| In this thesis, we show some works about the problems of the uniqueness of transcendental meromorphic functions of class F and the solutions of one differential equation with constant coefficients over non-Archimedean Fields κ, which I studied under the guidance of my supervisor Professor P.C.Hu. The main contents are as follow:In Chapter1,we introduce the research background and describe the basic Nevanlinna theory over non-Archimedean field κ.In Chapter2, we consider the problem of uniqueness of meromorphic functions of class F over non-Archimedean field κ. For f∈κ, we get the three simple value sets theorem by Yang’s methods (c.f [20]):Theorem1Let aj(j=1,2,3) and bj(j=1,2,3) be two groups of finite and non-zero numbers in κ, and ai≠aj,bi≠bj,(i≠j). Then any meromor-phic function f(z)∈F can be completely determined by E1)(aj,f)(j=1,2,3) or E1)(bj,f(k))(j=1,2,3).Next, we apply a corollary of the second main theorem and get the one simple value-set theorem:Theorem2Let f,g∈F, and a, b be two finite non-zero in κ and k∈N+, then(i) If E1}(a,f)=E1)(a,g), then f=g or f·g≡a2.(ii) If E1)(a,f(k))=E1)(a,g(k)), then f≡g or f(k)·g(k)≡a2.In Chapter3, we consider the problem of the form of admissible solutions of a special form of differential equations with constant coefficients. We improve the theorem3.4by foxing the form of differential equation and get theorem3:Theorem3Take n, k∈z+,aj∈k(0≤j≤k) with ak≠0. If the following differential equation has a non-constant meromorphic solution ω, then|n-k|is a factor of2n, and the equation assume the following form where c satisfies... |
