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We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases 0<α<1 and 1<α<2

Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness