Types Of Ordinary Differential Equations Of A Multi-point Boundary Value Problem For Existence Of Positive Solutions,

This thesis is devoted to mainly the existence of positive solutions and multiple solutions for multiple points boundary value problem of ordinary differential equations.It consists of six chapters.In Chapter one, the historical background and current conditions are introduced and summarized for boundary value problems of ordinary differential equations.In Chapter two,a kind of three symmetric positive solutions for p-Laplacian differential equations with integral boundary conditions is studiedwhereφ_p(s)=|s|^p-2s,1/p+1/q=1,p＞1,q＞1,φ_p^-1=φ_q;a(t)=a(1-t),it doesn't holdthat a(t)∈C([0,1],[0,+∞)) is always zero on any subinterval of interval [0,1], f is continuous from [0,1]×[0,+∞)×R to [0,+∞), moreover when t∈[0,1], f(t,u,v) is equivalent to f(1-t,u,-v). g∈C([0,+∞),[0,+∞)), (?) is a Riemann- Stieltjes integral.Under some conditions,The fixed point theorem due to Bai and Ge is applied to induce the conclusion that the boundary value problem (1) - (2) has at least three differentpositive symmetric solutions.In Chapter three, a kind of multiple points boundary value problem with p-Laplacian operator is discussedwhereφ_p(s)=|s|^p-2s,p＞1,0＜ξ₁＜ξ₂＜…＜ξ_m-2＜η₁＜η₂＜…＜η_m-2＜1,ξ_i+η_i= 1,α_i≥0,β_i＞0,a(t)∈C([0,1],[0,+∞)),a(t)=a(1-t),α(t) is not identically zero on any subinterval of interval [0,1],moreover 0＜(?)＜1.We obtain theconclusion that under suitable conditions,boundary value problem (3)-(4) has at least three symmetric positive solutions by using the fixed point theorem to Avery - Peterson.In Chapter four, we are devoted to a kind of forth order four points b-oundary value problem with p - Laplacianwhere p＞1,α＞0,0＜ξ＜η＜1,φ_p(s)=|s|^p-2s,ξ+η=1.This chapter mainly applysthe monotone iterative method,we get sufficient conditions of the existenceof approxiam solutions of boundary value problem (5) - (7) - (8) or (6)-(7)-(8).In Chapter five,we research the existence of positive solution for a kind of singular higer-order multiple points boundary value problem with a sign-changing nonlinear termwhere f∈C([0,1]×[0,+∞)×R,(-∞,+∞)),f is singular at t = 0,1 or u=1,α_i∈[0,+∞) , (?)＞0,0＜ξ₁＜ξ₂＜…＜ξ_m-2＜1. Using the fixed point indextheorem,we obtain the conclusion that boundary value problem (9) has at least a positive solution under suitable conditions.In Chapter six,we study the existence of a positive solution for the following second-order multiple points singular boundary value problems on the half-linewhere 0＜(?)＜1,η∈(0,+∞),q(t)∈C(R₀⁺,R) q(t) is infinite at t = 0,and q(t)＞0,t∈R₀⁺;moreover f∈C([0,+∞)×[0,+∞)×R,[0,+∞)). Combing the low- er and upper solutions methods and Leray - Schauder fixed point theorem, we get sufficient conditions are obtained for the existence of a positive solution.