Second-order initial value problems of singular type (Q1279943)

The authors study the second-order initial value problem \[ y''= \phi(t) f(t,y,y'),\quad t\in(0,T],\quad y(0)= y'(0)= 0.\tag{1} \] Here, the nonlinear term \(f\) may be singular at the following three cases: (a) \(y= 0\) but not \(y'= 0\), (b) \(y= 0\) and \(y'= 0\), (c) \(y'= 0\) but not \(y= 0\). Under some additional assumptions it is proved that the problem (1) has positive solutions at all the above quoted cases (a), (b) and (c).