The Fixed Point Theorems In Cone Metric Space And Existence Of Sign-changing Solutions For Third-order M-Point Boundary Value Problem

In recent decades,with the development of nonlinear analysis,the research on the existence of solutions of nonlinear differential equations and fixed point prob-lem of nonlinear operators has become more and more important.With the rapid progress of research in various fields of science,technology and engineering,a large number of practical problems can often be attributed to nonlinear fixed point prob-lems or nonlinear differential equation problems.This thesis studies two problems,one is to use the semi-order method to obtain several fixed point theorem of the mixed monotone operator in the cone metric space,the other is to use topological degree theory and cone theory to obtain the existence of sign-changing solutions of the m-point boundary value problem of third-order ordinary differential equations.The full thesis is divided into four chapters.The first chapter is the introduction.It introduces the research background and development status of the fixed point problem in nonlinear analysis and the boundary value problem of differential equations.At the same time,it lists some achievements have been made by some scholars in these two fields,and finally points out the main content of this thesis and the main theories and methods used.The second chapter introduces some basic definitions and theorems needed in this thesis.The third chapter studies the fixed point theorems of mixed monotone operators in cone metric space.Several new fixed point theorems are obtained when the nonlinear operator satisfies certain conditions.The forth chapter studies the existence of sign-changing solutions for third-order m-Point boundary value problem(?)where 0<?1<?2<…<?m-2<1,?i?[0,?),i=1,2,…,m-2,0<(?)?i<1,f ? C(R,R).By using cone theory and topological degree theory,the existence of sign-changing solutions for m-point boundary value problem is obtained.