Researches On The Growth Of Solutions Of Complex Differential Equations And Some Problems Of Normal Family

The value distribution theory founded by Rolf Ncvanlinna in the 1920's. Usually, we called Nevanlinna theory in honor of him. Nevanlinna theory can be seen the most important achievements in the preceding century to understand the properties of meromorphic functions. This theory is composed of two main theorems, which are called Nevanlinna's first and second main theorems that had been significant breakthroughs in the development of the classical function theory, since the later generalizes and extends the Picard's first theorem greatly, and hence it denoted the beginning of the theory of meromorphic functions. Moreover, Ncvanlinna theory and its extensive has numerous applications in some fields of mathematics, for example, potential theory, complex differential and difference equations, normal family, several complex variables and so on.In the study of complex differential equations, Nevanlinna theory has been applied to get insight into the properties of their solutions. The first such appli-cations were made, to the best of our knowledge, by F. Ncvanlinna in 1929. After that, in connection with Nevanlinna theory, the global complex differential equa-tions became more popular. Due to the contribution of many mathematicians from different countries, many results about linear complex differential equations and non-linear algebraic differential equations, such as Riccati equations, some of the Painleve equations, and the Schwarzian differential equations, have been gotten. Recent years, another popular studying field is the value distribution of the solutions of the difference equations and q-difference equations. Many results in the complex differential equations have the similar ones in difference equations and q-difference equations. In 1907, P. Montel introduced the concept of Normal Family. It plays an important role in complex dynamics. A family of meromorphic functions is called normal if every sequence in the family has a subsequence which converges (locally uniformly with respect to the spherical metric). The studying aims is to find normal family. One guiding principal in the study is so-called Bloch's Principle. In deed, Bloch's Principle is not true in general. But it's still an important guiding principle in the theory of normal family. Recent years, many papers focus on searching normal family by the famous Zaleman-Pang Lemma.The present thesis involves some results of the author on the growth hyper-order of solutions of certain linear complex differential equations, the growth hyper-order and the zeros of the solutions of second order linear complex dif-ferential equations, estimation of the growth order of meromorphic solutions of some q-difference equations and a normal family criterion about meromorphic functions and their differential polynomials sharing sets. It consists of five parts and the matters are explained as below.In Chapter 1, we introduce the general background of Nevanlinna Theory, the development of the studies in complex differential equations, Wiman-Valiron theory which is an indispensable device while considering the value distribution theory of entire solutions of complex differential equations and the development of the Normal Family Theory.In Chapter 2, we investigate the hyper-order of the solutions of the equation f(k)-eQf=a(1-eQ),where Q is an entire function, being a polynomial or not and a is an entire function, the order of which could be larger than 1. We improve some results by J. Wang and X. M. Li. In fact, we obtained the following results.Theorem 0.1.If f is a nonconstant solution of the differential equation f(k)-a=(f-a)eQ, where a is an entire function, Q is a polynomial with deg Q<σ(a)<∞and k is a positive integer, thenμ2(f)=σ2(f)=degQ. Theorem 0.2. Let Q be a transcendental entire function withσ(Q)<1/2, a be an entire function of finite order and k be a positive integer. If f is a solution of the equation thenσ2(f)=∞.In Chapter 3, we study the growth hyper-order of the solutions of second order linear complex differential equations, and get the upper bound of the hyper-order of the solutions. Also, we propose a question how about the lower bound of it. In fact, we getTheorem 0.3. Suppose that A0(?)0, A1(?)0, H are entire functions of order less than one, and the complex constants a, b satisfy ab≠0 and a≠b. Then the hyper-order of every nontrivial solution f of equation f"+A1(z)eazf'+A0(z)ebzf= H(z) is not greater than one.Theorem 0.4. Suppose that A0(?)0,A1(?)0,D0,D1,H are entire functions of order less than one,and the complex constants a, b satisfy ab≠0 and b/a<0. Then the hyper-order of every nontrivial solution f of equation f"+(A1(z)eaz+D1(z))f'+(A0(z)ebz+D0(z))f=H(z) is not greater than one.In Chapter 4, we estimate the growth order of the meromorphic solutions of some complex q-difference equations and investigate the convergence exponents of fixed points and zeros of the transcendental solutions of the second order q-difference equation. We also obtain a theorem about the q-difference equation mixing with difference.In the final Chapter 5, we investigate normal family criterions about mero-morphic functions and their differential polynomials sharing sets. In fact, we obtain the following theorems.Theorem 0.5. Let F be a family of meromorphic functions in a domain D, let n(≥2),k be two positive integers,and let S={a1,a2,...,an},where a1, a2,...,an are distinct finite complex numbers.If for each f(?)F,all zeros of f have multiplicity at least k+1,and f and G(f) share the set S in D, where G(f)= P(f(k))+H(f) be a differential polynomial of f satisfying q≥γH,andΓ/γ|H