Article Contents
Article Contents

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

• * Corresponding author: Samir Adly
• 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.

 Citation:

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

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