Journal of Optimization Theory and Applications

, Volume 135, Issue 3, pp 515–530 | Cite as

On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions

  • B. T. Kien
  • N. C. Wong
  • J. C. YaoEmail author


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.


Generalized quasivariational inequalities Lower semicontinuity Hausdorff upper semicontinuity Hausdorff lower semicontinuity Multifunctions Closed graphs Open graphs 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan, ROC

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