Positive solutions of some second-order nonlinear singular differential equations. (Q1416419)

The author studies the boundary value problem \[ x''+ f(t,x,x')= 0,\quad t\in (0,1),\qquad x(0)= x'(1)= 0,\tag{1} \] and the initial value problem \[ x''= f(t,x,x'),\quad t\in (0,1),\qquad x(0)= x'(0)= 0,\tag{2} \] where \(f(t,x,y)\) has the following singularities \[ \lim_{x\to 0+}\, f(\cdot,x,\cdot)= +\infty,\quad \lim_{y\to 0+}\, f(\cdot,\cdot,y)= +\infty. \] Under certain assumptions about \(f(t,x,x')\), the author proves that problems (1) and (2) have at least one positive solution.