Positive solutions of a Lidstone boundary value problem with variable coefficient function

Abstract

We establish the existence of positive solutions of the Lidstone boundary value problem

$$\begin{array}{rcl}(-1)^{n}u^{(2n)}&=&\lambda a(t)f(u),\quad 0<t<1,\\[3pt]u^{(2i)}(0)&=&u^{(2i)}(1)=0,\quad 0\leq i\leq n-1\end{array}$$

for all sufficiently small positive real λ, where the function a may change sign in [0,1] and the function f:[0,∞)→R satisfies f(0)>0. We also show that our assumption is not vacuous.