  We study the $s$-wave kaon-nucleon bound state with the strangeness $S=-1$ in
the Bethe-Salpeter formalism in the ladder and instantaneous approximations. We
solve the Bethe-Salpeter equation of the bound state and obtain the
Bethe-Salpeter amplitude. It is shown that the $K^-p$ bound state exists in
this formalism. We also study the decay width of the bound state based on the
Bethe-Salpeter techniques. The mass of this bound state is 1422 MeV and its
decay width is obviously smaller than that of $\Lambda(1405)$. These results
indicate that there may be some other structures in the observed resonance.
