## 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*→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.

This is a preview of subscription content,

to check access.### Similar content being viewed by others

## References

Chan, D., Pang, J.S.: The generalized quasivariational inequality problem. Math. Oper. Res.

**7**, 211–222 (1982)Cubiotti, P.: Finite-dimensional quasivariational inequalities associated with discontinuous functions. J. Optim. Theory Appl.

**72**, 577–582 (1992)Cubiotti, P.: An existence theorem for generalized quasivariational inequalities. Set-Valued Anal.

**1**, 81–87 (1993)Cubiotti, P., Yen, N.D.: A result related to Ricceri’s conjecture on generalized quasivariational inequalities. Arch. Math.

**69**, 507–514 (1997)Cubiotti, P.: Generalized quasivariational inequalities without continuities. J. Optim. Theory Appl.

**92**, 477–495 (1997)Cubiotti, P.: On the discontinuous infinite-dimensional generalized quasivariational inequality problem. J. Optim. Theory Appl.

**115**, 97–111 (2002)Cubiotti, P.: Existence theorem for the discontinuous generalized quasivariational inequality problem. J. Optim. Theory Appl.

**119**, 623–633 (2003)Cubiotti, P., Yao, J.C.: Discontinuous implicit quasivariational inequalities with applications to fuzzy mappings. Math. Methods Oper. Res.

**46**, 213–328 (1997)Yao, J.C., Guo, J.S.: Variational and generalized variational inequalities with discontinuous mappings. J. Math. Anal. Appl.

**182**, 371–382 (1994)Yao, J.C.: Generalized quasivariational inequality problems with discontinuous mappings. Math. Oper. Res.

**20**, 465–478 (1995)Yao, J.C., Guo, J.S.: Extension of strongly nonlinear quasivariational inequalities. Appl. Math. Lett.

**5**, 35–38 (1992)Yen, N.D.: On an existence theorem for generalized quasivariational inequalities. Set-Valued Anal.

**3**, 1–10 (1995)Yen, N.D.: On a class of discontinuous vector-valued functions and the associated quasivariational inequalities. Optimization

**30**, 197–202 (1994)Ricceri, O.N.: On the covering dimension of the fixed-point set of certain multifunctions. Comment. Math. Univ. Carolinae

**32**, 281–286 (1991)Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Basel (1990)

Marano, S.A.: Controllability of partial differential inclusions depending on a parameter and distributed-parameter control processes. Le Matematiche

**45**, 283–300 (1960)Michael, E.: Continuous Selections I. In: Annals of Mathematics, vol. 63, pp. 361–382 (1956)

## Author information

### Authors and Affiliations

### Corresponding author

## 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.

## Rights and permissions

## About this article

### Cite this article

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

Published:

Issue Date:

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