三阶无穷多点边值问题正解的存在性

Positive solutions of third-order ∞-point boundary value problems

本文研究了如下三阶微分方程的无穷多点边值问题{u'''+λa(t)f(u)=0,t∈(0,1),u(0)=βu′(0),u(1)=∑∞i=α1u(ξi),u′(1)=0正解的存在性,其中参数λ>0,ξi∈(0,1),αi∈(0,∞],且满足∑∞αi i=1> 1,0<∞∑αiξi(2-ξi)<1.a(t)∈C([0,1],[0,∞)),f∈C([0,∞),[0,∞)),运用锥拉伸与压缩不动点定理,在f满足超线性和次线性的情况下,本文不仅得到了该边值问题正解的存在性,同时还得到了使得问题有解的特征值λ的取值范围.

In this paper,we study the existence of positive solutions to the following third-order oo -point boundary value problem {u＇＇＇＋λa（t）f（u）=0,t∈（0,1）,u（0）=βu′（0）,u（1）=∑i=α^∞1u（ξi）,u′（1）=0 where λ〉0 is a parameter,ξi∈（0,1）,αi∈（0,∞],satisfying ∑i=1^∞αi 〉 1,0〈∑i=1^∞αiξi（2-ξi）〈1.a（t）∈C（[0,1],[0,∞））,f∈C（[0,∞）,[0,∞））.By using Krasnoselskii＇s fixed point theorem in cones, we obtain the existence of the positive solution and the eigenvalue intervals on which there exists a positive solution if f is either superlinear or sublinear.