A Filippov’s Theorem, Some Existence Results and the Compactness of Solution Sets of Impulsive Fractional Order Differential Inclusions

Abstract

In this paper, we first present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form, $$\begin{array}{lll} \quad \qquad D^{\alpha}_{*}y(t) & \in & F(t, y(t)), \quad\; {\rm a.e.}\ t\, \in \, J{\backslash} \{t_{1}, \ldots, t_{m}\}, \ \alpha\, \in \, (0,1], \\ y(t^{+}_{k}) - y(t^{-}_{k}) & = & I_{k}(y(t^{-}_{k})), \quad k = 1, \ldots, m, \\ \qquad \qquad y(0) & = & a,\end{array}$$ where J = [0, b], \({D^{\alpha}_{*}}\) denotes the Caputo fractional derivative and F is a set-valued map. The functions Ikcharacterize the jump of the solutions at impulse points tk (\({k = 1, \ldots , m}\)). In addition, several existence results are established, under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on a nonlinear alternative of Leray-Schauder type and on Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of solution sets is also investigated.