Impulsive neutral functional differential inclusions in Banach spaces

In this paper, we first present an impulsive version of Filippov’s Theorem for first-order neutral functional differential inclusions of the form, d dt [y(t)− g(t, yt)] ∈ F (t, yt), a.e. t ∈ J\{t1, . . . , tm}, y(t+k )− y(tk ) = Ik(y(tk )), k = 1, . . . , m, y(t) = φ(t), t ∈ [−r, 0], where J = [0, b], F is a set-valued map and g is a single-valued function. The functions Ik characterize the jump of the solutions at impulse points tk (k = 1, . . . , m). Then the convexified problem is considered and a Filippov-Ważewski result is proved. After several existence results, the topological structure of solution sets is also investigated. Some results from topological fixed point theory together with notions of measure of noncompactness are used. Finally, some geometric properties of solution sets are obtained. Applications to a problem from control theory are provided.