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Generalized Quasi-Variational Inequalities Without Continuities

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  • 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, 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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  • P. Cubiotti, 1997. "Generalized Quasi-Variational Inequalities Without Continuities," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 477-495, March.
    • Handle: RePEc:spr:joptap:v:92:y:1997:i:3:d:10.1023_a:1022699205336
    DOI: 10.1023/A:1022699205336
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    References listed on IDEAS

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    1. D. Chan & J. S. Pang, 1982. "The Generalized Quasi-Variational Inequality Problem," Mathematics of Operations Research, INFORMS, vol. 7(2), pages 211-222, May.
    2. Romesh Saigal, 1976. "Extension of the Generalized Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 260-266, August.
    3. Jen-Chih Yao, 1995. "Generalized-Quasi-Variational Inequality Problems with Discontinuous Mappings," Mathematics of Operations Research, INFORMS, vol. 20(2), pages 465-478, May.
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    1. P. Cubiotti, 2003. "Existence Theorem for the Discontinuous Generalized Quasivariational Inequality Problem," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 623-633, December.
    2. P. H. Sach, 2008. "On a Class of Generalized Vector Quasiequilibrium Problems with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 337-350, November.
    3. R. P. Agarwal & M. Balaj & D. O’Regan, 2012. "A Unifying Approach to Variational Relation Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 417-429, November.
    4. M. Balaj & L. J. Lin, 2013. "Existence Criteria for the Solutions of Two Types of Variational Relation Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 232-246, February.
    5. M. Fakhar & J. Zafarani, 2008. "Generalized Symmetric Vector Quasiequilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 136(3), pages 397-409, March.
    6. B. T. Kien & N. C. Wong & J. C. Yao, 2007. "On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 515-530, December.

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