Positive Solution Of N Order M-point Boundary Value Problem And Four Order Singular Boundary Value Problem

Boundary value problem for nonlinear ordinary differential equation has a long history and has extensive application in natural science and engineering technique.In recent years, the existence of positive solutions for higher order multi-point boundary value problem and singular boundary value problem attracts close attention.This paper studies that two kinds of boundary value problem.Firstly,following n order m-point boundary value problem is considered,assumptions are given following: (H1)βi,κj>0(i=0,1,...,n-2;j=0,1,...,m-2)；(H2)0=ξ0<ξ1<ξ2<…<ξm-2<ξm-1=1,1-(?)>0；(H3)f:[0,1]×Rn→R is continuous. In addition,there are constants a,c,d satisfying 00,a(t)is allowed to be singular at t=0 or t=1,and satisfies (?)s3(?)a(t)dt=0,f1,f2∈C([0,+∞)×[0,+∞)), f1≥0,f2≥0,for y∈[0,+∞),f1(x,y),f2(x,y)is uniform continuous in[α,β](?)[0,+∞). The existence results of classical positive solution are given by means of variational method and mountain pass lemma.