A Volterra inequality with the power type nonlinear kernel (Q1347685)

The author proves that there is a nonlinear solution to the integral inequality (and thus to the corresponding equation as well) \[ u(x) \leq \int_0^x k(x-s)u(s)^\beta ds,\quad x\geq 0, \] where \(\beta> 0\) and \(k\) is a positive locally integrable function if and only if \(0< \beta < 1\) and \[ \int_0^\delta {K^{-1}(s) \over s(-\ln(s))} ds < \infty, \] for some \(\delta > 0\) where \(K^{-1}\) is the inverse of the function \(K(x)=\int_0^x k(s) ds\).