一类二阶四点边值问题多个正解的存在性

Existence of Triple Positive Solutions of a Kind of Second-order Four-point Boundary Value Problem

研究了下面的二阶四点边值问题x″(t)+q(t)f(t,x(t),x′(t))=0,0<t<1x′(0)-αx(ξ)=0,x′(1)+βx(η)=0,其中0<α<1ξ,0<β<1-1η,0<ξ<η<1,αβη-αβξ+α+β>0.首先计算了相应齐次问题的Green函数,然后运用其Green函数的性质及Avery-Peterson不动点定理,我们得到了该边值问题至少存在三个正解.

We considered the following four-point boundary value problem x″（t）＋q（t）f（t,x（t）,x′（t））=0,0〈t〈1.x′（0）-αx（ε）=0,x′（1）＋βx（η）=0.where0〈α〈1/ζ,0〈β〈1/1-η,0〈ξ〈η〈1,αβη-αβξ＋α＋β〉0,We firstly give the corresponding Green＇s function, the by using a generalized Leggett-William% fixed point theorem due to Avery and Peterson, the existence of three positive solutions are obtained.