In this thesis,we mainly study two aspects of meromorphic function theory,based on the value distribution theory established by R.Nevanlinna.On the one hand,we study the uniqueness theorem of non-constant entire functions.Firstly,we study a class of special entire function of small order,discuss the nonobvious sufficient conditions for the non-constant entire functio n f?z? to be an even function,the following conclusion is obtained.?1?Let f?z?be an non-constant entire function,with order ?f<1,if f?z? and f?-z?have two finite IM sharing values,than f?z?is even function,i.e.f?z??f?-z?.In general case,we obtained the following result.?2?Let f?z?be an non-constant entire function,? be a real number.if f?z? and f(ei?z) have two finite IM sharing values,then there exists m?N+ such that (ei?)m=1.Secondly,we continue to study the uniqueness of non-constant entire functions with small order,The following result is established.?3?Let f?z? and g?z? be two non-constant entire functions,and satisfy?f<3/4,and let p1?z? and p2?z? are two distinct polynomials,if p1?z? and p2?z? are IM sharing functions of f?z? and g?z?,and f?z?-g?z? has infinitely many zeros with multiplicity,then f?z?? g?z?.On the other hand,in this paper,we discusses the existence of meromorphic function solutions for Fermat-type functional equation with four unknow functions,and the following conclusion is obtained.?4?If non-constant meromorphic function f?z?,g?z?,h?z?,w?z? at most have one simple common pole,and min{?f,?g,?h,?w}<1/2,then the functions of f?z?,g?z?,h?z?,w?z? dose not satisfy the functional equation f15?z?+ g15?z?+ h15?z?+w15?z?=1. |