State-Dependent Sweeping Processes with Stieltjes Derivative

Abstract

We prove the existence of solutions for a perturbed differential inclusion governed by a sweeping process with state dependent convex moving set

$$\begin{aligned}\left\{ \begin{array}{l} -u'_g(t)\in N_{C(t,u(t))}(u(t))+F(t,u(t)),\; \mu _g-a.e. \; t\in (0,T]\\ u(0)=u_0\in C(0,u_0). \end{array} \right. \end{aligned}$$

The novelty brought by our study is the involvement of the Stieltjes derivative \(u'_g\) with respect to a right-continuous nondecreasing function \(g:[0,T]\rightarrow {\mathbb {R}}\), thus establishing a very wide framework containing ODEs, impulsive differential problems, dynamic inclusions on time scales or generalized differential problems. Here \(\mu _g\) is the Stieltjes measure associated to g and \(N_{C(t,u(t))}(u(t))\) denotes the normal cone of C(t, u(t)) at the point u(t).