On evolution equations having hypomonotonicities of opposite sign governed by sweeping processes (Q1730404)

In this paper, the authors study the local existence of solutions for a class of evolution equations of the form \[ \dot x(t)\in -N_C(x(t))+F(x(t)), \quad x(0)=x_0\in C, \quad\text{a.e. } t\in [0,T], \] defined by a sweeping process involving a locally prox-regular set of constraints and an upper semicontinuous set-valued perturbation contained in the Clarke subdifferential of a nonconvex function. Using the existence and uniqueness of solutions for single-valued perturbation, a discretization method for the existence of solutions is provided, when the fixed set of constraints is possibly noncompact and satisfies a weaker assumption than uniform prox-regularity. An extension of the existence result to the case of shifted moving set is also discussed. Finally, the paper closes with some open problems.