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.
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