A second order differential inclusion with proximal normal cone in Banch spaces

Abstract

In the present paper we mainly
consider the second order evolution inclusion with proximal normal
cone:
$$
\begin{cases}
-\ddot{x}(t)\in N_{K(t)}(\dot{x}(t))+F(t,x(t),\dot{x}(t)), \quad \textmd{a.e.}\\
\dot x(t)\in K(t),\\
x(0)=x_0,\quad\dot x(0)=u_0,
\end{cases}
\leqno{(*)}
$$
where $t\in I=[0,T]$, $E$ is a separable reflexive Banach space, $K(t)$
a ball compact and $r$-prox-regular subset of $E$, $N_{K(t)}(\cdot)$ the proximal normal cone of $K(t)$ and $F$ an u.s.c. set-valued mapping
with nonempty closed convex values. First, we prove the existence of
solutions of $(*)$. After, we give an other existence result of
$(*)$ when $K(t)$ is replaced by $K(x(t))$.