A note on semipositone boundary value problems with nonlocal, nonlinear boundary conditions (Q461393)

The paper concerns the nonlinear BVP with nonlocal BCs on \([0,1]\): \[ \begin{aligned} -y''(t)&=f(t,u) 0<t<1\\ y(0)&=H(\varphi(y)),\\ y(1)&=0 ,\end{aligned}\tag{P} \] where \(f: [0,1]\times\mathbb{R}\longrightarrow\mathbb{R}\) and \(H: \mathbb{R}\longrightarrow\mathbb{R}\) are continuous, \(H([0,+\infty))\subset[0,+\infty)\), and \(\varphi: C([0,1])\longrightarrow\mathbb{R}\) is a linear functional such that \(\varphi=\varphi_1+\varphi_2\), with coercivity condition on \(\varphi_2\) and \(\varphi_1, \varphi_2\) realizable in the Stieltjes integral representation. Under asymptotic behavior condition on \(H\) and superlinear \(f\), the author shows that \((P)\) has at least one positive solution. The proof uses the fixed point index theory on a cone of a Banach space.