Existence Of Symmetric Positive Solutions To Several Class Singular Three Point Boundary Value Problems

This master thesis mainly discuss Existence of symmetric positive solutions to several class singular three point boundary value problems. The tools used in this thesis are the cone theory, the cone expansion and compression fixed point theorem, operator approximation theory, fixed point index theorems, Leggett-Williams fixed point theorem, Avery-Henderson fixed point theorem. This thesis consists of four chapters.In chapter 1 we introduce the historical background of problems which will be investigated and state the main results of this thesis. In addition, we list some preliminary knowledge which is needed in this thesis.In chapter 2 we mainly discuss the existence of symmetric positive solution of a class second order three point boundary value problems for nonlinear singular ordinary differential equations, by endowing with the conditions concerning the principal eigenvalues corresponding to the relevant linear operators for the nonlinear function, by employing the cone expansion and compression fixed point theorem, operator approximation theory, fixed point index theorems, we establish some optimal criteria for the existence of one or two symmetric positive solutions which involve the principal eigenvalue of a related linear operator to boundary value problems of singular nonlinear ordinary differential equations. We also consider the existence of multiple positive solutions of boundary value problems by applying Leggett-Williams fixed point theorem. The result extend and improve some known results. Moreover, the method we use in this thesis possesses general significance for the existence of symmetric positive solutions to boundary value problems of singular nonlinear ordinary differential equations.In chapter 3 we studied the existence of triple symmetric positive solution of second-order three point boundary value problems for nonlinear singular ordinary differential equations whose nonlinear term include the first-order differential u' by employing Avery-Henderson fixed point theorem, and obtain the sufficient conditions for the existence of triple symmetric positive solutions.In chapter 4 we consider the existence of multiple positive solutions of 2m-order three point boundary value problems by applying Leggett-Williams fixed point theorem and analytic method, and obtain the existence of three symmetric positive solution of the boundary value problems for nonlinear singular ordinary differential equations. We extend the results of chapter 2 to higher ever order three point boundary value problem of singular nonlinear ordinary differential equations. In particular, our results extend and improve some relate conclusions about differential equations in much recent literatures.