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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.

Abstract. We study the initial-boundary value problem for a nonlinear wave equation given bywhere η ≥ 0, q ≥ 2 are given constants and u 0 , u 1 , g, k, f are given functions.In this paper, we consider two main parts. In Part 1, under a certain local Lipschitzian condition on f with (e u 0 , ea global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part 2, the asymptotic behavior of the solution u as t → +∞ is studied, under more restrictive conditions, namelye σt F (t) 2 dt < +∞, with σ > 0, and (e u 0 , e, and some others ( · denotes the L 2 (0, 1) norm). It is proved that under these conditions, a unique solution u(t) exists on R + such that u / (t) + ux(t) decay exponentially to 0 as t → +∞. Finally, we present some numerical results.