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On the Discontinuous Infinite-Dimensional Generalized Quasivariational Inequality Problem

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  • P. Cubiotti

    (University of Messina)

Abstract

In this paper, we deal with the following generalized quasivariational inequality problem: given a real normed space E with topological dual E* and two multifunctions G: X→2 X and F: X→2 E*, find $$\left( {\hat x,\hat \phi } \right)$$ ∈X × E* such that $$\hat x \in G\left( {\hat x} \right),{\text{ }}\hat \phi \in F\left( {\hat x} \right),{\text{ }}\left\langle {\hat \phi ,\hat x - y} \right\rangle \leqslant 0,{\text{for all }}y \in G\left( {\hat x} \right).$$ We extend to such infinite-dimensional setting some existence results which have been obtained recently for the special case where E is finite dimensional. In particular, our assumptions do not imply any kind of continuity for the multifunction F.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:115:y:2002:i:1:d:10.1023_a:1019676929916
    DOI: 10.1023/A:1019676929916
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    Cited by:

    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. Paolo Cubiotti & Jen-Chih Yao, 2010. "Nash equilibria of generalized games in normed spaces without upper semicontinuity," Journal of Global Optimization, Springer, vol. 46(4), pages 509-519, April.
    3. S. Huang & J. C. Yao, 2006. "Technical Note Discontinuous Implicit Quasivariational Inequalities in Normed Spaces," Journal of Optimization Theory and Applications, Springer, vol. 129(1), pages 219-225, April.
    4. 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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