Abstract
In this paper, we consider a Cauchy problem for fractional evolution equations with Hilfer's fractional derivative on semi-infinite interval. An elementary fact shows that semi-infinite interval is not compact, the classical Ascoli-Arzela theorem is not valid. In order to establish the global existence criteria, we first generalize Ascoli-Arzela theorem into the semi-infinite interval. Next, we introduce a new concept of mild solutions based on cosine/sine family and probability density function and obtain several existence results of mild solutions on semi-infinite interval.