  In this paper, we investigate abstract time-fractional evolution equations
with nonlinear perturbations. We construct solutions of Lipschitz perturbation
problems in arbitrary large time interval independent of the Lipschitz
constants. We will extend well-known results for standard evolution equations
such as the blow-up alternative, to the time-fractional evolution equations. We
also prove the differentiability with respect to time of the solution when the
perturbation is sufficiently smooth. The differentiability enables us to use
the maximum principle. The theory on general Banach spaces enables us to deduce
space regularity result easily.
