Existence of multiple symmetric positive solutions of higher order Lidstone problems. (Q1406995)

For the singular nonlinear boundary value problem \[ \begin{gathered} y^{(2m)}(t)= f(t, y(t), y'(t),\dots, y^{(2m-2)}(t), y^{(2m-1)}(t)),\quad t\in (0,1),\\ y^{(2i)}(0)= y^{(2i)}(1)= 0,\qquad 0\leq i\leq m-1,\end{gathered} \] sufficient conditions are given for the existence of at least three symmetric solutions positive in \((0,1)\). The autonomous case without singularity is considered separately. The results are connected with those due to \textit{R. I. Avery} and \textit{J. Henderson} [Appl. Math. Lett. 13, 1--7 (2000; Zbl 0961.34014)].