The Existence Of Solutions Of A Type Of Difference Equations And The Properties Of Difference Polynomials

In this paper,Using the value distribution theory of Nevanlinna and its difference analogue,we investigate the existence of meromorphic solutions of some type of nonlinear difference equations,the characteristic function and the value distribution of certain linear difference polynomials.The thesis is divided into four chapters.In chapter 1,we briefly introduce the research background and the basic theory which are related to this thesis.Some relevant definitions,properties and standard symbols are also introduced.In chapter 2,The properties of meromorphic solutions of nonlinear difference equationsare inverstigated,where P_d?z?f?is a difference polynomial in f of degree d,P₁,P₂are non-vanishing small meromorphic functions of f,and ?₁?z?,?₂?z? are non-constant entire function with order less than 1.We obtain the determination conditions which guarantee the existence of meromorphic solutions of the above equations,and also give the expressions of these solutions.Our results improve and generalize some previous results.In chapter 3,we inverstigate the characteristic functions of linear difference polynomialand its more general formswheren?1 is an integer,C₀,···,C₉)are different constants,a₀,···,a₉)are non-vanishing small meromorphic functions off.We obtain precise asymptotic relations between the characteristic functions of the above difference polynomials andf.In chapter 4,basing on the chapter three,and using the characteristic function of linear difference polynomial H?z,f?,We study the distribution of small function-points and poles of H?z,f?.Under some conditions,we obtain the estimation on the deficiency of small functions with respect to linear difference polynomial H?z,f?.Some examples are given to show our results are best possible.