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Generalized Quasi-Variational Inequalities Without Continuities

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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, β φ: 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 φ is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case β(x(≡X.

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  1. P. Cubiotti
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Cubiotti, P. Generalized Quasi-Variational Inequalities Without Continuities. Journal of Optimization Theory and Applications 92, 477–495 (1997). https://doi.org/10.1023/A:1022699205336

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