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Implicit dynamical systems and quasi variational inequalities. (English) Zbl 1032.47041

The author considers the problem of finding an element \(u \in K(u)\) such that \(\langle A(u),v-u \rangle \geq 0\) for all \(v \in K(u)\). Here, \(A: \mathbb{R}^n \to \mathbb{R}^n\) is a nonlinear operator and \(K: u \mapsto K(u)\) a point-to-set mapping with closed convex \(K(u) \subset \mathbb{R}^n\) for all \(u \in \mathbb{R}^n\). The original inequality is approximated by the dynamical system \(du/dt=\lambda \{ P_{K(u)} [u-\rho A(u)]-u \}\), \(u(t_0)=u_0\), where \(P_K\) stands for the projection from \(H\) onto \(K \subset H\) and \(\rho>0\). The author proves that the system is stable and the trajectory \(u(t)\) globally converges as \(t\to +\infty\) to the solution subset of the inequality provided that the operator \(A\) is pseudomonotone and \(\|P_{K(u)}w-P_{K(v)}w \|\leq \nu \|u-v\|\) for all \(u,v,w \in H\).

MSC:

47J20 Variational and other types of inequalities involving nonlinear operators (general)
34G20 Nonlinear differential equations in abstract spaces
49J40 Variational inequalities
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