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Several notions of spectral radius arise in the study of nonlinearorder-preserving positively homogeneous self-maps of cones in Banach spaces. Wegive conditions that guarantee that all these notions lead to the same value.In particular, we give a Collatz-Wielandt type formula, which characterizes thegrowth rate of the orbits in terms of eigenvectors in the closed cone orsuper-eigenvectors in the interior of the cone. This characterization holdswhen the cone is normal and when a quasi-compactness condition, involving anessential spectral radius defined in terms of $k$-set-contractions, issatisfied. Some fixed point theorems for non-linear maps on cones are derivedas intermediate results. We finally apply these results to show that non-linearspectral radii commute with respect to suprema and infima of families of orderpreserving maps satisfying selection properties.

A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones