### 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 $K(x(t))$.

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 $K(x(t))$.

#### Keywords

Differential inclusion; uniformly smooth Banach space; sweeping process; proximal normal cone

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