  Investigating the existence, uniqueness, stability, continuous dependence of
data among other properties of solutions of fractional differential equations,
has been the object of study by an important range of researchers in the
scientific community, especially in fractional calculus. And over the years,
these properties have been investigated more vehemently, as they enable more
general and new results. In this paper, we investigate the existence and
uniqueness of a class of mild and classical solutions of the fractional
evolution differential equation in the Banach space $\Omega$. To obtain such
results, we use fundamental tools, namely: Banach contraction theorem, Gronwall
inequality and the $\beta$-times integrated $\beta$-times integrated
$\alpha$-resolvent operator function of an $(\alpha,\beta)$-resolvent operator
function.
