一类二阶常微分方程无穷多点边值问题正解的存在性

Existence of Positive Solutions for Second Order ∞-Point Boundary Value Problem

运用锥上的不动点定理研究一类非线性二阶常微分方程无穷多点边值问题u″(t)+f(t,u)=0,t∈(0,1),u′(0)=∑∞αiu(ξi),u′(1)+∑∞βiu(ξi)=0,i=1i=1正解的存在性,其中αi,βi∈(0,+∞),i=1,2,…,n,…,0<ξ1<ξ2<…<ξn<…<1为给定的常数,f:[0,1]×[0,+∞)→[0,+∞)连续.

In this paper,we investigate the existence of positive solutions to the following ∞-point boundary value problemu″（t）＋f（t,u）=0,t∈（0,1）, u′（0）=∑∞ i=1αiu（ξi）,u′（1）＋∑∞ i=1βiu（ξi）=0,where αi,βi are positive parameters,αi,βi∈（0,＋∞）,i=1,2,…,n,…,0ξ1ξ2…ξn…1,and f：×[0,＋∞）→[0,＋∞） is continuous.Based upon a fixed point theorem in cones,we show that the above problem has at least one positive solution if the nonlinearity f is either superlinear or sublinear.