# AN EXTENSION OF GENERALIZED VECTOR QUASI-VARIATIONAL INEQUALITY

• Kum Sang-Ho ;
• Kim Won-Kyu
• Published : 2006.04.01
• 70 19

#### Abstract

In this paper, we shall give an affirmative answer to the question raised by Kim and Tan [1] dealing with generalized vector quasi-variational inequalities which generalize many existence results on (VVI) and (GVQVI) in the literature. Using the maximal element theorem, we derive two theorems on the existence of weak solutions of (GVQVI), one theorem on the existence of strong solution of (GVQVI), and one theorem on strong solution in the 1-dimensional case.

#### Keywords

generalized vector quasi-variational inequality;equilibrium

#### References

1. N. Bourbaki, Topological Vector Spaces (translated by H. G. Eggleston and S. Madan), Springer-Verlag, New York, 1987
2. X. -P. Ding, W. K. Kim, and K. -K. Tan, Equilibria of non-compact generalized games with L$^*$-majorized preferences, J. Math. Anal. Appl. 164 (1992), 508-517 https://doi.org/10.1016/0022-247X(92)90130-6
3. H. Kneser, Sur un theoreme fondamental de la theorie des jeux, C. R. Acad. Sci. Paris 234 (1952), 2418-2420
4. P. Q. Khanh and L. M. Luu, On the existence of solutions to vector quasivariational inequalities and quasi-complementarity problems with applications to traffic network equilibria, J. Optim. Theory Appl. 123 (2004), 533-548 https://doi.org/10.1007/s10957-004-5722-3
5. Y. Chiang, O. Chadli, and J. C. Yao, Existence of solutions to implicit vector variational inequalities, J. Optim. Theory Appl, 116 (2003), 251-264 https://doi.org/10.1023/A:1022472103162
6. W. K. Kim and K. -K. Tan, On generalized vector quasi-variational inequalities, Optimization 46 (1999), 185-198 https://doi.org/10.1080/02331939908844451
7. X. -P. Ding, W. K. Kim, and K. -K. Tan, Equilibria of generalized games with L-majorized multifunctions, Int. J. Math. Math. Sci. 17 (1994), 783-790 https://doi.org/10.1155/S0161171294001092