Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 117, pp. 1-32.

Title: Initial value problems for Caputo fractional equations  
       with singular nonlinearities

Author: Jeffrey R. L. Webb (Univ. of Glasgow, Glasgow, UK)

Abstract:
  We consider initial value problems for Caputo fractional equations of the form
  $D_{C}^{\alpha}u=f$ where f can have a singularity.
  We consider all orders and prove equivalences with Volterra integral equations in
  classical spaces such as $C^{m}[0,T]$. In particular for the case $1<\alpha<2$
  we consider nonlinearities of the form $t^{-\gamma}f(t,u,D^{\beta}_{C}u)$ where
  $0<\beta \leq 1$ and $0\leq \gamma<1$ with f continuous, and we prove results on
  existence of global $C^1$ solutions under linear growth assumptions on f(t,u,p)
  in the u,p variables. With a Lipschitz condition we prove continuous dependence
  on the initial data and uniqueness. One tool we use is a Gronwall inequality for
  weakly singular problems with double singularities. We also prove some regularity
  results and discuss monotonicity and concavity properties.

Submitted November 20, 2018. Published October 30, 2019.
Math Subject Classifications: 34A08, 34A12, 26A33, 26D10.
Key Words: Fractional derivatives; Volterra integral equation;
           weakly singular kernel; Gronwall inequality.