Positive Solutions of a Weak Semipositone Third-Order Three-Point Boundary Value Problem
Received:July 12, 2007  Revised:March 08, 2008
Key Word: singular ordinary differential equation   multi-point boundary value problem   positive solution   existence   multiplicity.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10871059).
 Author Name Affiliation Qing Liu YAO Department of Applied Mathematics, Nanjing University of Finance and Economics, Jiangsu 210003, P. R. China
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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.