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We focus on eventually non-linear abstract Cauchy problems with a generalizedfractional derivative in time. First we prove a local existence and uniquenessresult, then we focus on a generalized Gr\"onwall inequality. Before addressingthe inequality, we study some properties of eigenvalues and eigenfunctions ofthe generalized fractional derivatives. Finally, we prove some consequences ofthe generalized Gr\"onwall inequality.

Abstract Cauchy problems for the generalized fractional calculus