The Properties Of Meromorphic Solutions Of Some Nonlinear Difference Equations And Linear Differential Equations

In this thesis,using Nevanlinna’s value distribution theory,we mainly investigate the properties of meromorphic solutions of some types of nonlinear difference equations and linear differential equations.The paper is divided into four chapters.In chapter 1,we briefly introduce the research background and the preliminary theoretical knowledge required in this paper.In chapter 2,we investigate the existence and growth of finite order meromorphic solutions of the nonlinear difference equation fn+q(z)eQ(z)Δcf=p1(z)eα1z+p2(z)eα2z,where n≥ 3 is an integer,q(z),p1(z),p2(z),Q(z)are nonzero polynomials,andα1,α2 are distinct nonzero constants.Particularly,when n≥4,we give the expressions of its finite order meromorphic solutions,and discuss its exponential polynomial solutions for the case n=3.In chapter 3,we investigate the growth of finite order meromorphic solutions of the nonlinear difference equations fn+q(z)eQ(z)Δcf=P(z)eα(z),where n≥ 2 is an integer,q(z),P(z)are nonzero polynomials,and α(z),Q(z)are nonconstant polynomials.It is proved that each finite order meromorphic solution of the above equation is an entire function.Some estimations on growth of its solutions are also obtained.In chapter 4,we investigate the existence of completely regular growth solutions of the higher order linear differential equation f(n)+An-1(z)f(n-1)+…+A1(z)f’+A0(z)f=0,where its dominant coefficient is an exponential polynomial.By using the Nevanlinna characteristic of exponential polynomials,some conditions which guarantee the non-existence of such solutions are obtained.At the same time,for the higher order linear differential equation with exponential polynomial solutions,the relationship between the expression of its solutions and the dominant coefficient is given.