Cauchy problem for non-autonomous fractional evolution equations

In this paper, we study the solvability of Cauchy problem for a class of non-autonomous fractional evolution equation with Caputo's fractional derivative of order alpha is an element of (1, 2), which can be applied to model the time dependent coefficients fractional differential systems. We first introduce an operator family and analyze its properties, by the iterative method, we construct a solution to an operator-valued Volterra equation, which is the most critical ingredient to prove solvability of the problem. Finally, based on the solution operators we establish the existence and uniqueness of classical solutions.