On the initial value problem of fractional evolution equations with noncompact semigroup

In this paper, we study the initial value problem of fractional semilinear evolution equations with noncompact semigroup in Banach spaces. The existence of saturated mild solutions and global mild solutions is obtained under weaker conditions. Particularly, we find that the results obtained are also valid for the initial value problem of fractional ordinary differential equations in Banach spaces. The theorems proved in this paper improve and extend some related conclusions on this topic. An example is given to illustrate that our results are valuable.