Existence of mild solutions for Hilfer fractional evolution equations in Banach space

The paper is concerned with the existence of a mild solution for Hilfer fractional evolution equations of order $\alpha \in (1,2)$ in Banach spaces. Firstly, by using the Laplace transform and Mainardi’s Wright-type function, a new concept of mild solution for this type of fractional equation is given based on the cosine family generated by the operator $A$ and the probability density function. Secondly, with the help of fractional calculus, the noncompactness measure and the Ascoli–Arzelà Theorem, the existence of mild solutions is established in the case of a noncompact cosine family. Our results improve and generalize some related conclusions on this topic. Finally, two examples are presented to illustrate the main results.