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) - lambda y(t) is an element of F(t,y(t)), a.e. t is an element of J\{t(1),..., t(m)}, y(t(k)(+)) - y(t(k)(-)) = I-k (yt(k)(-))), k = 1, ..., m, y(0) = y(b), where J = [0,b] and F : J X R-n -> p(R-n) is a set-valued map. The functions I-k 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) is an element of phi (t,y(t)), a.e. t is an element of J\{t(1), ..., t(m)}, y(t(k)(+)) - y(t(k)(-)) = I-k(y(t(k)(-))), k = 1,..., m, y(0) = y(b), where phi : J X R-n -> is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincare operator.