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Existence results for fractional differential inclusions with multivalued term depending on lower-order derivative. (English) Zbl 1264.34010

The paper studies a fractional-order differential inclusion of the form

D C α x(t)F(t,x(t),D C β x(t))fora.e.([0,T])

subject to two classes of boundary conditions

(1) a 1 x(0)+b 1 D C γ x(0)=c 1 , a 2 x(T)+b 2 D C γ x(T)=c 2 ;

(2) a 1 x(0)+b 1 x(T)=c 1 , a 2 D C γ x(0)+b 2 D C γ x(T)=c 2 .

F:[0,T]××𝒫() is a set-valued map, D C α denotes the Caputo fractional derivative of order α, α(1,2], β(0,1], γ(0,1) and a i ,b i ,c i , i=1,2.

Three existence results are obtained for the problems considered. The first result relies on the nonlinear alternative of Leray-Schauder type, the second result essentially uses the Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and the third result is based on the Covitz-Nadler contraction principle for set-valued maps.

34A08Fractional differential equations
34A60Differential inclusions
47N20Applications of operator theory to differential and integral equations