On Properties of Differential Inclusions with Prox-regular Sets

Abstract

In this paper, some regularity properties of solutions of the following differential inclusion \begin{equation}\nonumber \left\{ \begin{array}{l} \dot{x}(t) \in f\big(x(t)\big) -N_{C}\big(x(t)\big)\; {\rm a.e.} \; t \in [0,+\infty), x(0) = x_0\in C, \end{array}\right. \end{equation} are analyzed where $f: H\to H$ is Lipschitz continuous and $C$ is closed, uniformly prox-regular subset of a Hilbet space $H$. Here $N_{C}(\cdot)$ denotes the proximal normal cone of $C$. This work can be considered as an improvement of [Hantoute-Mazade 2013] since these properties are established without the additional tangential condition at each point in $C$.