# American Institute of Mathematical Sciences

## On evolution quasi-variational inequalities and implicit state-dependent sweeping processes

 1 Université de Limoges, Laboratoire XLIM UMR CNRS 7252,123, avenue Albert Thomas, 87060 Limoges, France 2 Laboratoire LMPEA, Faculté des Sciences Exactes et Informatique, Université de Jijel, B.P. 98, Jijel 18000, Algeria

Received  April 2018 Revised  December 2018 Published  September 2019

In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint ${C(\cdot, u)}$ 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 [4], 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.

Citation: Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020105
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##### References:
The moving set $C(u)$ of Example 1
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