  Using separable potentials for $\bar{K}N-\pi\Sigma$ interaction, we
investigated four-body kaonic nuclear systems such as $K^{-}ppn$ and
$K^{-}K^{-}pp$, with the Faddeev AGS method in the momentum representation. The
Faddeev calculations are based on the quasi-particle method and the method of
the energy dependent pole expansion was used to obtain the separable
representation for the integral kernels in the three- and four-body equations.
Different types of $\bar{K}N-\pi\Sigma$ potentials based on phenomenological
and chiral SU(3) approach are used and it was shown that the kaonic nuclear
systems under consideration are tightly bound.
