Existence principles and theory for singular Dirichlet boundary value problems (Q1355827)

The singular boundary value problem \[ y''+q(t) g(t,y)=0, \qquad 0<t<1, \qquad y(0)=a, \qquad y(1)=b,\tag{1} \] where \(q\in C(0,1)\), \(q>0\) on \((0,1)\), \(\int_0^1 t(1-t)q(t)dt<\infty\), \(g\in C([0,1]\times \mathbb{R})\) is solved using the lower and upper solutions. Problem (1) is solved also in the case when \(g\) has a singularity at \(y=0\) and \(a=b=0\). Then a sequence of modified problems is solved and by means of the Arzela-Ascoli theorem the searched solution is established.