Positive Solutions Of Three-point Boundary Value Problems For Second Order Differential Equations (Systems)

The study of boundary value problems for nonlinear ordinary differential equations has interested people for a long time and the research in this field is still very active. Recently , the existence of positive solutions of boundary value problems for second-order three-point nonlinear ordinary differential equations attracts close attention.This thesis consists of three 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 in needed in this thesis.In chapter 2, using the fixed point theorem of cone expansion and compression of norm type, we provide for the existence of at least one positive solution for the semipositone second-order three-point boundary value problem u″(t) +λf(t, u(t)) = 0, 0 < t < 1,u(0) =αu(η), u(1) =βu(η), where 0 <η< 1, 0 <β≤a < 1 andλ> 0 was a parameter.In chapter 3, we investigate the following system of nonlinear second-order three-point boundary value problems -u″= f(t, v),-v″= g(t, u), t∈(0,1), u(0) =αu(η), u(1) =βu(η),v(0) =αv(η), v(1) =βv(η), where 0 <η< 1 and 0 <β≤α< 1. First, Green's function for associated liner boundary value problem is constructed; and then, several useful properties of the Green's function are obtained; Finally, some existence and multiplicity criteria of positive solutions are established by using the well-known fixed point theorems of cone expansion and compression.