Positive Solutions Of Nonlinear Ordinary Differential Equation Neumann Problems And Its Related Four-points Boundary Value Problems

In this paper, by using the Krasnoselskii fixed point theorem, Krein-Rutman theorem and Rabinowitz’s global bifurcation theorem, we study the existence and multiplicity of positive solutions for some second-order nonlinear differential equa-tions with singularity. We describe them in detail as follows.1.By using the Krasnoselskii fixed point theorem, we establish the existence and multiplicity of positive solutions for second-order four-point boundary value problem where m e (0, π/2), e∈C[0,1], f∈C([0,1]×(0,∞),[0,∞)), and lim f(t,u)+∞uniformly for t∈[0,1]. These results extend the corresponding results of Jifeng Chu [J. Math. Anal. Appl.,2008] and Qingliu Yao [Appl. Math. Lett.,2012],2.By using the Krein-Rutman theorem and Rabinowitz’s global bifurcation theorem, we study the existence of positive solutions for four-point boundary value problems where A>0is a parameter, m∈(0,π/2|) is a constant, e∈(0,1/2), g∈C([0,1],(0,+∞)),f∈C([0,∞),[0,∞)), and f(u)u>0for all u∈(0,∞), we establish the existence of posi-tive solutions under the assumption f0:=limf(u)/u∈[0,∞],f∞=limf(u)∈[0,∞].3.We consider the existence of positive solutions for Neumann boundary value problems where A>0is a parameter, e∈C[0,l], h∈C([0,1],[0,∞)) and h(·)(?)0on any subinterval of [0,1], f∈C([0,1]×(0,∞),(0,∞)), and limf(t,u)=+∞uniformly for t∈[0,1],p,g∈L1[0,1]. By using the Krein-Rutman theorem and Rabinowitz’s global bifurcation theorem, we establish the existence of positive solutions under the assumption f∞∈(0,∞),f∞=0and f∞=∞, These results improve extend the corresponding results of Jifeng Chu [J. Math. Anal. Appl.,2008] and Qingliu Yac [Appl. Math. Lett.,2012].