## 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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### References:

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