Sobolev type fractional functional evolution equations have many
applications in the modeling of many physical processes. Therefore, we
investigate fractional-order time-delay evolution equation of Sobolev
type with multi-orders in a Banach space and introduce an analytical
representation of a mild solution via a new delayed Mittag-Leffler type
function which is generated by linear bounded operators. Furthermore, we
derive an exact analytical representation of solutions for
multi-dimensional fractional functional dynamical systems with
nonpermutable and permutable matrices. We also study stability analysis
of the given time-delay system in Ulam-Hyers sense with the help of
Laplace transform.