A coupled system consisting of an evolution inclusion with maximal monotone operators and a prox-regular sweeping process

A coupled system of an evolution inclusion and a sweeping process is considered in a separable Hilbert space. The evolution inclusion is described in terms of maximal monotone operators depending on time and the state variables of both the inclusion and the sweeping process. It involves a multivalued perturbation with nonconvex closed values. In the sweeping process the moving sets are prox-regular and the perturbation is single-valued. The perturbations in the evolution inclusion and sweeping process are related. An existence theorem is proved for absolutely continuous solutions. As a corollary, an existence theorem if proved for an evolution inclusion with maximal monotone operators. The compactness of the set of solutions for convex-valued perturbations is established for the first time. The proof is based on the author's comparison theorem for evolution inclusions with maximal monotone operators and Fan's fixed point theorem as applied to a direct product of multivalued maps. This approach made it possible to obtain new results.
Bibliography: 39 titles.