A Filippov%26apos;s Theorem, Some Existence Results and the Compactness of Solution Sets of Impulsive Fractional Order Differential Inclusions

作者：Henderson Johnny; Ouahab Abdelghani^*

来源：Mediterranean Journal of Mathematics, 2012, 9(3): 453-485.

DOI：10.1007/s00009-011-0141-9

• Fractional differential inclusions
• fractional derivative
• fractional integral
• Caputo fractional derivatives
• impulsive differential inclusions

In this paper, we first present an impulsive version of Filippov%26apos;s Theorem for fractional differential inclusions of the form, %26lt;br%26gt;D(*)(alpha)y(t) is an element of F(t, y(t)), a.e. t is an element of J\{t(1),..., t(m)}, alpha is an element of (0, 1], %26lt;br%26gt;y(t(k)(+)) - y(t(k)(-)) = I-k(y(t(k)(-))), k = 1,..., m, %26lt;br%26gt;y(0) = a, %26lt;br%26gt;where J = [0, b], 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) (). 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%26apos;s fixed point theorem for multivalued contractions. The compactness of solution sets is also investigated.

• 出版日期2012-8