Research On Uniqueness Of Functions And Properties Of Solutions Of Complex Differential Equations

The theory of value distribution provides a foundation for the study of the uniqueness of functions related to shared values and the properties of nontrivial solutions of complex differen-tial equations.Firstly,based on the basic results of the theory of Nevanlinna value distribution and the concepts of shared value,we obtain the conclusion and proof of the identity of the fi-nite order transcendental entire function and its differential polynomials when they share small functions.We investigate the uniqueness of nonconstant entire function and its difference oper-ators when they share(a,1)*and(b,1)*.Meanwhile,we improve the conclusion to A_c~nf(z)by using the properties of the difference operators of entire functions.According to the properties of weakly weighted sharing,we consider the uniqueness of the weakly weighted sharing of the zero order of nonconstant meromorphic function and its q differential polynomial.We give the case of sharing(1,m).According to the three different ranges of m,we get the conclusion of uniqueness.According to the properties of q difference operators,we extend the conclusion to?_qf(z).Secondly,we consider the growth of nontrivial solutions of a kind of linear differen-tial equation of order n and obtain that the lower order is infinite.Then we study the measure of its limit direction set of the Julia set.We study the properties of nontrivial solutions of a special linear differential equation whose coefficients A(z),B(z)are the solutions of another second order differential equation.When we limit the number of accumulation rays of the zero sequence of the coefficient A(z),we obtain the hyper-order of the solutions of the equation by using the properties of the solutions of the second order differential equation.We analyze the measure of the limit direction set of the solution whose lower order is infinite,and we get the lower bound of the measure.We further discuss the case that the coefficients A(z)and B(z)are solutions of two different second order linear differential equations.When the coefficients of two second order linear differential equations have different degrees,we obtain that the lower order is infinite and give the proof of the lower bound of the measure of limit direction set.At the same time we give the measure of limit direction set of the solution whose lower is infinite when the degrees are the same.