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On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions

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  • B. T. Kien

    (National Sun Yat-Sen University)

  • N. C. Wong

    (National Sun Yat-Sen University)

  • J. C. Yao

    (National Sun Yat-Sen University)

Abstract

We study the following generalized quasivariational inequality problem: given a closed convex set X in a normed space E with the dual E *, a multifunction $\Phi :X\rightarrow 2^{E^{*}}$ and a multifunction Γ:X→2 X , find a point $(\hat{x},\hat{z})\in X\times E^{*}$ such that $\hat{x}\in \Gamma(\hat{x}),\hat{z}\in \Phi (\hat{x}),\langle \hat{z},\hat{x}-y\rangle \leq 0$ , $\forall y\in \Gamma(\hat{x})$ . We prove some existence theorems in which Φ may be discontinuous, X may be unbounded, and Γ is not assumed to be Hausdorff lower semicontinuous.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:135:y:2007:i:3:d:10.1007_s10957-007-9239-4
    DOI: 10.1007/s10957-007-9239-4
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    References listed on IDEAS

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    1. P. Cubiotti, 1997. "Generalized Quasi-Variational Inequalities Without Continuities," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 477-495, March.
    2. 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.
    3. Paolo Cubiotti & Jen-Chih Yao, 1997. "Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 213-228, June.
    4. Jen-Chih Yao, 1995. "Generalized-Quasi-Variational Inequality Problems with Discontinuous Mappings," Mathematics of Operations Research, INFORMS, vol. 20(2), pages 465-478, May.
    5. P. Cubiotti, 2002. "On the Discontinuous Infinite-Dimensional Generalized Quasivariational Inequality Problem," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 97-111, October.
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    Cited by:

    1. D. Aussel & J. Cotrina, 2013. "Quasimonotone Quasivariational Inequalities: Existence Results and Applications," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 637-652, September.
    2. Didier Aussel & Asrifa Sultana & Vellaichamy Vetrivel, 2016. "On the Existence of Projected Solutions of Quasi-Variational Inequalities and Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 818-837, September.
    3. C. S. Lalitha & Guneet Bhatia, 2011. "Stability of Parametric Quasivariational Inequality of the Minty Type," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 281-300, February.

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