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

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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 I k characterize the jump of the solutions at impulse points t k (\({k = 1, \ldots , m}\)). 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.

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Authors and Affiliations

  1. Department of Mathematics, Baylor University, Waco, Texas, 76798-7328, USA

    Johnny Henderson

  2. Laboratory of Mathematics, Sidi-Bel-Abbès University, P.O. Box 89, 22000, Sidi-Bel-Abbès, Algeria

    Abdelghani Ouahab

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Henderson, J., Ouahab, A. A Filippov’s Theorem, Some Existence Results and the Compactness of Solution Sets of Impulsive Fractional Order Differential Inclusions. Mediterr. J. Math. 9, 453–485 (2012). https://doi.org/10.1007/s00009-011-0141-9

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