  In the present paper, it is assumed that there exist two dark matter
particles: superheavy dark matter particles (SHDM), whose mass $\sim$ inflaton
mass, and lighter fermion dark matter particles (LFDM) which are the
ultra-relativistic products of its decay. The event rates of LFDM measured by
the Pierre Auger observatory (Auger) are evaluated in the energy range between
1 EeV and 1 ZeV at the zenith angles between 0$^{\circ}$ and 60$^{\circ}$ when
the different lifetimes of decay of SHDM. Thus it is proved that the
possibility of measurement of these LFDM at Auger. Based on the assumption that
the detection efficiency for a shower caused by LFDM is set to be 100\%, the
LFDM contribution to the energy spectrum of extreme energy cosmic rays (EECRs)
is discussed. And it is found that there are some hints on the existence of
SHDM and LFDM in the past EECRs observations.
