We provide conditions for the existence of measurable solutions to the equation
ξ
(
T
ω
)
=
f
(
ω
,
ξ
(
ω
)
)
\xi (T\omega )=f(\omega ,\xi (\omega ))
, where
T
:
Ω
→
Ω
T:\Omega \rightarrow \Omega
is an automorphism of the probability space
Ω
\Omega
and
f
(
ω
,
⋅
)
f(\omega ,\cdot )
is a strictly nonexpansive 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
(
ω
)
D(\omega )
of a random closed cone
K
(
ω
)
K(\omega )
in a finite-dimensional linear space into the cone
K
(
T
ω
)
K(T\omega )
. Under the assumptions of monotonicity and homogeneity of
D
(
ω
)
D(\omega )
, we prove the existence of scalar and vector measurable functions
α
(
ω
)
>
0
\alpha (\omega )>0
and
x
(
ω
)
∈
K
(
ω
)
x(\omega )\in K(\omega )
satisfying the equation
α
(
ω
)
x
(
T
ω
)
=
D
(
ω
)
x
(
ω
)
\alpha (\omega )x(T\omega )=D(\omega )x(\omega )
almost surely.