Positive Solutions of a Weak Semipositone Third-Order Three-Point Boundary Value Problem

DOI：10.3770/j.issn:1000-341X.2010.01.017

 作者 单位 姚庆六 南京财经大学应用数学系，江苏 南京 210003

研究了非线性三阶三点边值问题$\begin{array}{c}u'''(t)=f(t,u(t)),~a.e.~t\in [0,1],\quad u(0)=u'(\eta)=u''(1)=0\end{array}$的正解, 其中非线性项 $f(t,u)$ 是一个 Carath\'eodory函数并且存在非负函数 $h\in L^{1}[0,1]$ 使得 $f(t,u)\geq -h(t)$.通过利用“高度函数”的积分和锥上的 Krasnosel'skii 不动点定理证明了$n$ 个正解的存在性.

The positive solutions are studied for the nonlinear third-order three-point boundary value problem $$\begin{array}{c}u'''(t)=f(t,u(t)),~\mbox{a.e.}~t\in [0,1],\quad u(0)=u'(\eta)=u''(1)=0,\end{array}$$where the nonlinear term $f(t,u)$ is a Carath\'eodory function and there exists a nonnegative function $h\in L^{1}[0,1]$ such that$f(t,u)\geq -h(t)$. The existence of $n$ positive solutions is proved by considering the integrations of height functions'' and applying the Krasnosel'skii fixed point theorem on cone.