A necessary and sufficient condition for the existence of positive solutions of fourth-order singular boundary value problems (Q1431926)

The authors study the boundary value problem \[ u^{(4)}(t)=a(t)f(u(t)), \quad t\in (0,1), \quad u(0)=u(1)=0,\quad u'' (0)=u''(1)=0, \] where \(a \in C((0,1),[0,\infty))\), \(a\) may be singular at the endpoints of \((0,1)\) and \(f\in C([0,\infty),[0,\infty))\). Using the approximation theory of operators and a fixed-point theorem, a necessary and sufficient condition for the existence of \(C^3([0,1],[0,\infty))\) positive solutions is obtained in the cases that \(f\) is superlinear or sublinear.