Positive and monotone solutions of an \(m\)-point boundary-value problem (Q2778484)

The existence of a positive solution (and monotone in some cases) to the second-order ordinary differential equation NEWLINE\[NEWLINE y''(t)=-f(t,y(t),y'(t)),\quad 0\leq t\leq 1, NEWLINE\]NEWLINE subject to the multipoint boundary conditions NEWLINE\[NEWLINE \alpha y(0)\pm \beta y'(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i), NEWLINE\]NEWLINE is proved under superlinear and/or sublinear growth rate of \(f\). The approach by the author is based on an analysis of the corresponding vector field in the \((y,y')\) phase-plane and on Kneser's property for the solution's funnel. Motivation comes from elliptic problems in annular domains.