Study On The Properties Of Meromorphic Solutions Of Special Equations

At the beginning of the 20th century,the famous mathematician Nevanlinna introduced the characteristic function and established Nevanlinna's first fundamental theorem and Nevanlinna's second fundamental theorem.Then he established the value distribution theory.This theory is known as the most beautiful branch of mathematics in the 20th century.Using value distribution theory to study the properties of meromorphic solutions of special equations is a hot spot.There are many special equations to study.Tumura-Clunie type nonlinear differential equation is one of them.Many scholars researched Tumura-Clunie type nonlinear differential equations and made many significant achievements[15-27].In addition,some progress has been made in the study of meromorphic solutions of nonlinear complex differential equations similar to Tumura-Clunie type[29].In this paper,I study the properties of meromorphic solutions of two kinds of nonlinear differential equations.The main contents are as follows:In Chapter 1,the theory of value distribution is briefly introduced,and some important definitions and theorems in this paper are summarized.In Chapter 2,we study the properties of meromorphic solutions for a class of Tumura-Clunie type equations.Some research results of Janne Heittckangas,Zinela-abidine Latreuch,Jun Wang and Mohamed Amine Zemirni[27]are extended from h-(z)satisfying second order differential equation to h(z)satisfying m order differential equation.M is a natural number greater than or equal to two.The main results are as follows:Theorem 2.1.3 Let f be the transcendental meromorphic solution of the following equation:fn(z)+P(z.f)=h(z)P(z,f)is a differential polynomial off and its coefficient is a small function of f,?P?n-1.And h(z)is the meromorphic solution of the following equation:h(m)+rm-1(z)h(m-1)+…+r1(z)h(1)+r0(z)h=rm(z)Meromorphic function ri have finite zeros,finite poles,and T(r,ri)=S(r,f),i=0,1,…,m.n?m?2.Then one of the following conclusions must be true:(1)Iffhas only finite zeros and the order of f is finite,then f(z)=q(Z)e?(z)where q(z)is a rational function,a is a non-constant polynomial,and T(r,h)=nT(r,f)+S(r,f)(2)If n?j+1 and ?p ?n-j for some integer j?1,then jT(r,f)?jN(r,f)+mN(r,f)+mN(r,1/f)+S(r,f)In Chapter 3,we study the properties of meromorphic solutions of another kind of nonlinear complex differential equation similar to Tumura-Clunie type.The main conclusion is as follows:Theorem 3.1.1 Let f be the transcendental meromorphic solution of the following equation;fnfl1fl2…flq+P(z,f))=h(Z)l1,l2,…,lq are positive integers and satisfy l1?l2?…?lq,q?1.P(z,f)is a differential polynomial of f and its coefficient is a small function of f,?p?n.And h(z)is the meromorphic solution of the following equation:h(m)+rm-1(z)h(m-1)+…+r1(z)h(1)+r0(z)h=rm(z)Meromorphic function ri have finite zeros,finite poles,and T(r,ri)=S(r,f),i=0,1,…,m.n?m?2.Then one of the following conclusions must be true:(1)If f has only finite zeros and the order of f is finite,then f(z)=?(z)e?(z),where ?(z)is a rational function,? is a non-constant polynomial,and T(r,h)=(n+q)T(r,f)+S(r,f)Furthermore,if ri is an entire function,i=0,1,…,m,then ?(z)is a constant.(2)If n?j+1 and ?p?n-j for some integer j?1,then(j+q)T(r,f)?(j+q)N(r,f)+(m+l1+l2+…+lq)N(r,f)+(m+l1+l2+…+lq)N(r,1/f)+s(r,f)...