# American Institute of Mathematical Sciences

July  2002, 8(3): 519-562. doi: 10.3934/dcds.2002.8.519

## Eigenvalues for a class of homogeneous cone maps arising from max-plus operators

 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, United States 2 Department of Mathematics, Rutgers University, Piscataway, NJ 08854, United States

Received  March 2001 Revised  March 2002 Published  April 2002

We study the nonlinear eigenvalue problem $f(x) = \lambda x$ for a class of maps $f: K\to K$ which are homogeneous of degree one and order-preserving, where $K\subset X$ is a closed convex cone in a Banach space X. Solutions are obtained, in part, using a theory of the "cone spectral radius" which we develop. Principal technical tools are the generalized measure of noncompactness and related degree-theoretic techniques. We apply our results to a class of problems max

$\max_{t\in J(s)} a(s, t)x(t) = \lambda x(s)$

arising from so-called "max-plus operators," where we seek a nonnegative eigenfunction $x\in C[0, \mu]$ and eigenvalue $\lambda$. Here $J(s) = [\alpha(s), \beta(s)] \subset [0, \mu]$ for $s\in [0, \mu]$, with $a, \alpha$, and $\beta$ given functions, and the function $a$ nonnegative.

Citation: John Mallet-Paret, Roger D. Nussbaum. Eigenvalues for a class of homogeneous cone maps arising from max-plus operators. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 519-562. doi: 10.3934/dcds.2002.8.519
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