Research On The Analytical Theory Of Some Differential Equations In The Complex Domain

The research of some theoretical and practical problems usually needs to find the roots of certain entire functions or meromorphic functions,that is:for a entire function or meromorphic function f(z)and any complex number a,we would like to study the existence or the distribution of roots of equation f(z)=a.Famous mathematicians such as Picard and Borel have obtained some outstanding results,that forming the value distribution theory of entire function and meromorphic function.In 1925,the Finnish mathematician R.Nevanlinna introduced the charac-teristic function of meromorphic functions and established two basic theorems,which called Nevanlinna value distribution theory.This theory makes Picard's theorem and Borel's theorem become its special cases and it caused a great response in the field of mathematics.In the past century,the theory has been deeply developed and applied to many other complex analysis fields,such as uniqueness of meromorphic func-tions,normal families,complex dynamical systems,complex differential equa-tions,etc.As we all know,the complex diferential equations theory has been widely applied to mathematical and physical problems,research in the fields of special functions by scholars from all over the world.Moreover,Nevanlinna theory and the theory of complex differential equations have a profound influ-ence on each other.In 1929,Nevanlinna studied the second-order differential equation f"(z)+A(z)f(z)=0,where A(z)is a polynomial.Subsequent-ly,many mathematicians have carried out a series of researc on the complex differential equations of the form P(z,f,f',…,f(n))=0.Later,some schol-ars applied the Nevanlinna theory to the study of difference equations in the complex domain.Among them,Halburd and Korhonen established the dif-ference form of the basic theorem of Nevanlinna.Especially,for studying theproperties of solution of the differential difference(or difference)equation in the complex domain,the logarithmic derivative lemma of difference equation plays an immeasurable role.On the basis of the existing literature,by using Nevanlinna theory to find the existence of solutions of meromorphic solutions and other analytical properties of some delay differential equations of the form P(z,f,f',…,f(n),fc1,f'c1,…,fc1(l1),fck,f'ck,…,fck(lk))=0 have attracted the high interest of a large number of scholars,which is one of the hottest topics in recent years.In this dissertation,we mainly study the properties of meromorphic solu-tions of some types of delay differential equations and differential functional equations,such as existence,growth order ect..The dissertation consists of four parts and the matters are the following.In the first chapter,we will mainly give a brief introduction to Nevan-linna's theory,including some notations,definitions,results and also some theorems of difference equations.In the second chapter,we discuss the following delay differential equation fk(z)(?)e?(z)f(z+c?)+a(z)f(n)(z)/f(z)=R(z,f(z)),a more simple form of the delay differential equations is obtained under some proper conditions,the result can be regard as the extend of a famous theorem of Manliquist.Moreover,to show the accurate of our results we will present some concrete examples.In the third chapter,by using Nevanlinna theory and Nevanlinna-Borel theorem,we will investigate two types of non-linear delay differential posses two advantage terms.For non-linear delay differential equation fn(z)+fn-2(z)f'(z)+Pd(z,f)=p1(z)e?1(z)+p2(z)e?2(z)some proper conditions are found to ensure it admits transcendental meromor-phic solutions.In addition,the form of the exponent solutions of non-linear delay differential equations f2+q1eQ1fc1(t1)+q2eQ2fc2(t2)=P is found,some concrete examples are given to show that the condition in our theorem can not be dropped.In the forth chapter,by combining the differential equations with func-tional equations,the existence of meromorphic solution on a class of higher order differential functional equations(f')nf(n)=afn+1(g)+bf+d is investigated,we obtain if f(z)is rational,then degz(g)?2.If f(z)is a transcendental,then g must be linear.Especially,we find the form of f(z)and g(z)when degz(g)=2.Moreover,we also give an estimate of the integral functions of f for a type of differential functional equations?1(z,f)=(?)ai(f(p))i under certain conditions,some examples are given to show that our condition is best possible.