The authors consider the Cauchy problem for a fractional impulsive differential inclusion:
the case of fractional differential equations and the following periodic one:
where denotes the Caputo fractional derivative (), and is a set-valued map. The functions characterize the jump of the solutions at impulse point (). Under Lipschitz and Nagumo-type growth conditions on the authors prove existence of solutions via fixed point theory for multivalued mappings. Also, they study the topological structure of solution sets (compactness, ; acyclicity; contractibility). The proofs use the general theory on topological structure of fixed point set for multi-valued operators. Note in case of first order and periodic differential inclusions, these results have been obtained in [S. Djebali, L. Górniewicz, A. Ouahab, Topol. Methods Nonlinear Anal. 32, No. 2, 261–312 (2008; Zbl 1182.34087)] for initial problems with delays and in [S. Djebali, L. Górniewicz, A. Ouahab, Math. Comput. Model. 52, 683–714 (2010)] for first-order periodic problems.