  The production of $K^+$ mesons in proton-nucleus collisions from 1.0 to 2.3
GeV is analyzed with respect to one-step nucleon-nucleon $(NN\to N Y K^+$) and
two-step $\Delta$-nucleon $(\Delta N \to K^+ Y N$) or pion-nucleon $(\pi N \to
K^+ Y $) production channels on the basis of a coupled-channel transport
approach (CBUU) including the kaon final-state-interactions (FSI).
Momentum-dependent potentials for the nucleon, hyperon and kaon in the final
state are included as well as $K^+$ elastic rescattering in the target nucleus.
The transport calculations are compared to the experimental $K^+$ spectra taken
at COSY-J\"ulich. Our systematic analysis of $K^+$ spectra from $^{12}C$,
$^{63}Cu$, $^{107}Ag$ and $^{197}Au$ targets as well as their momentum
differential ratios gives a repulsive $K^+$ potential of $20\pm 5 $ MeV at
normal nuclear matter density.
