From the text: The object of this paper is to study the fractional evolution equation
in a Banach space , where , is an -valued function on , and is a given abstract function on with values in . We assume that is a linear closed operator defined on a dense set in into , is a family of linear bounded operators defined on into .
We prove the existence of optimal mild solutions for linear fractional evolution equations with an analytic semigroup in a Banach space. We use the Gelfand-Shilov principle to prove existence, and then the Bochner almost periodicity condition to show that solutions are weakly almost periodic. As an application, we study a fractional partial differential equation of parabolic type.