Research Of The Property And Existence Of Solutions Of Complex Difference Equations

In1920s, Finland mathematician Rolf Nevanlinna introduced the charac-teristic functions of meromorphic functions of complex plane, and proved two fundamental theorems, to investigate the value distribution of meromorphic functions. The meihod of Nevanlinna is of great significance for the research of the properties of meromorphic functions and value distribution theory. Af-ter the continuous improvements by many scholars over the past tens of years. Nevanlinna theory has been highly mature. Also, it has been widely applied to many branches of mathematics, such as the uniqueness theory of mero-morphic functions, complex differential equations, several complex variables theory, minimal surface, and also the recent hot Nevanlinna difference theory.In this paper, the author mainly applied the Nevanlinna theory to lin-ear homogeneous difference equations and linear non-homogeneous difference equations to study the growth of solutions of these difference equations, and obtain some results. At the same time, we also obtain some results related to the Nevanlinna exceptional values of their solutions. The paper is divided into three chapters.In chapter1, we simply introduce the background of this paper, which some contains notations of Nevanlinna theory, basic concepts and some prima-ry results of meromorphic functions theory.In chapter2, we continue to investigate the problem of growth of solutions of linear difference equations αn(z)f(z+n)+…+a1(z)f(z+1)+α0(z)f(z)=b(z),(2.1) αn(z)f(z+n)+…+a1(z)f(z+1)+α0(z)f(z)=0,(2.2) which is based on the research of Chiang-Feng([3]), and obtain the following results.Theorem2.1: Lot an(z),...,a-l(z),a0(z) bo polynomials. Then the dif-ference equation (2.1) and (2.2) do not admit entire1solutions of finite order that satisfy σ(f)≥2and A(f)<1.Theorem2.2: Let an(z),...,a1(z).a0(z),b(z) be polynomials. Then the difference equation (2.1) and (2.2) do not have entire solutions J(z) that sat-isfy σ(f)≥2and max{λ(f), A(f/1)}<1.Theorem2.3: Let αn(z),...,a1(z),a0(z) be meromorphic functions and σ(αj)<1(j=0,1,..., n). Then the difference equation (2.2) do not have finite order meromorphic solution f(z) that satisfy σ(f)≥1and max{λ(f).λ(f/1)}<1Theorem2.4: Lot an(z),...,α1(z),α0(z),b(z)((?)0) be finite order entire functions satisfying λ(aj)<1≤σ(aj),σ(b)≤1,j=0,1,...,n. Then any meromorphic solution f(z) of (2.1) satisfies σ(f)≥1.Theorem2.5: Let f(z) be a transcendental meromorphic solution of finite order σ(f)≥1of (2.1), an{z),...,a1(z),a0(z) be entire functions that satisfy an(z)a0(z)(?)0and max{σ(α0).σ(α1),...σ(αn),σ(b)}≤λf, where λf-max{λ(f).λ(f/1)}. Then1≤σ(f)≤1+λf.We also consider solutions of linear difference equations when no dominant coefficients are provided. And we obtain the following results.Theorem2.6: Lot αj(z)=α0jzn+α1jzn-1+…+αnj(j=0,1,...,n) be nonzero polynomials of degree n that satisfy α00+α01+…+α0n≠0.Then, cither the linear difference equation ao(z)f(z+n)+α1(z)f(z+n-1)+…+αn(z)f(z)=0,(2.3) has no transcendental meromorphic solutions or any transcendental meromor-phic solutions f(z) of (2.3) satisfy σ(f)>1.Theorem2.7:Let a(z) be a nonconstant entire function of order less than one and aj(z)=α(z+j) for j=0,1,…, n. Then, either the linear difference equation (2.3) has no transcendental meromorphic solutions or any transcendental meromorphic solutions f(z) of (2.3) satisfy σ(f)>1.In the following, we investigate the deficient value problem of solutions of linear difference equations.Theorem2.8:Let aj(z)(j=0,1,..., n) be meromorphic functions that satisfy Σjn=0αj(z)(?)0,f(z)be a transcendental meromorphic solution of finite order of (2.3). Then, f(z) has no nonzero and finite Nevanlinna exceptional values.In chapter3, we continue to study the difference equations. However, we mainly focus on the nonlinear difference equations fn＋P(z)(△cf)m＝Q(z),(3.1) fn+(△cf)n=1,(3.2) fn+P(z)f(z+c)m=Q(z),(3.3) and their related forms. We obtain the following results.Theorem3.1:Let P and Q be polynomials, c is a nonzero constant. Suppose f(z) is a meromorphic solution of the nonlinear difference equation fn+P(z)f(z+c)n=Q(z).(3.4) Then: (ⅰ) Any transcendental meroinorphic solution f(z) of (3.4) satisfies a(f)>1:(ⅱ) If f{z) is transcendental and Q(z)(?)0, then λ(f)=σ(f);(ⅲ) If λ(f)<σ(f) and Q(z)(?)0, then (3.4) has no entire solutions of finite order.(ⅳ) If n≥3and Q(z)(?)0, then (3.4) has no transcendental entire solu-tions.Theorem3.2: Let f(z) he a transcendental meromorphic solution of (3.2). If f(z) satisfies any one of the following conditions(ⅰ) f(z) has only finitely many poles;(ⅱ) f(z) satisfies A(f/1)<λ(f). Then σ(f)≤1.Theorem3.3: Let f(z) be a transcendental meromorphic solution of (3.1), let P(z) and Q(z)((?)0) be polynomials, then λ(f)-σ(f). Theorem3.4: The nonlinear difference; equation f2+P(z)(△cf)2-Q(z)(3.5) has no transcendental entire solution of finite order if N(r,f)>T(r,△cf), where P(z) and Q(z)((?)0) arc polynomialsTheorem3.5: The nonlinear difference: equation f2+P(z)△cf=Q(z) has no entire solution of infinite order if N(r,△cf/1)<T(r,f),where P(z) and Q(z)((?)0) arc polynomials.