## Taiwanese Journal of Mathematics

### ON VECTOR EQUILIBRIUM PROBLEM WITH MULTIFUNCTIONS

#### Abstract

In this paper, a vector equilibrium problem (VEP) with multifunctions is considered. Using the asymptotic cone of the solution set of (VEP), we give conditions under which the solution set is nonempty and compact, and then extend it to a random vector equilibrium problem with multifunctions.

#### Article information

Source
Taiwanese J. Math., Volume 10, Number 2 (2006), 399-407.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500403832

Digital Object Identifier
doi:10.11650/twjm/1500403832

Mathematical Reviews number (MathSciNet)
MR2208274

Zentralblatt MATH identifier
1106.49015

#### Citation

Lee, Gue Myung; Bu, In Ja. ON VECTOR EQUILIBRIUM PROBLEM WITH MULTIFUNCTIONS. Taiwanese J. Math. 10 (2006), no. 2, 399--407. doi:10.11650/twjm/1500403832. https://projecteuclid.org/euclid.twjm/1500403832

#### References

• [1.] Q. H. Ansari and F. Flores-Baz$\acute {\rm a}$n, Recession methods for generalized vector equilibrium problems, preprint.
• [2.] Q. H. Ansari, I. V. Konnov and J. C. Yao, On generalized vector equilibrium problems, Nonlinear Analysis 47 (2001), 543-554.
• [3.] Q. H. Ansari, W. Oettli and D. Schl$\ddot a$ger, A generalization of vectorial equilibrium, Mathematical Methods of Operations Research 46 (1997), 147-152.
• [4.] Q. H. Ansari and J. C. Yao, An existence result for the generalized vector equilibrium problem, Applied Mathematics Letters 12 (1999), 53-56.
• [5.] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkh$\ddot a$ser Boston, 1990.
• [6.] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer-Verlag New York, Inc., 2003.
• [7.] Fabi$\acute {\rm a}$n Flores-Baz$\acute {\rm a}$n and Fernando Flores-Baz$\acute {\rm a}$n, Vector equilibrium problems under asymptotic analysis, Journal of Global Optimization 26 (2003), 141-166.
• [8.] P. G. Georgiev and T. Tanaka, Vector-valued set-valued variants of Ky Fan's inequality, Journal of Nonlinear and Convex Analysis 1 (2000), 245-254.
• [9.] E. M. Kalmoun, Some deterministic and random vector equilibrium problems, Journal of Mathematical Analysis and Applications 267(2002), 62-75.
• [10.] E. M. Kalmoun, From deterministic to random vector equilibrium problems, Journal of Nonlinear and Convex Analysis 4 (2003), 77-85.
• [11.] I. V. Konnov and J. C. Yao, Existence of solutions for generalized vector equilibrium problems, Journal of Mathematical Analysis and Applications 233 (1999), 328-335.
• [12.] G. M. Lee and S. Kum, Vector variational inequalities in a Hausdorff topological vector space in “Vector Variational Inequalities and Vector Equilibria", edited by F. Giannessi, Kluwer Academic Publishers, 2000, pp. 307-320.
• [13.] D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.
• [14.] W. Song, Vector equilibrium problems with set-valued mappings in “Vector Variational Inequalities and Vector Equilibria", Ed. F. Giannessi, Kluwer Academic Publishers, 2000, pp. 403-421.
• [15.] N. X. Tan, On the existence of solutions to systems of vector quasi-optimization problems, Mathematical Methods of Operations Research 60 (2004), 53-71.