二阶常微分方程无穷多点边值问题的正解

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'+a (t ) f (u)=0, t∈(0, 1),u(0)=0, u(1)=∑^∞_{i =1}α _i u ( ξ _i ) 正解的存在性. 其中ξ _i∈ (0,1),α _i∈ [0,∞), 且满足∑^∞_i=1α_iξ _i <1.α∈C([0,1], [0,)),f∈C ([0,∞), [0,∞)).

Positive Solutions of Nonlinear ∞-point Boundary Value Problems

Let α _i ∈[0,  ∞), ξ _i ∈ (0,1) and ∑^∞_i=1α _i ξ _i <1.  We study the existence of positive solutions to  the  boundary value problemu'+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:[0, ∞) → [0, ∞) is continuous. We show the existence of at least one positive solution if f is either superlinear or sublinear by applying a fixed-point theorem in cones.