Existence Of Solutions Of Second Order Singular Differential(Difference) Systems

In this paper,we mainly study the existence of solutions for three kinds of second order singular differential(difference)systems.The proofs are based on topological degree theory,fixed point theorem and Leray-Schauder alternative principle.At the same time,we also discuss the different roles of weak singularities and strong singularities in the existence of solutions for singular differential(difference)systems.The main contents of this paper are divided into the following five chapters:In the first chapter,we introduce the research background,significance and current situation of singular differential equations.In the second chapter,based on topological degree theory and Schauder's fixed point theorem,we study the existence of periodic orbits of radially symmetric systems with a repulsive singularity.x"+a(t)x=(f(t,|x|)+e(t))x/|x|,x?R2\{0}.In the third chapter,by using a fixed point theorem in cones and Leray-Schauder alternative principle,we study the existence and multiplicity of non-collision periodic solutions for the n-dimensional nonlinear system-x"+A(t)x'+B(t)x=F(t,x).In the fourth chapter,by applying Leray-Schauder alternative principle,we study the existence positive solutions for the following nonlinear difference systems-?[p(n-1)?x(n-1)]+q(n)x(n)=f(n,x(n))+e(n).In the fifth chapter,we summarize the content of this paper,and prospect the future research direction.