ON EVOLUTION QUASI-VARIATIONAL INEQUALITIES AND IMPLICIT STATE-DEPENDENT SWEEPING PROCESSES

International audienceIn this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint C(·, u) depends upon the unknown state u, which causes one of the main difficulties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a fixed point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set C does not depend explicitly on the state u (i.e. C := C(t)) given in [3], we prove a new existence result of solutions of the quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required