Using the Fixed Center Approximation to Faddeev equations we have investigated the $DKK$ and $DK\bar{K}$ three-body systems, considering that the $D^*_{s0}(2317)$ acts as the heavy cluster in both cases, generated from the $DK$ interaction in isospin 0. For the $DK\bar{K}$ system we have found evidence of a state with $I(J^P)=1/2(0^-)$ and mass about $2833 - 2858$ MeV, above the threshold of $D^*_{s0}(2317)\bar{K}$. Our results indicate that this state is dominated by a $Df_0(980)$ component, then it could be searched for in the $\pi \pi D$ invariant mass. On the other hand, no clear evidence related to a state from the $DKK$ interaction is found.