About asymptotical properties of solutions of difference equations with random parameters

We investigate the asymptotic behavior of solutions of difference equations. Their right-hand sides at given time depend not only on the value of state at the previous moment, but also on a random value from a given set $\Omega$. We obtain conditions of Lyapunov stability and asymptotic stability of the equilibrium for all values of random parameters and with probability one. We show that the problem of coexistence of stochastic cycles of different periods has a solution, which strongly differs from a known Sharkovsky result for a determined difference equation. Under some conditions, the existence of a stochastic cycle of length $k$ implies the existence of a cycle of any length $\ell>k$.