# American Institute of Mathematical Sciences

October  2017, 37(10): 5337-5354. doi: 10.3934/dcds.2017232

## On the Bonsall cone spectral radius and the approximate point spectrum

 1 Institute of Mathematics, Czech Academy of Sciences, Žitna 25, 115 67 Prague, Czech Republic 2 Faculty of Mechanical Engineering, University of Ljubljana Aškerčeva 6, SI-1000 Ljubljana, Slovenia 3 Institute of Mathematics, Physics and Mechanics Jadranska 19, SI-1000 Ljubljana, Slovenia

* Corresponding author: Aljoša Peperko

Received  March 2017 Revised  May 2017 Published  June 2017

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.

Citation: Vladimir Müller, Aljoša Peperko. On the Bonsall cone spectral radius and the approximate point spectrum. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5337-5354. doi: 10.3934/dcds.2017232
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