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

Samir Adly; Tahar Haddad

doi:10.3934/dcdss.2020105

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2020, Volume 13, Issue 6: 1791-1801. Doi: 10.3934/dcdss.2020105

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

Samir Adly^1, , and
Tahar Haddad^2,

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

^* Corresponding author: Samir Adly
^* Corresponding author: Samir Adly

Received: April 2018

Revised: December 2018

Published online: June 2020

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Abstract

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.

Keywords:

Moreau's sweeping process,
evolution quasi-variational inequalities,
state-dependent sweeping process.

Mathematics Subject Classification: Primary: 49J52, 34A60, 49J40; Secondary: 47J20, 34G25.

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Figure 1. The moving set $ C(u) $ of Example 1

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References

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