Impulsive differential inclusions with fractional order

Abstract

In this paper, we first present an impulsive version of the Filippov–Ważewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form D∗αy(t)∈F(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, where J=[0,b],D∗α denotes the Caputo fractional derivative, and F is a set-valued map. The functions Ik,I¯k characterize the jump of the solutions at impulse points tk(k=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 Nadler’s fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, Rδ sets, acyclicity and contractibility, corresponding to Aronszajn–Browder–Gupta type results, are obtained. We also consider the impulsive fractional differential equations D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=Īk(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, and D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(0,1],y(tk+)=Ik(y(tk−)),k=1,…,m,y(0)=a, where f:J×R→R is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions, D∗αy(t)∈φ(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=y(b),y′(0)=y′(b), where φ: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.