  We study Lipschitz, positively homogeneous and finite suprema preserving
mappings defined on a max-cone of positive elements in a normed vector lattice.
We prove that the lower spectral radius of such a mapping is always a minimum
value of its approximate point spectrum. We apply this result to show that the
spectral mapping theorem holds for the approximate point spectrum of such a
mapping. By applying this spectral mapping theorem we obtain new inequalites
for the Bonsall cone spectral radius of max type kernel operators.
