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 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(\omega ) of a random closed cone K(\omega ) in a finite-dimensional linear space into the cone K(T\omega ). Under the 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.