Second and higher order boundary value problems of nonsingular type
The \(n\)th-order focal problems of the form \[ \begin{aligned} &(-1)^{n-p} y^{(n)}(t) = \phi(t) f\left(t, y(t), y'(t), \dots,\right), \quad t\in (0,1),\\ & y^{(i)}(\sigma_i) = 0, \quad i\in\{0,1,\dots, n-1\}, \end{aligned}\tag{*} \] are considered, where \(p\in\{1,2,\dots,n-1\}\) is fixed, \(\sigma_i := \min\left\{\text{ sign}\left(i-p\right), 0\right\} + 1\), \(0\leq i < n\), \(\phi\) is a summable scalar function on \([0,1]\) which is continuous and positive on \((0,1)\), and \(f : \left[0,1\right] \times \left[0, +\infty\right)^p \to [0,+\infty)\) is continuous and strictly positive on \(\left[0,1\right] \times \left(0, +\infty\right)^p\). A previous result of the authors [Aequationes Math. 57, No.~2-3, 233-240 (1999; Zbl 0931.34011)] is used to prove the existence of a positive solution. Under the conditions assumed, the identically zero function may be a solution to the problem considered.
scientific article; zbMATH DE number 1423095