Impulsive differential inclusions with fractional order

In this paper, we first present an impulsive version of the Filippov Wa»ewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form D y.t/ 2 F .t; y.t//; a.e. t 2 J n ft1; : : : ; tmg; 2 .1; 2U; y.tC k / D Ik.y.t􀀀 k //; k D 1; : : : ;m; y0.tC k / D Ik.y.t􀀀 k //; k D 1; : : : ;m; y.0/ D a; y0.0/ D c; where J D T0; bU; D denotes the Caputo fractional derivative, and F is a set-valued map. The functions Ik; Ik characterize the jump of the solutions at impulse points tk .k D 1; : : : ;m/. Additional existence results are obtained under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on the nonlinear alternative of Leray Schauder type, a Bressan Colombo selection theorem, and Covitz and Nadlerʹs fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, R sets, acyclicity and contractibility, corresponding to Aronszajn Browder Gupta type results, are obtained. We also consider the impulsive fractional differential equations. D y.t/ D f .t; y.t//; a.e. t 2 J n ft1; : : : ; tmg; 2 .1; 2U; y.tC k / D Ik.y.t􀀀 k //; k D 1; : : : ;m; y0.tC k / D NIk.y.t􀀀 k //; k D 1; : : : ;m; y.0/ D a; y0.0/ D c; and D y.t/ D f .t; y.t//; a.e. t 2 J n ft1; : : : ; tmg; 2 .0; 1U; y.tC k / D Ik.y.t􀀀 k //; k D 1; : : : ;m; y.0/ D a; where f V J R ! R is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions,D y.t/ 2 ʹ.t; y.t//; a.e. t 2 J n ft1; : : : ; tmg; 2 .1; 2U; y.tC k / D Ik.y.t􀀀 k //; k D 1; : : : ;m; y0.tC k / D Ik.y.t􀀀 k //; k D 1; : : : ;m; y.0/ D y.b/; y0.0/ D y0.b/; where ʹ V J R ! P.R/ is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator.