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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

John Mallet-Paret 1, and  Roger D. Nussbaum 2,

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.

Keywords: nonlinear eigenvalue problem, Cone, spectral radius., measure of noncompactness, max-plus operator.
Mathematics Subject Classification: 47H07, 47H09, 47H10, 47J1.
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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