The Properties Of Solutions Of Complex Differential Equations And The Uniqueness Of Meromorphic Functions Sharing Values With Their Q-shifts

In this thesis,using the Nevanlinna's theory,we mainly investigate the growth and the value distribution of solutions to some classes of linear differential equa-tions,and the uniqueness of meromorphic functions sharing values with their q-shifts.It consists of four chapters.In chapter 1,we give a brief introduction of the Nevanlinna's theory and the research background of this thesis.We also give some relevant notations and definitions used in this thesis.In chapter 2,we investigate the growth of solutions to higher order linear differential equation f(k)+Ak-1f(k-1)+…+A0f = 0,where Aj(j = 0,…,k-1)are entire functions.When there exists some coefficient A,being a nonzero solution of ?"+ P(z)? = 0,where P(z)is a non-constant polynomial,we give some conditions to guarantee that every nontrivial solution of the above equation has infinite order.We also obtain some estimation on hyper-order of its solutions.When the polynomial P(z)takes some special expression,A8 may be an Airy function,Weber-Hermite function or exponential function.In chapter 3,we investigate the relations between the small functions ? and the solutions,their 1st,2nd derivatives to the second order linear differential equation f" A1(z)f' + A0(z)f = 0,where A1(z),A0(z)are entire functions with different orders.Under certain conditions,we obtain that the exponent of convergence of ?-points of the solutions and their 1st,2nd derivatives are all infinite.In chapter 4,we investigate the uniqueness problems on meromorphic func-tion f(z)of zero order sharing values with their q-shift f(qz+c).It is shown that if f(z)and f(qz+c)share one values CM and IM respectively,or share four values partially,then they axe identical under an appropriate deficiency assumption.