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The main purpose of this paper is to study a class of boundary value problem governed by a fractional differential inclusion in a separable Banach space E D α u (t) + λ D α − 1 u (t) ∈ F (t , u (t) , D α − 1 u (t)) , t ∈ [ 0 , 1 ] I 0 + β u (t) t = 0 = 0 , u (1) = I 0 + γ u (1) $$\begin{array}{} \displaystyle \left\{ \begin{array}{lll} D ^\alpha u(t) +\lambda D^{\alpha-1 }u(t) \in F(t, u(t), D ^{\alpha-1}u(t)), \hskip 2pt t \in [0, 1] \\ I_{0^+}^{\beta }u(t)\left\vert _{t=0}\right. = 0, \quad u(1)=I_{0^+}^{\gamma }u(1) \end{array} \right. \end{array}$$ in both Bochner and Pettis settings, where α ∈ ]1, 2], β ∈ [0, 2 – α], λ ≥ 0, γ > 0 are given constants, Dα is the standard Riemann-Liouville fractional derivative, and F : [0, 1] × E × E → 2E is a closed valued multifunction. Topological properties of the solution set are presented. Applications to control problems and subdifferential operators are provided. [ABSTRACT FROM AUTHOR]