Initial value problems for Caputo fractional equations with singular nonlinearities

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.

Keywords

• fractional derivatives
• volterra integral equation
• weakly singular kernel
• gronwall inequality