一类非线性m-点边值问题正解的存在性

Existence of Positive Solutions for a Nonlinear m-Point Boundary Value Problem

设α∈C[0,1],b∈C([0,1],(-∞,0)).设φ(t)为线性边值问题 u″+a(t)u′+b(t)u=0, u′(0)=0,u(1)=1的唯一正解.本文研究非线性二阶常微分方程m-点边值问题 u″+a(t)u′+b(t)u+h(t)f(u)=0, u′(0)=0,u(1)-sum from i=1 to(m-2)((a_i)u(ξ_i))=0正解的存在性.其中ξ_i∈(0,1),a_i∈(0,∞)为满足∑_(i=1)^(m-2)a_iφ_1(ξ_i)<1的常数,i∈{1,…,m-2}.通过运用锥上的不动点定理,在f超线性增长或次线性增长的前提下证明了正解的存在性结果.

Let a ∈ C[0,1], b ∈ C([0,1],(-∞,0)). Let φ1(t) be the unique positive solution of the linear problem u' + a(t)u' + b(t)u = 0, u'(0) = 0, u(1) = 1. We study the existence of positive solutions for the nonlinear m-point boundary value problem u' + a(t)u' + b(t)u + h(t)f(u) = 0, where & ∈ (0,1) and αi ∈ (0, ∞) are given constants satisfying i ∈{1,...,m - 2}. We show the existence of positive solutions if f is either superlinear or sublinear by applying the fixed point theorem in cones.