A condition for finite blow-up time for a Volterra integral equation (Q1323240)

The author studies the nonlinear Volterra equation \[ u(x) = \int_ 0^ x (x - s)^{\alpha - 1} g \bigl( u(s) \bigr) ds, \quad x > 0, \] where \(\alpha>0\) and \(g\) is continuous and nondecreasing with \(g(x) > g(0) = 0\) for \(x>0\), and shows how to obtain estimates on the derivative of the maximal solution. Furthermore, it is proved that the maximal solution blows up in a finite time if and only if \[ \int^ \infty_ 0 \left( {s \over g(s)} \right)^{1/ \alpha} {ds \over s} < \infty. \]