We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem $$ \begin{aligned} (-1)^mu^{(2m)}(t)&=\lambda a(t)f(u(t)),\ \ \ \ \ 0<t<1, \\ u^{(2i)}(0)&=u^{(2i)}(1)=0,\ \ \ \ i=0,1,2,\cdots ,m-1 . \end{aligned} $$ where $a\in C([0,1], [0,\infty ))$ and $a(t_0)>0$ for some $t_0\in [0,1]$, $f\in C([0,\infty ),[0,\infty ))$ and $f(s)>0$ for $s>0$, and $f_0=\infty $, where $f_0=\lim _{s\rightarrow 0^+}f(s)/s$. We investigate the global structure of positive solutions by using Rabinowitz's global bifurcation theorem.
In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem \[ u^{(2n)}(t)=f(t,u(t),u^{\prime \prime }(t),\dots ,u^{(2(n-1))}(t)),\quad 0<t<1, u^{(2i)}(0)=u^{(2i)}(1)=0,\quad 0\le i\le n-1, \] where $f\:[0,1]\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ is continuous. We obtain sufficient conditions on $f$ to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.