Initial value problems for Caputo fractional equations with singular nonlinearities

Abstract

We consider initial value problems for Caputo fractional equations of the form DαCu = ƒ where ƒ can have a singularity. We consider all orders and prove equivalences with Volterra integral equations in classical spaces such as Cm [0, T]. In particular for the case 1 < α < 2 we consider nonlinearities of the form t-γ ƒ(t, u, DβCu) where 0 < β ≤ 1 and 0 ≤ γ < 1 with ƒ continuous, and we prove results on existence of global C1 solutions under linear growth assumptions on ƒ(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.