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

  • * Corresponding author: Samir Adly

    * Corresponding author: Samir Adly 
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  • 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.

    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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