  The energy-, density-, and isospin-dependent $\Delta$-formation cross section
$\sigma_{N\pi \rightarrow \Delta}^*$ and $\Delta$-decay width are calculated
based on the relativistic BUU approach in which the effective mass splitting of
nucleon and $\Delta$ baryons in isospin-asymmetric matter is considered by the
inclusion of the $\delta$ meson exchange in the effective Lagrangian density.
With the density-dependent couplings for baryons of F. Hofmann et al., the
$\sigma_{N\pi \rightarrow \Delta}^*$ is decreased (increased) moderately with
increasing density with (without) the consideration of the density dependent
pion effective mass. Meanwhile, if the invariant mass of the system is not far
from the $\Delta$ pole mass, the $\Delta$-decay width is also weakly dependent
on density. The mass splitting effect of differently charged nucleon and
$\Delta$ baryons on $\sigma_{N\pi \rightarrow \Delta}^*$ are found to be more
obvious than that of pion mesons but much weaker than the mass splitting in the
hard $\Delta$ production channel $NN\rightarrow N\Delta$. Further, the largest
mass-splitting influence is seen in the $\pi^-p\rightarrow \Delta^0$ and
$\pi^+n\rightarrow \Delta^+$ channels but not in the production of $\Delta^-$
and $\Delta^{++}$ isobars.
