces,Nevanlinna theory and Phragmen-Lindelof indicator functions to investigate the properties of solutions of some complex differential and differential-difference equations.This thesis mainly contains four chapters.In chapter 1,we briefly introduce historical backgrounds of the problems discussed in the thesis and the main results we have got.In chapter 2,we introduce some definitions of function spaces,Nevanlinna theory and some results of difference value distribution.In chapter 3,we mainly investigate the properties of solutions of second order complex linear diferential equationsf"+A(z)f=0,(1)where A(z)is analyitic in the unit disc D.We give some sufficient conditions such that the solutions of equation(1)belong to Bloch(small Bloch)spaces by using reproducing kernel formula of exponential type weighted Bergman spaces.Moreover,we find some sufficient conditions which guarantee any solution of equation(1)to be in analytic Morrey spaces by using the representation formula h(z)=1/2π∫02π h(eis)/1-ze-is ds,where h∈H(D),z∈D.In chapter 4,we firstly discuss the growth of meromorphic solutions of higher order linear diference equations An(z)f(z+ηn)+…+A1(z)f(z+η1)+A0(z)f(z)=0(2)where Aj(z)(j=0,1,…,n)are entire functions,ηj(j=1,2,…,n)are complex con-stants.By using Phragmen-Lindelof indicator functions,we obtain an estimation of lower bound of the growth of meromorphic solutions of equation(2)if it has multiple dominant coefficients with the same type.Secondly,we investigate the properties of solutions of finite order of nonlinear diferential-diference equations f(z)n+an-1f(z)n-1+…+a1f(z)+q(z)eQ(z)f(k)z+c)=P(z),(3)where q(z),Q(z),P(z)are polynomial,c G∈C\{0},ai∈C(i=1,2,…,n-1).We will classify these solutions of equation(3)according to growth and zero distribution.Particularly,we show that exponential polynomial solutions satisfying certain conditions must reduce to rather special forms when n=2 and a1≠0.