## Topological Methods in Nonlinear Analysis

### A second order differential inclusion with proximal normal cone in Banach spaces

#### Abstract

In the present paper we mainly consider the second order evolution inclusion with proximal normal cone: $$\begin{cases} -\ddot{x}(t)\in N_{K(t)}(\dot{x}(t))+F(t,x(t),\dot{x}(t)), \quad \text{a.e.}\\ \dot x(t)\in K(t),\\ x(0)=x_0,\quad\dot x(0)=u_0, \end{cases} \tag{*}$$ 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 $K(x(t))$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 44, Number 1 (2014), 143-160.

Dates
First available in Project Euclid: 11 April 2016

https://projecteuclid.org/euclid.tmna/1460381474

Mathematical Reviews number (MathSciNet)
MR3289012

Zentralblatt MATH identifier
1360.34132

#### Citation

Aliouane, Fatine; Azzam-Laouir, Dalila. A second order differential inclusion with proximal normal cone in Banach spaces. Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 143--160. https://projecteuclid.org/euclid.tmna/1460381474

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