| The uniqueness theory of meromorphic functions originates from some works in R. Nevanlinna. The Nevanlinna theory is important not only because it is the basis of modern meormorphic function theory, but also because it has quite an effect on the development of mathematical branches, and on the interaction among them. Especially, the Nevanlinna theory supplies a very powerful tool to the research of complex differential equations. In 1929, R.Nevanlinna studied the conditions with which a meromorphic function can be determined and obtained two celebrated uniqueness theorems for meromorphic functions, which are usully called Nevanlinna's four-value theorem and Nevanlinna's five-value theorem. This launched the investigation of uniqueness theory of meromorphic functions and in particular the shared values of meromorphic functions.In chapter one, we briefly introduce some main concepts, fundamental results and usual notations concerned with this thesis in the value distribution theory of meromorphic functions.In chapter two, we study on the transcendental entire solutions of a certain differential equation and fixed-points and obtain some result.Theorem 2.1 Let f be an entire function, n ((?) 2) be a positive integer,be an entere function,then the equation F'- z= e(?) (F- z)has transcendental entire solution f ,and f assumes the form: f = cen, Where c is a non-zero constant. Corollary 2.1 Let f be a non-constant entire function,n be a positive integer,if fn( (?) 2) fn and ( f n)(?) share z CM,then the conclusion of Theorem 2.1 is valid. Theorem 2.2 Let f be a transcendental entire function,n ,k be positive integers,and n (?) k+2,if fn and ( fn )k share zIM,then wzf = ce(?),where and are non-zero constants,and wk =1In chapter three, we study the growth of a certain complex differential equation.In chapter four, we study fixed-points and uniqueness of meromorphic functions and obtain some results.Theorem 4.1 Let f and g e two nonconstant meromorphic functions,and let n, k be two positive integers with n>3k+10.( fn)k and (gn)k share ) CM, P(?)is a polynomial with degree m ,f and (?)share P(?) CM,P(?) where q(?)=(?),c1,c2 and c are constants satisfying ( m=1) n2 (c1 c2 )nc2+=- ,or f=tg for a constant t such t hat tn=1.In chapter five, we consider the uniqueness of meromorphic functions or entire functions and their differential polynomials and obtain some results.Theorem 5.1 Let n and m be two positive integers,l =min{2, m},Let f and g be two non-constant meromorphic functions such that (?).If Ek ( S m , f n ( f - 1) f ')= Ek ( S m, g n( g-1) g') and one of the following conditions is satisfied: ( a ) k (?) 3 and n>(?)+3 ,(b)k= 2 and n>(?)+3,(c)k=1 and n>(?)+2, Then f=g. |
