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A second order differential inclusion with proximal normal cone in Banch spaces
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 , is a separable reflexive Banach space,
a ball compact and -prox-regular subset of , the proximal normal cone of and 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 is replaced by