On the regularity of solutions to a class of nonlinear Volterra integral equations with singularities

We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval $[0,b]$. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on $(0,b]$, with possible singularities of the derivatives of the solution at the left endpoint of the interval $[0,b]$.