Existence criteria for singular initial value problems with sign changing nonlinearities (Q1612060)

The authors establish the existence of solutions to the singular initial value problem \[ y'(t) = q(t)f(t,y(t)),\quad 0<t<T,\quad y(0) =0, \] where \( f: [0,T]\times (0,\infty) \to \mathbb{R}\) is continuous. The solution is obtained as a limit of solutions to a family of nonsingular problems for which the existence of suitable lower and upper solutions is assumed. Easily verifiable criteria which guarantee that the problem has a solution are also presented.