Abstract
In this paper, we first present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form,
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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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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DOI: https://doi.org/10.1007/s00009-011-0141-9