State dependent nonconvex sweeping processes in smooth Banach spaces

Abstract

In the setting of 2-uniformly smooth and $q$-uniformly convex Banach spaces, we prove the existence of solutions of the following multivalued differential equation:

This inclusion is called State Dependent Nonconvex Sweeping Process (SDNSP). Here $N^C(C(t,u(t));u(t))$ stands for the Clarke normal cone. The perturbed (SDNSPP) is also considered. Our results extend recent existing results from the setting of Hilbert spaces to the setting of Banach spaces. In our proofs we use some new results on $V$-uniformly generalized prox-regular sets in Banach spaces.