The Properties Of The Entire Solutions Of Nonlinear Differential-difference Equations Of Certain Types

In the early 1920s,R.Nevanlinna first introduction the characteristic function of meromorphic functions,and created the two basic theorems,which is known as the Nevanlinna theory.The famous Picard and Borel theorems would be a simple deduction of it.Through the unremitting efforts of the scholars,the theory has been continuously improved and developed rapidly,and has been widely used in the field of compound analysis.Among them,many far-reaching consequences have been obtained in the fields of uniqueness theory,complex difference equation and complex differential equationIn this paper,using the Nevanlinna value distribution theory and the main conclusions,we study the properties of solutions of certain types of non-linear differential and difference equations and generalize some important results,which is divided into four chaptersfirstly,we summarize the theory of Nevanlinna value distribution,usual notations,as well as some basic concepts and related knowledge about the properties of functionssecondly,we deduce a uniqueness result of entire functions that share a small entire function with their two difference operators,which generalizes some previous theorems of Chen[11]and A.El Farissi[12]by omitting the assumption that the shared small entire function is periodic and proof of theorem.Considering the condition of p(a)<1,a new theorem is derivedOnce again,we consider the uniqueness problems of finite-order meromorphic solutions to Painlevé equation,focuses on researching the solution of the equationH(z,f)=[b2?2+b1?+b0][?(z+1)+?(z-1)]-[a2?2+a1?+a0]=0 Our result says that such solutions w are uniquely determined by their poles and the zeros of ?-ej(counting multiplicities)for two finite complex numbers e1?e2.And we get two different theorem,and the theorem was proved and the applicationLast chapter,we studied the existence of the solutions of nonlinear differential-difference equation problem,increased the complexity of the equation,simplify the existence conditions of understanding,and satisfactory results are obtained.After index increased to equation of qualified to get a simple reasoning.