  In this paper we explore the problem of antiparticles in DSR1 and
$\kappa$-Minkowski space-time following three different approaches inspired by
the Lorentz invariant case: a) the dispersion relation, b) the Dirac equation
in space-time and c) the Dirac equation in momentum space. We find that it is
possible to define a map $S_{dsr}$ which gives the antiparticle sector from the
negative frequency solutions of the wave equation. In $\kappa$-Poincar\'e, the
corresponding map $S_{kp}$ is the antipodal mapping, which is different from
$S_{dsr}$. The difference is related to the composition law, which is crucial
to define the multiparticle sector of the theory. This discussion permits to
show that the energy of the antiparticle in DSR is the positive root of the
dispersion relation, which is consistent with phenomenological approaches.
