Study On The Meromorphic Solutions Of Differential Difference Equations

In the 1920s,Rolf Nevanlinna.based on early results by Picard and Borel on entire functions,developed the value distribution theory of meromorphic functions,Nevanlinna's theory on the value distribution of meromorphic func-tions is far reaching and has important impact on many areas of mathematics.Therefore,the value distribution theory is also called Nevanlinna's theory.The essential parts are Nevanlinna's first,and second fundamental theorems.Lat-er,Ahlfors found the geometric explanations of Nevanlinna's theory,which put the theory on a firm base.Nevanlinna theorry also has numerous appli-cations in some other complex branches,for example uniqueness theory of meromorphic functions,normal family,complex dynamical systems,complex differential equations,and so on.The study on the existence and properties of meromorphic solutions of complex differential equations is one of the important applications of Nevan-linna theory.In recent,years,many researchers began to apply Nevanlinna's theory to the study of difference equations.For example,Halburd and Ko-rhonen[16-18]set up the difference analogues of Nevanlinna theory.And the difference analogue of the classical logarithmic derivative lemma plays a very important role on the study of the property of meomorphic solutions of complex differential difference(or difference)equations.In this dissertation,we study the unicity,growth,existence and the form of the meromorphic solutions of differential difference equations.It consists of four parts and the matters are the following.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 the second chapter,we give a uniqueness theorem on meromorphic solutions f of finite order of a class of differential-difference equations such that solutions f are uniquely determined by their poles and two distinct values.Meanwhile,to show the sharpness of our results we will present some concrete examples.In the third chapter,firstly,by using Nevanlima theory and linear al-gebra.we show that the number one is a lower bound of the hyper-order of any meromorphic solution of a non-linear delay differential(i.e.differential-difference)equation under certain conditions.Our result extended the existing result.Meanwhile,to show the sharpness of our results we will present,some concrete examples.Then we give the form of the entire solutions of another type of non-linear delay differential(i.e.differential-difference)equation under certain conditions.We continue the study of meromorphic solution of differential-difference equation in the fourth chapter.By using Nevanlinna theory,we generalize a result given by Wittich to complex differential-difference equations.The result,obtained is that the differential-difference equation in f which is of only one dominant term,has no admissible meromorphic solution with hyperorder less than 1 provided N(r,f)= S(r,f).We also give two examples show that conditions P(f)has only one dominant term and N(r,f)= S(r,f)in theorem 4.1 can not be dropped.