First Order Impulsive Differential Inclusions with Periodic Conditions

In this paper, we present an impulsive version of Filippov's Theorem for the first-order nonresonance impulsive differential inclusion y (t) − λy(t) ∈ F (t, y(t)), a.e. characterize the jump of the solutions at impulse points t k (k = 1,. .. , m.). Then the relaxed problem is considered and a Filippov-Wasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion y (t) ∈ ϕ(t, y(t)), a.e. t ∈ J\{t 1 ,. .. , t m }, y(t + k) − y(t − k) = I k (y(t − k)), k = 1,. .. , m, y(0) = y(b), where ϕ : J × R n → P(R n) is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator.