Coexistence of periodic solutions with various periods of impulsive differential equations and inclusions on tori via Poincare operators

The coexistence of subharmonic periodic solutions of various orders is investigated to the first-order vector system of impulsive (upper-) Caratheodory differential equations and inclusions on tori. As the main tool, our recent Sharkovsky-type results for multivalued maps on tori are applied via the associated Poincare translation operators along the trajectories of given systems. The solvability criteria are formulated, under natural bi-periodicity assumptions imposed on the right-hand sides, in terms of the Lefschetz numbers of admissible impulsive maps. Since the criteria become effective on the circle, the main general theorem can be improved and reformulated there in a more transparent way. The obtained results can be regarded in a certain sense as a nontrivial extension of those due to Poincare [28], Denjoy [17] and van Kampen [24]. (C) 2019 Published by Elsevier B.V.