Researches On The Meromorphic Solutions Of Some Types Of Differential-difference Equations In The Complex Domain

In the 1920s,the mathematician R.Nevanlinna,a Finlander,established the value distribution theory of meromorphic functions,namely Nevanlinna theory.The theory is known as one of the greatest mathematical achieve-ments of the 20th century.For nearly 100 years,Nevanlinna theory has been completed and developed,and has been widely applied to other fields of com-plex analysis such as the uniqueness of meromorphic functions,normal families,complex dynamical systems,complex differential and difference equations,etc..Many outstanding mathematicians such as Ahlfors,Cartan,Wittich,Hayman,Qinglai Xiong,Le Yang,Gua.nghou Zhang and so on,have all made significant contributions to Nevanlinna theory.In 1929,F.Nevanlinna[42]first applied Nevanlinna theory to the study of differential equation f"+A(z)f=0 in the complex domain.Later,by using Nevanlinna theory,Wittich,Gol'dberg[16]investigated the solutions of complex differential equations and complex algebraic differential equations,respectively.Over the next 60 years,many scholars used Nevanlinna theory to study the problems related to the solutions and other analytic theories of complex differential equations and got rich research results.In the 1980s,Shimomura[46],Yanagihara[51-53]obtained the solutions of a class of complex difference equations from the viewpoint of the Nevanlinna theory.Until the recent 10 years,Nevanlinna theory has been used as a power-ful tool to study complex difference equations.The key result is the difference analogue of the lemma on the logarithmic derivative obtained by Halburd and Korhonen[20,23],Chiangand Feng[8].independently.Besides,the difference analogue of the classical value distribution theory of Nevanlinna,such as the Nevanlinna second main Theorem,Clunie Theorem,Wiman-Valiron Theorem,and Picard Theorem and so on,have also been established gradually.These theories have been widely applied to the study of complex difference polyno-mials,complex difference equations,complex differential-difference equations,making them one of the hot research issues in the field of complex analysis.In this dissertation,we investigate the meromorphic solutions of some types of complex differential-difference equations by using Nevanlinna theory.The structure of the dissertation is as follows:Chapter 1:This chapter briefly introduces the basic knowledge and clas-sical theories of Nevanlinna theory and Nevanlinna theory on difference oper-ators.Chapter 2:This chapter mainly studies the following differential-difference equation of Fermat type in f(z)Fn(z)+m?F(z)G(z)+Gn(z)=e?z+?,where F=a0f(z)+aif'(z)+a2f(z+c),G=b0f(z)+b1f'(z)+b2f(z+c),{?,?,?,ai,bi}?C,i=0,1,2.We give the precise form of the finite order transcendental meromorphic solutions of the above equation for the two cases that n=3,m=0andn=2,m=2,?2?0,1,respectively.Our results generalize some results by Saleeby[45]?Liu and Yang[36]?Lu and Han[25,39].Chapter 3:First,we consider the following complex differential-difference equation?wherefni=f(z+?i),fe=f(z+c),{c,?0,?1,…,?n}?C,{n,k} C Z+,Po(z)is an entire function,P1(z),ho(z),h1(z),…,hn(z)are small functions of f(z).We obtain the order and the exponent of convergence of zeros and the Borel values of finite order transcendental entire solutions of the above equation,which improve the results in Liu and Song[37].In addition,motivated by the consideration of exponential polynomial solutions in Wen et al.[48],we discuss exponential polynomial solutions of the following complex differential-difference equation#12 where Po(z)is a polynomial,P1(z),a0(z),a1(z),…,an(z)are exponential poly-nomials of orders at most deg(P0)—1.We have that the order of every expo-nential polynomial solution of the above equation is not more than deg(P0)+1.Chapter 4:We first study the complex difference Riccati equation with rational coefficients?(z+1)=a?(z)+b/c?(z)+d'and prove that the order of every transcendental meromorphic solution of the above equation under certain conditions is not less than 1.Our result partially answers a question proposed by Chen Zongxun[7].Then,we describe the form of rational solutions of celay differential equation as follows?(z+1)-?(z-1)+a?'(z)/?(z)=P(z,?(z))/Q(z,?(z)),where a?C,P(z),Q(z)are polynomials in ?(z)with coeffcients in z.This result is a supplement to the results of Halburd and Korhonen[22].Finally,we discuss the more general delay differential equation a1?(z+c1)+…+an?(z+cn)+a?'(z)/H(z,?(z))=P(z,?(z))/Q(z,?(z))whereci?C,a(z),ai(z)(i?0,1,…,n)are rational functions,H(z),P(z),Q(z)are polynomials in ?(z)with rational coefficients in z,Q(z)has zeros but are not zeros of P(z).We prove that if the byper-order of the transcenden-tal meromorphic solution of the above equation is strictly less than 1,then deg(P)?deg(Q)+1.Our result generalizes and improves the result of Hal-burd and Korhonen[22],and some examples are given to illustrate the accuracy of our results.