Positive solutions of singular fourth-order boundary-value problems

In this paper, we present necessary and sufficient conditions for the existence of positive $C^3[0,1]cap C^4(0,1)$ solutions for the singular boundary-value problem $$displaylines{ x''''(t)=p(t)f(x(t)),quad tin(0,1);cr x(0)=x(1)=x'(0)=x'(1)=0, }$$ where $f(x)$ is either superlinear or sublinear, $p:(0,1)o [0,+infty)$ may be singular at both ends $t=0$ and $t=1$. For this goal, we use fixed-point index results.