On Entire Function Solutions Of Nonlinear Complex Differential Difference Equations

In the 1920s,the famous Finnish mathematician Nevanlinna established the first and second fundamental theorem of Nevanlinna by introducing the characteristic function,created the value distribution theory,and opened the beginning of studying the complex variable function theory with the value distribution theory The value distribution theory is one of the greatest achievements in mathematics in the 20th century and has a great impact on the development of mathematics.In order to commemorate Nevanlinna’s outstanding contributions,the value distribution theory is also known as Nevanlinna’s theory Since the establishment of the value distribution theory,it has been continuously improved and developed,and has been widely used in various fields of complex analysis,such as complex differential equations,complex difference equations,complex dynamic systems,and so on.The value distribution theory has greatly promoted the study of periodicity of difference equations,especially in recent years,the establishment of the difference form of the value distribution theory has greatly promoted the study of periodicity of difference polynomials.The value distribution theory also plays a great role in the research of complex differential equations.Many scholars have made many good results in this field.With the development of the value distribution theory,the research on the properties of complex differential equations and their integral function solutions has gradually become a research hotspot.On the basis of the above two aspects,this paper further studies the periodicity of the difference equation and the solution of the whole function of the differential equation.The structure of this paper is as follows:Chapter 1:A brief introduction to the value distribution theory and an overview of the important definitions and theorems in the value distribution theory used in this paper.Chapter 2:The periodicity of the following difference polynomials is mainly studied:where f(z)=p(z)+eh(z),p(z)is a polynomial,h(z)is an entire function,ρ2(f)<1.The same conclusion is obtained by extending f(z)2 to f(z)n in the original theorem.Chapter 3:The following differential equations are mainly studied:a(z)f f"+b(z)(f’)2=c(z)e2d(z).The concrete form of the solution of the entire function of,where a(z),b(z),c(z),d(z)are polynomials,c(z)≠0,d(z)non-constant numbers.The coefficients of the original theorem are generalized,thus the conclusions of the original theorem are generalized.