On Fractional Differential Inclusions with Nonlocal Boundary Conditions

Abstract

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.

Some results associated to first-order set-valued evolution problems with subdifferentials

“…Remark 5.1. The existence result in Theorem 5.1 may find applications in optimal control theory by using Young measures, Filippov's theorems and the problems with delay in the line of recent works related to fractional differential and integral equations with applications; see, e.g., [39,40,45]. This needs further investigations and may be the subject of forthcoming research.…”

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