From the text: The object of this paper is to study the fractional evolution equation
\[ \frac{d^\alpha u(t)}{dt^\alpha}+(A-B(t))u(t)= f(t), \quad t>t_0, \]
in a Banach space \(X\), where \(0<\alpha\leq 1\), \(u\) is an \(X\)-valued function on \(\mathbb R^+=[0,\infty)\), and \(f\) is a given abstract function on \(\mathbb R^+\) with values in \(X\). We assume that \(-A\) is a linear closed operator defined on a dense set \(S\) in \(X\) into \(X\), \(\{B(t): t\in\mathbb R^+\}\) is a family of linear bounded operators defined on \(X\) into \(X\).
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.