  In this paper, we prove a theorem of linearized asymptotic stability for
fractional differential equations with a time delay. More precisely, using the
method of linearization of a nonlinear equation along an orbit (Lyapunov's
first method), we show that an equilibrium of a nonlinear Caputo fractional
differential equation with a time delay is asymptotically stable if its
linearization at the equilibrium is asymptotically stable. Our approach based
on a technique which converts the linear part of the equation into a diagonal
one. Then using properties of generalized Mittag-Leffler functions, the
construction of an associated Lyapunov--Perron operator and the Banach
contraction mapping theorem, we obtain the desired result.
