  We provide conditions for the existence of measurable solutions to the
equation $\xi(T\omega)=f(\omega,\xi(\omega))$, where $T:\Omega
\rightarrow\Omega$ is an automorphism of the probability space $\Omega$ and
$f(\omega,\cdot)$ is a strictly non-expansive mapping. We use results of this
kind to establish a stochastic nonlinear analogue of the Perron-Frobenius
theorem on eigenvalues and eigenvectors of a positive matrix. We consider a
random mapping $D(\omega)$ of a random closed cone $K(\omega)$ in a
finite-dimensional linear space into the cone $K(T\omega)$. Under assumptions
of monotonicity and homogeneity of $D(\omega)$, we prove the existence of
scalar and vector measurable functions $\alpha(\omega)>0$ and $x(\omega)\in
K(\omega)$ satisfying the equation $\alpha(\omega)x(T\omega)=D(\omega
)x(\omega)$ almost surely.
