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Generalized Quasi-Variational Inequalities Without Continuities

Authors:	P Cubiotti

Abstract:	Given a nonempty set $X \subseteq \mathbb{R}^n$ and two multifunctions $\beta :X \to 2^X ,\phi :X \to 2^{\mathbb{R}^n }$ , we consider the following generalized quasi-variational inequality problem associated with X, : Find $(\bar x,\bar z) \in X \times \mathbb{R}^n$ such that $\bar x \in \beta (\bar x),\bar z \in \phi (\bar x){\text{, and sup}}_{y \in \beta (\bar x)} \left\langle {\bar z,\bar x - y} \right\rangle \leqslant 0$ . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.

Keywords:	Generalized quasi-variational inequalities upper semicontinuous multifunctions lower semicontinuous multifunctions fixed points
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