  Using a gauged flavor SU(3) invariant hadronic Lagrangian, we study $\Xi$
production from $\bar K$ induced reactions on $\Lambda$ and $\Sigma$ in a
coupled-channel approach. Including the four channels of $\bar K\Lambda$, $\bar
K\Sigma$, $\pi\Xi$, and $\eta\Xi$, we solve the Bethe-Salpeter equation in the
$K$-matrix approximation by neglecting the off-shell contribution in the
intermediate states. For the transition potential which drives the higher-order
contribution, we consider all allowed Born diagrams in the $s$, $t$, and $u$
channels. With coupling constants determined from the SU(3) symmetry with
empirical input, we find that the cross sections for the reactions $\bar
K\Lambda\to\pi\Xi$, $\bar K\Sigma\to\pi\Xi$, $\bar K\Lambda\to\eta\Xi$, and
$\bar K\Sigma\to\eta\Xi$ all have values of a few mb. In contrast to the
results from the lowest-order Born approximation, the magnitude of these cross
sections is less sensitive to the values of the cut-off parameters in the form
factors. From the transition matrix in the coupled-channel approach, we have
further evaluated the cross sections for the elastic process $\bar
K\Lambda\to\bar K\Lambda$, $\bar K\Sigma\to\bar K\Sigma$, $\pi\Xi\to\pi\Xi$,
and $\eta\Xi\to\eta\Xi$ as well as for the inelastic processes $\bar
K\Lambda\to\bar K\Sigma$ and $\pi\Xi\to\eta\Xi$. Implications of the reactions
studied here in $\Xi$ production from relativistic heavy ion collisions are
discussed.
