Some Studies On Meromorphic Function Theory And Complex Difference Equations

Function theory is a branch of management mathematics,which provides a powerful tool for management.The theory of meromorphic functions belongs to the classical subject of complex analysis in the field of function theory,especially the value distribution theory of meromorphic functions(also known as Nevanlinna theory),which was founded by the famous mathematician R.Nevanlinna in 1920 s,greatly promoted the development of complex analysis and is applied into the uniqueness theory of meromorphic functions and complex differential equations.Since 2006,the difference operators are introduced into the value distribution theory by some researchers and then applied to complex difference equations.In this dissertation,we mainly consider to improve and extend the difference analogous of the logarithmic derivative lemma for meromorphic functions,and apply it to the complex difference equation.This thesis consists of eight chapters.In the first chapter,we briefly introduce the basic knowledge of Nevanlinna theory for meromorphic functions of one and several complex variables and the uniqueness problem.In the second chapter,we introduce our results on the difference form of the logarithmic derivative lemma of meromorphic functions,by the growth lemma due to Zheng and Korhonen and the Borel type lemma due to Hinkkanen,respectively.These are improvement and extensions of the earlier results in one dimension and higher dimensions.The condition on the hyperorder strictly less than one for meromorphic functions is extended to lim supr??log T(r,f)/r= 0,which is the best estimate at present.In the third chapter,we mainly introduce our work on the difference Riccati equation involving f(qz + c),which is a generalization of the recent results of Z.X.Chen and K.H.Shon.In Chapter 4,we mainly introduce our work on one-dimensional Fermat type difference equations.We obtain the expression forms of all entire solutions of Fermat type difference equations by finding a new way instead of the difference analogous of the logarithmic derivative lemma.In Chapter 5,we mainly introduce our work on high dimensional Fermat type partial difference equations.The meromorphic function solutions of Fermat type partial difference equations are discussed by introducing difference operators into Fermat type functional equations and applying the difference analogous of the logarithmic derivative lemma.Our theorems generalize the related results of K.Liu,T.B.Cao and H.Z.Cao.In Chapter 6,we apply the logarithmic derivative lemma of difference operators in Chapter 2 to study the complex difference equations.The meromorphic properties of linear partial difference equations and nonlinear partial difference equations such as Kd V type and Fermat type are studied for the first time.In Chapter 7,based on the fact that entire functions and meromorphic functions have many different properties involving the total derivatives,we extend an uniqueness theorem of entire functions in several complex variables due to L.Jin to the case of meromorphic functions,which is also the generalization of a related result of one complex variable due to H.X.Yi.In the last chapter,we give a short summary for this thesis.