BV Sweeping Process Involving Prox-Regular Sets and a Composed Perturbation

Abstract

In this paper we study the existence and properties of solutions for a discontinuous sweeping process involving prox-regular sets in a Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is the sum of two multivalued mappings. The values of the first one are closed, bounded, not necessarily convex sets. It is measurable in the time variable, Lipschitz continuous in the phase variable, and it satisfies a conventional growth condition. The values of the second one are closed, convex, not necessarily bounded sets. We assume that this mapping has a closed with respect to the phase variable graph.

Other assumptions concern the intersection of the second mapping with the multivalued mapping defined by the growth conditions. We suppose that this intersection has a measurable selector and it possesses some compactness properties.

We prove the existence of right-continuous solutions of bounded variation for our inclusion. If the values of the first inclusion are closed convex sets, then the solution set is a closed subset of the space of right-continuous functions of bounded variation with sup-norm. If, in addition, the values of the moving sets are compact sets, then the solution set is compact in the space of right-continuous functions of bounded variation endowed with the topology of uniform convergence on an interval.

The proofs are based on the author’s theorem on continuous with respect to a parameter selectors passing through fixed points of contraction multivalued maps with closed, nonconvex, decomposable values depending on the parameter and some compactness criteria (an analog of the Arzelà–Ascoli theorem) for sets in the space of right-continuous functions of bounded variation with sup-norm. The classical Ky Fan fixed point theorem is also used. The results that we obtain are new.