Positive solutions for third-order boundary value problems with change of signs (Q425467)

The paper deals with the boundary value problem NEWLINE\[NEWLINEu'''(t)+h(t)f(u(t))=0, 0\leq t<1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0)=\alpha u(\eta), u''(0)=u(1)=0,NEWLINE\]NEWLINE where \(\eta\in(0,1)\) and \(\alpha\in(0,{{1}\over{1-\eta}}).\) \(h\) is a function such that \(h(t)\geq 0,\) for \(0\leq t\leq \eta\) and \(h(t)\leq 0,\) for \(\eta\leq t\leq 1\) and does not vanish identically on any subinterval of (0,1). The authors provide sufficient conditions on \(g\) and \(f\) and apply Krasnoselskii's and the Avery-Henderson fixed point theorems on cones in Banach spaces to show the existence of at least one or two positive solutions of the problem.