  In this article, we give some results for fractional-order delay differential
equations. In the first result, we prove the existence and uniqueness of
solution by using Bielecki norm effectively. In the second result, we consider
a constant delay form of this problem. Then we apply Burton's method to this
special form to prove that there is only one solution. Finally, we prove a
result regarding the Hyers-Ulam stability of this problem. Moreover, in these
results, we omit the conditions for contraction constants seen in many papers.
