Geodesic
The method of upper and lower solutions for a Lidstone boundary value problem
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 639-652 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 \[ u^{(2n)}(t)=f(t,u(t),u^{\prime \prime }(t),\dots ,u^{(2(n-1))}(t)),\quad 01, u^{(2i)}(0)=u^{(2i)}(1)=0,\quad 0\le i\le n-1, \] 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.
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 \[ u^{(2n)}(t)=f(t,u(t),u^{\prime \prime }(t),\dots ,u^{(2(n-1))}(t)),\quad 01, u^{(2i)}(0)=u^{(2i)}(1)=0,\quad 0\le i\le n-1, \] 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.
Classification : 34B15
Keywords: $n$-parameter eigenvalue problem; Lidstone boundary value problem; lower solution; upper solution 
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     author = {Guo, Yanping and Gao, Ying},
     title = {The method of upper and lower solutions for a {Lidstone} boundary value problem},
     journal = {Czechoslovak Mathematical Journal},
     pages = {639--652},
     year = {2005},
     volume = {55},
     number = {3},
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     zbl = {1081.34019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a5/}
}
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Guo, Yanping; Gao, Ying. The method of upper and lower solutions for a Lidstone boundary value problem. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 639-652. http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a5/