Positive solutions to \((n-1,1)\) \(m\)-point boundary value problems with dependence on the first order derivative (Q1030385)

The paper deals with the \(n\)th order \(m\)-point boundary value problem \[ u^{(n)}(t)+f(t,u,u')=0,\quad t\in (0,1), \leqno(1) \] \[ u(0)=0,\quad u'(0)=\dots =u^{(n-2)}(0)=0,\quad u(1)=\sum_{i=1}^{m-2}k_iu(\xi_i), \leqno(2) \] where \(f:[0,1]\times [0,\infty)\times \mathbb{R}\to [0,\infty)\) is continuous, \(k_i>0\), \(i=1,2,\dots,m-2\), \(0=\xi_0<\xi_1<\dots <\xi_{m-2}<\xi_{m-1}=1\), and \(0<\sum_{i=1}^{m-2}k_i\xi_i^{n-1}<1\). Under certain growth conditions imposed on \(f\), the existence of at least one positive solution to problem (1), (2) is proved. The proofs are based on the extension of the Krasnoselskii fixed point theorem in cones and the Green function associated to (1), (2). An example illustrating the result is given.