  This paper deals with initial value problems for fractional functional
differential equations with bounded delay. The fractional derivative is defined
in the Caputo sense. By using the Schauder fixed point theorem and the
properties of the Mittag-Leffler function, new existence and uniqueness results
for global solutions of the initial value problems are established. In
particular, the unique existence of global solution is proved under the
condition close to the Nagumo-type condition.
