Existence of solutions to a singular initial value problem (Q2465535)

The authors have obtained sufficient conditions which guarantee the global existence of solutions of the initial value problem \[ x^{\prime}= f(t,x,x^{\prime}),\, x(0)= A, \] where \(f(t,x,p)\) is defined for \((t,x,p)\in D_{t} \times (D_x -\{A\}) \times D_{p},D_{t},D_{x},D_{p} \subseteq R\) such that \(A\) is an interior point of \(D_x\) and the sets \(D_{p}^{+} = D_{p} \cap (0, \infty)\) and \(D_{p}^{-} = D_{p}\cap (- \infty, 0)\) are not empty. Further, \(f\) may be such that for at least one \((t_{0}, p_{0}) \in D_{t} \times D_{p}\), the function \(f(t_0,x,p_0)\) is unbounded when \(\text{x}\rightarrow A^{-}\).