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On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions

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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→2X, 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.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC

    B. T. Kien, N. C. Wong & J. C. Yao

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  1. B. T. Kien
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  2. N. C. Wong
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  3. J. C. Yao
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Corresponding author

Correspondence to J. C. Yao.

Additional information

Communicated by X.Q. Yang.

The authors express their sincere gratitude to the referees for helpful suggestions and comments.

This research was partially supported by a grant from the National Science Council of Taiwan, ROC.

B.T. Kien was on leave from National University of Civil Engineering, Hanoi, Vietnam.

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Kien, B.T., Wong, N.C. & Yao, J.C. On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions. J Optim Theory Appl 135, 515–530 (2007). https://doi.org/10.1007/s10957-007-9239-4

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  • DOI: https://doi.org/10.1007/s10957-007-9239-4

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