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$n$-parameter eigenvalue problem; Lidstone boundary value problem; lower solution; upper solution
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.
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