On a fractional differential inclusion with integral boundary conditions in Banach space

Abstract

We consider a class of boundary value problem in a separable Banach space E, involving a nonlinear differential inclusion of fractional order with integral boundary conditions, of the form Dαu(t)∈F(t,u(t),Dα−1u(t)),a.e.,t∈[0,1],Iβu(t)|t=0=0,u(1)=∫01u(t)dt, (*) where Dα is the standard Riemann-Liouville fractional derivative, F is a closed valued mapping. Under suitable conditions we prove that the solutions set of (*) is nonempty and is a retract in WEα,1(I). An application in control theory is also provided by using the Young measures.