On Zeros And Fixed Points Of Difference Operators

In the 1920s,Rolf Nevanlinna,based on early results by Picard and Borel on entire functions,developed the value distribution theory of meromorphic functions,which since then has led to intensive discussions and generalizations.Nevanlinna's theory on the value distribution of meromorphic functions is far reaching and has important impact on many areas of mathematics.Nowadays,the value distribution theory is also called Nevanlinna's theory.The essential parts are Nevanlinna's first and second fundamental theorems.Later Ahlfors found geometric explanations of Nevanlinna's theory,which put the theory on a firm base.Nevanlinna's theory is deep.Except for itself,the theory has been applied to many other areas of methematics.One of them is using Nevanlinna theory to study the existence and properties of meromorphic solutions of complex differential equations,which turns out to be useful.The theory of complex differential equations has been upgated since then.In recent years,many researchers began to apply Nevanlinna's theory to the study of difference op-erators and difference equations.For instance,a difference analogue of the classical logarithmic derivative lemma has been established,the existence of zeros and fixed points of difference operators has been given under certain con-ditions,and some properties of meromorphic solutions of difference equations have also been found.Nevanlinna's theory is also applied to complex dynamics and dynamics of semi-groups.We will not discuss them here.In the first chapter,we will mainly give a brief introduction to Nevanlin-na's theory,including some definitions,results and also some commonly used symbols.From now on we assume that f is a function meromorphic in the plane.In 2007,Bergweiler and Langley first studied the existence of zeros and fixed points of the difference operator ?f = f(z + c)-f(z)and the difference quotient ?f/f,Their results were later generalized by others.In the second chapter,we continue their research on this respect,giving conditions which ensure the existence of zeros and fixed points of difference operators and dif-ference quotients when the meromorphic function in question has exactly order of growth one.Meanwhile,to show the sharpness of our results we will present some simple and concrete examples.In the third chapter,we will not restrict ourselves to difference operators and difference quotients.Instead we consider linear combinations of difference shifts and their corresponding quotients and study the existence of zeros and fixed points of these linear combinations.Our results in this chapter gen-eralizes those in the last chapter and has close connections with difference equations.We will mainly give conditions on coefficients,under which the lin-ear combinations have infinitely many zeros and infinitely many fixed points.A key point in the proof of these results is to study the transcendency of these combinations.We continue the study of Chapter three in the fourth Chapter.We give much weaker conditions,but still have conclusions as in the last chapter.However,the method used here is different.In the last chapter,the use of Hadamard's factorization theorem plays an important role.But in this chap-ter,we rely more on the properties of the function in question and hence are able to give weaker conditions.In the last chapter,we will mainly discuss possible questions concerning difference operators and possible forthcoming research on this respect.In the previous chapters we give conditions under which the difference operators have zeros and fixed points.Here we will propose the question concerning conditions which imply that the difference operators have no or only finitely many zeros or fixed points.