Positive solutions for semipositone \((n,p)\) boundary value problems (Q1428802)

Let \(n\geq 2\) and \(1\leq p\leq n-1\) be fixed. The author studies the \((n,\,p)\) boundary value problem \[ \begin{gathered} u^{(n)}+\lambda f(t,\,u)=0,\quad 0<t<1,\\ u^{(i)}(0)=0,\;0\leq i\leq n-2,\;u^{(p)}(1)=0, \end{gathered} \] where the nonlinearity \(f\) is allowed to be negative somewhere. Sufficient conditions which ensure the existence of positive solutions of the above \((n,p)\)-problem for \(\lambda\) on a suitable interval are established in this paper. The proof is based on a fixed-point theorem in a cone.