Summary: In our paper [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 10, 3462–3471 (2010; Zbl 1229.34004
)] we discussed an error in the recent literature on abstract fractional differential equations and we proposed a different approach to treat these type of problems. By noting that the results in [loc. cit.] are not applicable for problems with nonlocal conditions, in this paper we study the existence of mild solutions for a class of abstract fractional differential equations with nonlocal conditions. An application involving a partial differential equation with a fractional temporal derivative and nonlocal conditions is considered.