This paper studies the solvability of Cauchy problems for fractional evolution equations with uncertainty. By using a new approach based on the concept of non-compactness measure and the principle of condensing mappings in the spaces without linearity, we prove the existence of $C^0$−solutions without assuming the Lipschitz continuity of the function on the right-hand side. The present results extend previous results when external forces are always required to satisfy some kinds of generalized Lipschitz conditions. Moreover, the principle of condensing mappings for fuzzy-valued functions or set-valued functions found as an application of noncompactness measure is a useful result when studying dynamical systems containing uncertainties.