  We study the Bonsall cone spectral radius and the approximate point spectrum
of (in general non-linear) positively homogeneous, bounded and supremum
preserving maps, defined on a max-cone in a given normed vector lattice. We
prove that the Bonsall cone spectral radius of such maps is always included in
its approximate point spectrum. Moreover, the approximate point spectrum always
contains a (possibly trivial) interval. Our results apply to a large class of
(nonlinear) max-type operators.
  We also generalize a known result that the spectral radius of a positive
(linear) operator on a Banach lattice is contained in the approximate point
spectrum. Under additional generalized compactness type assumptions our results
imply Krein-Rutman type results.
