Research Of Solutions Of A Kind Of Differential And Difference Equations In Complex Plane

In 1920s. Finland mathematician R. Nevanlinna introduced the character-istic functions of meromorphic functions in the complex plane, and obtained the Nevanlinna theory〔[23]〕. Nevanlinna theory is one of the most important mathematics achievements in the 20th century, which played basic roles for re-searches of meromorphic functions. Nevanlinna theory keep on developing and has been widely applied in many branches of complex analysis, for example, theory of functions of a complex variable, uniqueness theorems on meromor-phic functions, complex differential and difference equations, several complex variables, minimal surface, and so on.Under the guidance of professor Lianzhong Yang, the author researched the existence and the growth of solutions of some differential and difference equations with the Nevanlinna theory. The paper consists of three chapters.In chapterl, we introduce the background of this paper, which contains notations of Nevanlinna theory, basic concepts and results of meromorphic functions.In chapter2, we investigate the growth of solutions and the existence of subnormal solutions of a class of higher order linear differential equations, which extend the results of Chen-Shon([4]), Qi-Yang([24]) and Liu-Yang([19]), and obtain the following main results:Let Pj(z),Qj{z) (j=0.1,..., n - 1) be polynomials in z, and A(z)be a transcendental entire function. If degP0> degPj or degQ0 >degQj, j=1,...,n-1, then the differential equation: f(n)+[Pn-1(eA(z))+Qn-1(e-A(z)]f(n-1)+···+[P0(eA(z))+Q0(e-A(z)]f=0 has no nontrivial subnormal solutions, and every nontrivial solution satisfiesσ2(f)=∞.In chapter3, we investigate the growth of solutions of a class of linear difference equation without dominated coefficients in growth.and obtain some results which supplement some results of Chiang and Feng([6]),and obtain the following main results:1.Let Pj(z)and Qj(z)(j=0.1,..,n-1)be polynomials that satisfy deg(P0)>deg(Pj) or deg(Q0)>deg(Qj),j=1,...,n-1, then, each nontrivial entire solution f(z) of finite order of the difference equa-tion: f(z+n)+(?){Pj(ez)+Qj(e-z)}f(z+j)=0 satisfiesσ(f) =λ(f-α)≥2,and so f(z)assumes every nonzero complex value a∈C infinitely often.2.Let Pj(z) and Qj(z)(j=0,1,...,n-1)be polynomials in z and A(z)be a transcendental entire function.If deg(P0)>deg(Pj) or deg(Q0)>deg(Qj),j=1,..,n-1, then every solution of the difference equation f(z+n)+(?){Pj(eA(z)+Qj(e-A(z))}f(z+j)=0 is of infinite order andσ2(f(z))≥σ(A(z)).