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Existence Theorem for the Discontinuous Generalized Quasivariational Inequality Problem

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  • P. Cubiotti

    (University of Messina)

Abstract

We consider the following generalized quasivariational inequality problem: given a real Banach space E with topological dual E* and given two multifunctions G:X→2 X and F:X→2 E *, find $$(\hat x,\hat \varphi ) \in X \times E*$$ such that $$\hat x \in G(\hat x),{\text{ }}\hat \varphi \in F(\hat x),{\text{ }}\left\langle {\hat \varphi ,\hat x - y} \right\rangle \leqslant 0,{\text{ for all }}y \in G(\hat x).$$ We prove an existence theorem where F is not assumed to have any continuity or monotonicity property. Making use of a different technical construction, our result improves some aspects of a recent existence result (Theorem 3.1 of Ref. 1). In particular, the coercivity assumption of this latter result is weakened meaningfully.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:119:y:2003:i:3:d:10.1023_b:jota.0000006960.70743.63
    DOI: 10.1023/B:JOTA.0000006960.70743.63
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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, 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. 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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