Some Researches On Solutions Of Complex Differential Difference Equations And Complex Difference Equations

In this paper,we use the value distribution theory of meromorphic functions as a main tool.Firstly,we study the existence of solutions of a certain type of nonlinear complex differential difference equations.Secondly,we study the uniqueness of mero-morphic solutions of a certain type of complex difference equations.In recent years,these are the questions that lots of experts in complex analysis are interested in.Paper is divided into four chapters,the concrete arrangement is as follows:In Chapter 1,we introduce some basic notations and results on the value distri-bution theory of meromorphic functions,the difference version of Nevanlinna theory and the uniqueness theory of meromorphic functions briefly.In Chapter 2,we investigate the existence of solutions of a certain type of nonlin-ear complex differential difference equations.Firstly,we consider the transcendental entire solutions of a certain type of nonlinear complex differential difference equations base on the results of Liu et al and Chen et al.Several sufficient conditions on the non-existence of transcendental entire solutions of such equations are supplied.Secondly,based on the results obtained by Zeng et al.,we study transcendental entire solutions of the following more general higher order nonlinear complex differential difference equationwhere p(z)(?)0 is a polynomial,r(z)is a non-constant polynomial,??0 and? are constants,k is a positive integer,and??C\{0}.We get the precise form of finite order transcendental entire solutions of the above equation.Examples show that the conditions in the theorem are accurate.In Chapter 3,we mainly study the uniqueness of two finite order transcendental meromorphic solutions f(z)and g(z)of the following complex difference equationwhen they share 0,?CM,where A₁(z),A₀(z),F(Z)are non-zero polynomials,?(z)is a polynomial.Our result generalizes and complements some known results given by Cui and Chen,Li and Chen.Meanwhile,we give some examples to demonstrate that the conditions are necessary.In Chapter 4,we make a summary and pose some problems.