Existence of positive solutions to a singular second order boundary value problem (Q1612625)

The author considers the scalar nonlinear boundary value problem \[ x''=f(t,x,x'),\;t\in[0,1], \quad x(0)=a,\;x(1)=b,\;a,b\geq 0, \] where \(f\) is singular at \(x=0\), and at least one of the numbers \(a,b\) is zero. Sufficient conditions are obtained for the existence of at least one solution \(x\) such that \(x(t)\) is positive for \(t\in(0,1)\). The results are illustrated by two examples.