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 formwhere D α 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 W α,1 E (I). An application in control theory is also provided by using the Young measures.

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 $$\begin{array}{} \displaystyle \left\{ \begin{array}{lll} D ^\alpha u(t) +\lambda D^{\alpha-1 }u(t) \in F(t, u(t), D ^{\alpha-1}u(t)), \hskip 2pt t \in [0, 1] \\ I_{0^+}^{\beta }u(t)\left\vert _{t=0}\right. = 0, \quad u(1)=I_{0^+}^{\gamma }u(1) \end{array} \right. \end{array}$$ 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.

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We study the existence of solutions to a nonlinear fractional differential equation in Hilbert spaces associated with three-point boundary conditions at resonanceby using Mawhin's continuation theorem. We propose a new technique to improve the conditions on A which have been used previously. In addition, a necessary and sufficient condition for that the fractional differential operator is Fredholm with zero-index is established, especially for the first time when the fractional differential operator takes values in an arbitrary Hilbert space.

This paper is devoted to the study of the following nonlocal p -Laplacian functional differential equationsubject to multi point boundary conditions. We obtain some results on the existence of at least one (when n ∈ Z + ) or triple (when n = 0 ) pseudo-symmetric positive solutions by using fixedpoint theory in cone.