State-dependent prox-regular sweeping process with a general nonconvex composed perturbation

Abstract

We consider a sweeping process with a triple perturbation defined on a separable Hilbert space. The values of the moving set are time- and state-dependent prox-regular sets. The perturbation is given by the sum of three multivalued mappings having different semicontinuity properties with respect to the state variable. The first mapping with closed, possibly, nonconvex values is lower semicontinuous. The second one with closed convex values has weakly sequentially closed graph. The values of the third mapping can be both convex and nonconvex closed sets. This mapping has closed graph at the points where it is convex-valued. At a point therein its value is a nonconvex set, the mapping is lower semicontinuous on a neighborhood of this point. Usually, the latter mapping is called a mapping with mixed semicontinuity properties. We prove the existence of a solution to our sweeping process. To this aim, we propose a new method that is not related to the catching-up algorithm or its modifications often used in the existence proofs for sweeping processes. We use classical approaches based on a priori estimates and a fixed-point argument for multivalued mappings. Our existence result is completely new and it implies the existing results for the considered class of sweeping processes with state-dependent moving sets.