二阶常微分方程无穷多点边值问题的正解

Positive Solutions of Nonlinear ∞-point Boundary Value Problems

该文运用锥上的不动点定理研究非线性二阶常微分方程无穷多点边值问题u″+α(t)f(u)=0,t∈(0,1), u(0)=0,u(1)=sum from i=1 to∞(α_iu(ξ_i)正解的存在性。其中ξ_i∈(0,1),α_i∈[0,∞),且满足sum from i=1 to∞(α_iξ_i)<1.a∈C([0,1],[0,∞)),f∈C([0,∞),[0,∞)).

Let αi∈[0,∞],￡i∈（0,1） and ∑∞i=1α i ξ i 〈1. We study the existence of positive solutions to the boundary value problem u＇＇＋a （t ） f （u）=0, t ∈（0, 1）,u（0）=0, u（1）= ∑∞i =1 α i u （ ξ i ） where α ∈ C（[0,1], [0,∞））,and f ∈ C （[0, ∞）, [0, ∞））.is continuous. We show the existence ofat least one positive solution if f is either superlinear or sublinear by applying a fixed-point theorem in cones.