一类四阶常微分方程非线性边值问题正解的存在性

Existence of positive solutions for a class of fourth-order ordinary differential equations with nonlinear boundary value

本文研究了一类四阶常微分方程非线性边值问题u″″=rf(t,u(t)),0<t<1,u(0)=u′(0)=u′(1)=u″′(1)+ψ(u(1))=0正解的存在性,其中r是一个正参数,ψ(s)=sc(s),c∈C([0,∞),[0,12)∪(12,∞)),且当u→0+时f(t,u)=au+o(u),ψ(s)=a 1s+o(s);当u→∞时,f(t,u)=bu+o(u),ψ(s)=b 1s+o(s).主要结果的证明基于Dancer全局分歧理论.

We study the existence of positive solutions for a class of fourth-order ordinary differential equations with nonlinear boundary value u″″=rf(t,u(t)),0<t<1,u(0)=u′(0)=u′(1)=u″′(1)+ψ(u(1))=0,where r is a positive parameter,ψ(s)=sc(s),c∈C([0,∞),[0,12)∪(12,∞)).When u→0+,f(t,u)=au+o(u),ψ(s)=a 1s+o(s).When u→∞,f(t,u)=bu+o(u),ψ(s)=b 1s+o(s).The proof of the main results is based on the Dancer global bifurcation technique.