Nonlinear \(n\)-order \(m\)-point semipositive boundary value problems and applications (Q6099105)

This paper concerned with the following nonlinear \(n\)-order \(m\)-point semi-positive problem \[ (L\phi)(t)=f(t,\phi(t)), \quad 0<t<1 \] with the following boundary value conditions \[ \left\{ \begin{array}{ll} \phi(1)=\sum_{i=1}^{m-2}a_i\phi(\eta_i),\\ \phi^{(i)}(0)=\phi^{(j)}(1)=0, \end{array} \right. \] where \((L\phi)(t)=(-1)^{n-k}\phi^{(n)}(t)\), \(0\leq i\leq k-1\), \(0\leq j\leq n-k-1\), \(n\geq 2\), \(1<k<n-1;\) \(a_i\in [0,+\infty)\), \(i=1,2,\ldots,m-2\), \(0<\eta_1<\eta_2<\cdots<\eta_{m-2}<1\), are constants, \(m\geq 3\). In this paper, the author obtains the existence of positive solutions and many positive solutions under the condition that \(f\) is superlinear or sublinear. The main tool is the fixed point index theory. It is worth mentioning that the nonlinear term \(f\) can change the sign for \(0<t<1\).