Set-valued perturbation to second-order evolution problems with time-dependent subdifferential operators

The main purpose of this work is to study the existence of solutions for a perturbed second-order evolution inclusion involving time-dependent subdifferential operators. Under suitable conditions on the set-valued perturbation, the main result of the paper is proved in the context of a separable Hilbert space. A second-order evolution quasi-variational inequality is also investigated.

On fractional differential inclusions with Nonlocal boundary conditions

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.

Subdifferential inclusions for stress formulations of unilateral contact problems

We consider two classes of inclusions involving subdifferential operators, both in the sense of Clarke and in the sense of convex analysis. An inclusion that belongs to the first class is stationary while an inclusion that belongs to the second class is history-dependent. For each class, we prove existence and uniqueness of the solution. The proofs are based on arguments of pseudomonotonicity and fixed points in reflexive Banach spaces. Then we consider two mathematical models that describe the frictionless unilateral contact of a deformable body with a foundation. The constitutive law of the material is expressed in terms of a subdifferential of a nonconvex potential function and, in the second model, involves a memory term. For each model, we list assumptions on the data and derive a variational formulation, expressed in terms of a multivalued variational inequality for the stress tensor. Then we use our abstract existence and uniqueness results on the subdifferential inclusions and prove the unique weak solvability of each contact model. We end this paper with some examples of one-dimensional constitutive laws for which our results can be applied.