Convex Sweeping Processes with Noncompact Perturbations and with Delay in Banach Spaces

Abstract

We prove two results concerning the existence of solutions for functional differential inclusions that are governed by sweeping processes, with noncompact valued perturbations in Banach spaces. Indeed, we have two goals. The first is to give a technique that allows considering sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time. The second is to give a technique to overcome the arising problem from the nonlinearity of the normalized mappings, when we deal with sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time and state.

State-Dependent Delayed Sweeping Process With a Noncompact Perturbation in Banach Spaces

“…In [17], Ibrahim and AL-Adsani, considered the following sweeping process with noncompact valued perturbation and with delay            u(t) = ψ(t) for t ∈ [−r, 0]; u(t) = ψ(0) + t 0 u (s)ds, for t ∈ I = [0, T ]; u(t) ∈ C(t), for t ∈ I; u * (t) = J(u(t)), for t ∈ I; (u * ) (t) ∈ −N C(t) (u(t) + F (t, τ (t)u), a.e. for t ∈ I,…”

“…Motivated by the above studies, and inspired by [5] and [17], in this paper, we give an existence result for another sweeping process with a noncompact perturbation in Banach spaces. Indeed, we find the sufficient conditions that guarantee the existence of two continuous functions u : [−r, T ] → X and w : [−r, T ] → X, I = [0, T ] (T > 0) such that u and w are absolutely continuous functions on I and that the following M.S.…”