Solvability of singular second-order initial value problems (Q2832414)

The author considers the initial value problem NEWLINE\[NEWLINE \begin{aligned} x'' & = f(t,x,x'), \\ x(0) & = A, x'(0) = B \end{aligned} NEWLINE\]NEWLINE with a continuous function \(f : \mathbb R^3 \to \mathbb R\), which is unbounded as \(t \to 0\).NEWLINENEWLINESufficient conditions for the existence of monotone and positive solutions are given. The results are obtained via sign conditions on \(f\) together with uniform boundedness on some bounded set. The proof relies on the regularization method, i.e. on construction of a sequence of auxiliary regular problems.