Eigenvalues for a class of homogeneous cone maps arising from maxplus operators doi:10.3934/dcds.2002.8.519 Abstract Full Text (409.9K) Related Articles
John MalletParet  Division of Applied Mathematics, Brown University, Providence, RI 02912, 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 orderpreserving, 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 degreetheoretic 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 socalled "maxplus 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, maxplus operator, measure of noncompactness, nonlinear eigenvalue
problem, spectral radius.
Received: March 2001; Revised: March 2002; Published: April 2002. 
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