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

摘要:

本文研究了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) =∑αiu(ξi), u'(1) = 0正解的存在性，其中 λ > 0 是一个参数， ξi∈ (0,1), αi∈ [0,+∞], 且满足∑αi>1,0<∑αiξi(2−ξi) < 1. a(t) ∈ C([0,1],[0,∞)), f ∈ C([0,∞),[0,∞)).我们运用锥拉伸与压缩不动点定理，在f满足超线性或次线性的情况下，不仅得到了该边值问题的正解，同时还得到了使得该问题有解的特征值 λ的取值范围。

Abstract:

In this paper, we study the existence of positive solutions to the third-order ∞-point boundary value problem u'''+ λa(t)f(u) = 0, t ∈ (0,1),u(0) = βu'(0), u(1) =∑αiu(ξi), u'(1) = 0,where λ > 0 is a parameter, ξi∈ (0,1), αi∈ [0,+∞], and satisfy ∑αi>1,0<∑αiξi(2−ξi) < 1. a(t) ∈ C([0,1],[0,∞)), f ∈ C([0,∞),[0,∞)).By using Krasnoselskii’s fixed point theorem in cones, we can 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.