Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 519 - 562, Volume 8, Issue 3, July 2002

doi:10.3934/dcds.2002.8.519       Abstract        Full Text (409.9K)       Related Articles

John Mallet-Paret - Division of Applied Mathematics, Brown University, Providence, RI 02912, United States (email)
Roger D. Nussbaum - Department of Mathematics, Rutgers University, Piscataway, NJ 08854, United States (email)

Abstract: 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:  Cone, max-plus operator, measure of noncompactness, nonlinear eigenvalue problem, spectral radius.
Mathematics Subject Classification:  47H07, 47H09, 47H10, 47J10.

Received: March 2001;      Revised: March 2002;      Published: April 2002.