m点边值共振问题解的存在性与上下解方法

研究一类共振情形下二阶m点边值问题(ρ(t)x′)′=f(t,x(t),x′(t)),t∈[0,1],x′(0)=0,x(1)=∑m-2i=1αix(ηi),其中mi 3为整数,αi 0,ηi∈(0,1)(i=1,2,…,m-2)为常数,满足∑m-2i=1αi=1,0<η1<η2<…<ηm-2<1.本文的研究工具主要依赖于一个新的增算子不动点定理,本质不同于以往文献中使用的Mawhin重合度定理.

Existence and Method of Low and Upper Solutions for m Point Boundary Value Problems at Resonance

We study m point boundary value problem at resonance in the following form{(ρ(t)x′)′=f(t,x(t),x′(t)),t∈,x′(0)=0,x(1)=∑m-2i=1αix(ηi),where mi3 is an integer,αi0,ηi∈(0,1)(i=1,2,…,m-2) are constants satisfying ∑m-2i=1αi=1,0<η1<η2<…<ηm-2<1.Our main tool is a new fixed point theorem of increasing operators.The method is different essentially from the previous papers used by Mawhin′s coincidence degree theory.