The limits of maximal means and no autonomous differential inclusions

The existence theorem of the limit of the maximum average for almost periodic functions on a stand-alone solutions of no autonomous differential inclusion, the right part of which depends on the time periodically is proved.
The main condition is a condition of multiple attainability for a differential inclusion. This condition is satisfied, for example, for a constant right-hand side, which does not belong to the eigenspace of speeds. The result is related to the theory of averaging of differential inclusions with slow and fast variables.