This paper is concerned with nonlinear fractional differential equations with the Caputo fractional derivatives in Banach spaces. Local existence results are obtained for initial value problems with initial conditions at inner points for the cases that the nonlinear parts are Lipschitz and non-Lipschitz, respectively. Hausdorff measure of non-compactness and Darbo-Sadovskii fixed point theorem are employed to deal with the non-Lipschitz case. The results obtained in this paper extend the classical Peano’s existence theorem for first order differential equations partly to fractional cases.
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