Implicit subdifferential inclusions with nonconvex-valued perturbations

This paper addresses an evolution inclusion of subdifferential type with a multivalued perturbation. The values of the latter are closed, not necessarily convex sets. Our inclusion is implicit in the sense that the velocity enters it implicitly: the subdifferential is evaluated not at the state, but at a function depending both on the state and the velocity. We prove the existence of a solution to the inclusion by using a fixed point theorem for an auxiliary multivalued mapping with closed, nonconvex, decomposable values. This multivalued mapping is related to an ordinary differential equation containing the resolvent of the subdifferential operator. In the case when the perturbation is single-valued the solution is unique. We also introduce an explicit ordinary differential equation with the solution set coinciding with that for our implicit evolution inclusion. The application of our general result to implicit sweeping processes with nonconvex perturbations yields the existence of solutions to these processes generalizing a number of recent existence results for implicit sweeping processes. This existence result is further illustrated for a quasistatic evolution variational inequality arising in contact mechanics. Our results on implicit subdifferential inclusions are completely new and have no analogs in the existence literature.