Periodic solutions for nonlinear differential inclusions with multivalued perturbations

In this paper, the periodic solutions for nonlinear differential inclusion governed by convex subdifferential and different perturbations are studied. It is firstly proved that the differential inclusion has unique periodic solution, if the perturbation function is a single-valued function. Then, by Schauder's fixed point theorem and Kakutani's fixed point theorem, we prove that the differential inclusion has at least a periodic solution, when the perturbation function is an upper semicontinuous (or lower semicontinuous) multifunction. Moreover, the existence of the extremal solution for the differential inclusion is also studied. Finally, based on one-sided Lipschitz (OSL) assumption, we prove the related relaxation theorem.