Let , , a Banach space, a family of linear closed operators defined on dense set in with values in , a function, , a given function and be the Banach space of linear bounded operators endowed with the topology defined by the operator norm. Assume that is independent of . Suppose that the operator
exists in for any with and
The author considers the fractional integral evolution equation
where is the Gamma-function, constructs the fundamental solution of the homogeneous fractional differential equation
and proves the existence and uniqueness of the solution of (2) with the initial condition .
The author also proves the continuous dependence of the solutions of equation (1) on the element and the function and gives an application to a mixed problem of a parabolic partial differential equation of fractional order.