Fractional differential equations with a Krasnoselskii–Krein type condition

We consider an initial value problem for a fractional differential equation of Caputo type. The convergence of the Picard successive approximations is established by first showing that the Caputo derivatives of these approximations converge. The method utilized, originally introduced in [O. Kooi, The method of successive approximations and a uniqueness theorem of Krasnoselskii–Krein in the theory of differential equations, Nederi. Akad. Wetensch, Proc. Ser. A61; Indag. Math. 20 (1958) 322–327], is interesting in itself.