Geodesic
Positive solutions of singular fourth-order boundary-value problems
Electronic journal of differential equations, Tome 2006 (2006)
In this paper, we present necessary and sufficient conditions for the existence of positive

$\displaylines{ x''''(t)=p(t)f(x(t)),\quad t\in(0,1);\cr x(0)=x(1)=x'(0)=x'(1)=0, }$

where $f(x)$ is either superlinear or sublinear, $p:(0,1)\to [0,+\infty)$ may be singular at both ends $t=0$ and $t=1$. For this goal, we use fixed-point index results.
Classification : 34A34, 34B15, 45G15
Keywords: singular boundary value problem, fixed point theorem, positive solution 
@article{EJDE_2006__2006__a30,
     author = {Cui,  Yujun and Zou,  Yumei},
     title = {Positive solutions of singular fourth-order boundary-value problems},
     journal = {Electronic journal of differential equations},
     year = {2006},
     volume = {2006},
     zbl = {1096.34012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a30/}
}
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Cui,  Yujun; Zou,  Yumei. Positive solutions of singular fourth-order boundary-value problems. Electronic journal of differential equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a30/