Weakly singular Gronwall inequalities and applications to fractional differential equations

We obtain some new Gronwall type inequalities which are applicable to some weakly singular Volterra integral equations similar to the ones first studied by D. Henry. The main interest is that we consider cases with a double singularity and we obtain explicit L∞ bounds rather than L1 bounds. Furthermore our bounds involve the exponential function and not the Mittag-Leffler function as in some previous works. We give applications to some Volterra integral equations with a doubly singular kernel that arise from Caputo fractional differential equations where, as opposed to previous papers, we have a singularity in the nonlinearity.