二阶常微分方程边值问题变号解的存在性和唯一性

Existence and uniqueness of sign-changing solutions for second order ordinary differential boundary value problem

考虑非线性两点常微分方程边值问题-u″(t)=λf(u(t)),0<t<1,u(0)=u(1)=0变号解的存在性,其中λ>0,f∈C(R,R),f(s)s>0,s≠0。基于时间映像分析法,证明在C+l空间中,当非线性项f满足一些合理的条件下,该问题有唯一确定的解,这里C+l:={在(0,1)中有l-1零点,且y'(0)>0,y∈C1y[0,1]。y的所有零点都是简单的,y(0)=y(1)=0}

Consider the existence of sign-changing solutions for nonlinear two - point ordinary differential boundary value problem -u″（t）=λf（u（t））,0〈t〈1, u（0）=u（1）=0 where λ〉0,f∈C（R,R）,f（s）s〉0,s≠0.Based upon the time-map method, it is shown that there is exactly one solution with large norm in Cl^＋ under some reasonable conditions about f, where Cl^＋：={y∈C^1[0,1]｜y has exactly l-1 zeros in （0,1）,y′（0） 〉0,all zeros of y are simple ,y（0） = y（ 1 ） = 0}.