Research Of Positive Solutions For Several Even Order Multi-point Boundary Value Problems Of Ordinary Differential Equations

In this paper, we mainly research on the positive solutions for several even order multi-point boundary value problems of ordinary differential equations by using the Legget-Williams theorem, the fixed point index theorem and the related inequality technique on cone. The whole paper includes five chapters, and my main work is concentrated in the chapter2,3,4.In the first chapter, we outline the historical background, research status of the problems and some basic knowledge which will be used in this paper.In the second chapter, we discuss the positive solutions for the boundary value problems of ordinary equation systems with p-Laplacian operatorby using fixed points index theorem, The existence theorems are given when f and g with the condition of superlinear or sublinear. Theφp(s) is p-Laplacian operator,φp(s)=|s|P-2s,p>1, a,>0, βi>0,0<ε1,<ε2<…<εn,<1/2,εi+ηi=1, i=l,2,...n.In the third chapter, we will study the existence of symmetric positive solutions for a four order three point boundary value problemby means of the Legget-Williams theorem and inequality technique, where α,β∈(0,1),η∈(0,l/2],h(t)L1(0,1)is nonnegative and symmetric on(0,1), f:[0,1]×(0,∞)â†'[0,+∞)is continuous and f(·,u)is symmetric on[0,1]for all u∈[0,∞).In the fourth chapter,we will study the even order four-point boundary value problemswhere0<αi<1/ξ,0<ξ<η<1,αiβiη-αiβiξ+αi+βi>0,i=1,2…m, g:(0,1)â†')[0,+∞)is continuous on(0,1),and may be singular at t=0or t=1, f:[0,1]×(0,∞)â†'[0,+∞) is continuous.by means of the Legget-Williams theorem and inequality techniques. Sufficient conditions are established to guarantee the existence of triple positive or arbitrary odd solutions of the boundary value problem.The fifth chapter puts an end to this whole paper and expounds the disadvantage of my paper and the future work experience and ideas.