Topological Properties of Solution Sets for Sweeping Processes with Delay

Charles Castaing and Manuel D.P. Monteiro Marques

Abstract: Let $r>0$ be a finite delay and ${\cal C}_0={\cal C}([-r,0],H)$ the Banach space of continuous vector-valued functions defined on $[-r,0]$ taking values in a real separable Hilbert space $H$. This paper is concerned with topological properties of solution sets for the functional differential inclusion of sweeping process type: $$ {du\over dt} \in -N_{K(t)}(u(t))+F(t,u_t), $$ where $K$ is a $\gamma$-Lipschitzean multifunction from $[0,T]$ to the set of nonempty compact convex subsets of $H\!$, $N_{K(t)}(u(t))$ is the normal cone to $K(t)$ at $u(t)$ and $F:[0,T]\times{\cal C}_0\rightarrow H$ is an upper semicontinuous convex weakly compact valued multifunction. As an application, we obtain periodic solutions to such functional differential inclusions, when $K$ is $T$-periodic, i.e. when $K(0)=K(T)$ with $T\geq r$.

Keywords: Functional differential inclusion; normal cone; Lipschitzean mapping; sweeping process; perturbation; delay.

Classification (MSC2000): 35K22, 34A60

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