The Properties Of Solutions Of Some Types Of Nonlinear Delay-differential Equations And Higher Order Linear Differential Equations

In this paper,using Nevanlinna theory of meromorphic function and the corresponding difference analogue,we investigate the properties of entire solutions of some class of nonlinear delay-differential equation and the zero distribution of solutions of higher order linear differential equations.This paper is divided into three chapters.In chapter 1,we briefly introduce the development history in the field of complex difference equations and linear differential equations and the research background of this paper.Basic theory and common notation related to this paper are also introduced.In chapter 2,the existence and growth of entire solutions of the nonlinear delay-differential equations fn(z)f’(z)+P(z)f(k)(z+η)=H1(z)eω1zq+…+Hm(z)eωmzq are studied,where n,k,m,q are positive integers,ω1,...ωm are distinct nonzero complex constants,P(z),H1(z),...,Hm(z)are nonzero entire functions with order less than q.Especially for n≥m+2,we obtain the exact expression of its entire solutions with hyper-order less than 1.In chapter 3,The zero distribution of solutions of the higher order linear differential equation f(k)+∑j=0k-2 Aj(z)f(j)=0 is studied,where k≥3 is an integer,Ak-2(z),…,A0(z)are entire functions such that for some l ∈{1,…,k2},the order of Al(z)dominates the orders of the other coefficients and its lower order is less than 1/2.We obtain a result concerning the effect of perturbation of Al(z)on the complex oscillation of solutions of such equation.