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Closedness of the Solution Map in Quasivariational Inequalities of Ky Fan Type

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  • Massimiliano Giuli

    (University of L’Aquila)

Abstract

This paper is mainly concerned with the stability analysis of the set-valued solution mapping for a parametric quasivariational inequality of Ky Fan type. Perturbations are here considered both on the bifunction and on the constraint map which define the problem. The bifunction is assumed to be either pseudomonotone or quasimonotone. This fact leads to the definition of four different types of solution: two when the bifunction is pseudomonotone, and two for the quasimonotone case. These solution sets are connected each other through two Minty-type Lemmas, where a very weak form of continuity for the bifunction is employed. Using these results, we are able to establish some sufficient conditions, which ensure the closedness and the upper semicontinuity of the maps corresponding to the four solution sets.

Suggested Citation

  • Massimiliano Giuli, 2013. "Closedness of the Solution Map in Quasivariational Inequalities of Ky Fan Type," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 130-144, July.
    • Handle: RePEc:spr:joptap:v:158:y:2013:i:1:d:10.1007_s10957-012-0221-4
    DOI: 10.1007/s10957-012-0221-4
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    References listed on IDEAS

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    1. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    2. D. Aussel & J. Cotrina, 2011. "Semicontinuity of the solution map of quasivariational inequalities," Journal of Global Optimization, Springer, vol. 50(1), pages 93-105, May.
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    5. Nils Langenberg, 2012. "An Interior Proximal Method for a Class of Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 902-922, December.
    6. Lam Anh & Phan Khanh, 2010. "Continuity of solution maps of parametric quasiequilibrium problems," Journal of Global Optimization, Springer, vol. 46(2), pages 247-259, February.
    7. M. Bianchi & R. Pini, 2005. "Coercivity Conditions for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 79-92, January.
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