Discrete-time strongly monotone dynamical systems on a Banach space
$X$ are considered. The main theorems give the following information:

- For any bounded set $B \subset X$, the minimal periods of stable periodic
solutions contained in $B$ are bounded above by a constant depending only
on $B$.

- This constant is not increased by a small $C^1$ perturbation of the map.

Applying the above theorems, various results on the asymptotic behavior
of solutions of periodic parabolic equations are obtained.