  The BRST invariance of the open pure spinor superstring is examined in the
presence of background superfields on a Dp-brane. We note that the background
superfields introduced in this paper depend on boundary fermions. The BRST
invariance leads to supersymmetric Dirac-Born-Infeld (DBI) equations for
background superfields depending on boundary fermions as well as boundary
conditions on spacetime coordinates. After quantizing boundary fermions,
background superfields are promoted to non-abelian ones. As a result, we obtain
the supersymmetric non-abelian DBI equations from the supersymmetric DBI
equations depending on boundary fermions. It is shown that these non-abelian
DBI equations reduce to the super-Yang-Mills equations in the limit alpha' ->
0. We also show the nilpotency of the BRST transformation of boundary fermions.
