The Complex Oscillation Of Solutions Of Differential Equations And The Uniqueness Theorems Of Meromorphic Functions

In this thesis, we mainly investigate the complex oscillation properties of solutions of differential equations and the uniqueness problems of meromorphic functions sharing a small function. It consists of four chapters.In chapter 1, we briefly introduce some basic results of the Nevanlinna’s the-ory, which are the powerful tools in the research of the complex oscillation theory of differential equations and the uniqueness theory of meromorphic functions.In chapter 2, we investigate the distribution of zeros of solutions for certain higher order homogeneous linear differential equations f(κ)＋Aκ-1 f(κ-1)＋…＋Aof＝0 with entire coefficients. By using the fundamental theorems of Nevanlinna’s value distribution theory, we obtain that the exponent of convergence of zeros of its every transcendental solutions is infinite when Aκ-1 is the dominant coefficient.In chapter 3, we investigate the growth of solutions of higher order linear differential equations f (κ) + Aκ-1f (κ-1)＋…十Aof=0, whereAj(z)(0≤j≤ κ - 1) are meromorphic functions. It is shown that every nonzero meromorphic solution of such equation has infinite order, provided that A0(z) has a deficient value ∞ and Aj(z)(0≤ j≤ k - 1) satisfies certain conditions. The lower bound of hyper order of meromorphic solutions of such equation is also estimated.In chapter 4, we study the uniqueness problems of meromorphic functions(or entire functions) sharing a polynomial weakly, and obtain two results which extend those results obtained by Li and Yi, Chen and Zhang.