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

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 (*)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left\{ \begin{gathered} D^\alpha u(t) \in F(t,u(t),D^{\alpha - 1} u(t)),a.e.,t \in [0,1], \hfill \\ I^\beta u(t)|_{t = 0} = 0,u(1) = \int\limits_0^1 {u(t)dt,} \hfill \\ \end{gathered} \right. $\end{document} 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.