On Meromorphic Solutions Of Certain Nonlinear Differential Difference Equations

In the 1920s,Rolf.Nevanlinna,a famous Finnish mathematician,generalized the earlier work of Picard and Borel on entire functions.By introducing the characteristic functions of meromorphicfunctions,he established the first fundamental theorem and the second fundamental theorem,and began the study of value distribution theory.The value distribution theory is one of the most important mathematical results in the last century,which has had a great influence on other branches of mathematics.In honor of his outstanding contributions,value distribution theory is also called Nevanlinna theory.In the past 100 years,Nevanlinna theory has been continuously improved and developed,and has been widely used in other complex analysis fields,such as uniqueness theory of meromorphic functions,complex differential equations,normal families,complex difference equations and complex dynamical systems.Nevanlinna theory has greatly promoted the study on the existence and other analytical properties of meromorphic solutions of complex differential equations,and has become a powerful tool for the study of complex differential equations.In particular,the logarithmic derivative lemma plays a crucial role on the study of complex differential equationsIn recent years,the difference variant of Nevanlinna theory has been established in[1,2,3]by Halburd-Korhonen and Chiang-Feng,respectively.Using these theories,some mathmeticians in the world began to consider the study of difference polynomials,difference equations,and differential difference equations in complex fields,producing many fine results[4,5,6,7,8].In this paper,we will use Cartan’s second mian theorem to consider the existence,growth and form of meromorphic solutions of certain nonlinear complex differential difference equations.The structure of the paper is arranged as follows:In the firstchapter,we will mianly give a brief introduction to Nevanlinna theory and the difference variant of the Nevanlinna theory,including some definitions and classical results.In the second chapter,we consider the existence,growth of meromorphic solutions of the following nonlinear differential difference equqtion:fn(z)fk(z)+p(z)f(z+η)=H0(z)+(?)Hj(z)eωjzq,where n,m,q,k are positive integers,p(z)is a poly nimial,w1,w2,…,wm are m distinct nonzero complex numbers,and H0(z),H1(z),…,Hm(z)are entire funtions of order less than q such that H1(z)H2(z)…Hm(z)(?)0,ηis a finite complex number.We get several conclusions that generalize previous related results.Firstly,we consider the case p(z)≡ 0,and obtain the following result.Theorem2.5.Let n,m,q,k are positive integers,w1,w2,…,wm are m distinct nonzero complex numbers,and H0(z),H1(z),…,Hm(z)are entire funtions of order less than q such that H1(z)H2(z)…Hm(z)(?)0.Suppose that f is an entire function of the differential equqtion:fn(z)f(k)(z)=H0(z)+(?)Hj(z)eωjzq.Then,the following assertions hold.(ⅰ)When H0≡0,we have two possibilities:(1)m-1,f(z)=u(z)eω1zq/n+1,where T(r,u)=S(r,f).(2)λ(f)=ρ(f)=q and n<m.(ⅱ)When H0(?)0,we have λ(f)=ρ(f)=q and n≤m.Next,we consider the case p(z)(?)0,H0(z)≡0,and obtain the following result.Theorem2.8.Let n,m,q,k are positive integers and n≥2,p(z)(?)0 is a polynomial,η is a finite complex number,w1,w2,…,wm are m distinct nonzero complex numbers,and H1(z),H2(z),…,Hm(z)are entire funtions of order less than q such that H1(z)H2(z)…Hm(z)(?)0.Suppose that f is a meromorphic function of the nonlinear differential difference equqtion fn(z)f(k)(z)+p(z)f(z+η)=H1(z)eω1zq+H2(z)eω2zq+…Hm(z)eωmzq satisfying ρ2(f)<1.Then,f reduces to a transcendental entire function,and the following assertions hold.(i)m=2,f(z)=H1(z-η)/p(z-η)eω1(z-η)q=L(z)eω1zq,ω2=(n+1)ω1,where(?),L(z)=H1(z-η)/p(z-η)eω1M(z),#12(ⅱ)λ(f)=ρ(f)=q and n≤m+1.Further,we consider the case p(z)(?)0,H0(z)(?)0,and obtain the following result.Theorem2.9.Let n,m,q,k are positive integers and n≥ 2,p(z)(?)0 is a polynomial,η is a finite complex number,w1,w2,…,wm are m distinct nonzero complex numbers,and Hj(z)(0≤j≤m)are entire funtions of order less than q such that H0(z)H1(z)…Hm(z)(?)0.Suppose that f is a meromorphic function of the nonlinear differential difference equqtion fn(z)f(k)(z)+p(z)f(z+η)=H0(z)+(?)Hj(z)eωjzq satisfying ρ2(f)<1.Then,f reduces to a transcendental entire function,and we have λ(f)=ρ(f)=q and n≤m+2.In the third chapter,we summarize the whole paper and point out directions of further study.