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 $(NNto N Y K^+$) 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-Julich. 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 $20pm 5 $ MeV at normal nuclear matter density.
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