The paper deals with the local existence and uniqueness of a mild solution for the Cauchy problem formed by a fractional integro-differential equation with time-delay and a nonlocal initial condition:
Here, , , is a generator of a solution operator on a complex Banach space , is a function representing the delay, is an operator. The convolution integral in the equation is of the Riemann-Liouville fractional integral type. The existence of a global solution is also proven here. An illustrative example is presented at the end of the paper.