Abstract
This paper aims to study the solvability of vector Ky Fan inequalities and the compactness of its solution sets. For vector-valued functions with the cone semicontinuity and the cone quasiconvexity in infinite dimensional spaces, the authors prove some existence results of the solutions and the compactness of the solution sets. Especially, some results for the vector Ky Fan inequalities on noncompact sets are built and the compactness of its solution sets are also discussed. As applications, some existence theorems of the solutions of vector variational inequalities are obtained.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Fan K, A minimax inequality and applications, Inequality III (ed. by Shisha O), Academic Press, New York, 1972, 103–113.
Blum E and Oettli W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 1994, 63(1–4): 123–145.
Tan K K, Yu J, and Yuan X Z, The stability of Ky Fan’s points, Proc. Amer. Math. Soc., 1995, 123(5): 1511–1519.
Ansari Q H, Vector equilibrium problems and vector variational inequalities, Presented at 2nd World Congress of Nonlinear Analysts, Athens, 1996.
Bianchi M, Hadjisawas N, and Schaible S, Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 1997, 92(3): 527–542.
Oettli W, A remark on vector valued equilibria and generalized monotonicity, Acta Math. Vietnamica, 1997, 22(1): 213–221.
Yang X Q and Goh C J, On vector variational inequalities application to vector equilibria, J. Optim. Theory Appl., 1997, 95(2): 431–443.
Chen G Y, Goh C J, and Yang X Q, Vector network equilibrium problems and nonlinear scalarization methods, Math. Meth. Oper. Res., 1999, 49: 239–253.
Giannessi F, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publication, London, 2000.
Giannessi F, Theorems of the alternative, quadratic programs, and complementarity problems, Variational Inequalities and Complementarity Problems (ed. by Cottle R W, Giannessi F, and Lions J L), John Wiley and Sons, New York, 1980, 151–186.
Chen G Y and Cheng G M, Vector Variational Inequality and Vector Optimization, Springer-Verlag, Berlin, 1987.
Luc D T, Theory of Vector Optimization, Springer-Verlag, Berlin, 1989.
Chen G Y, Huang X X, and Yang X Q, Vector Optimization: Set-Valued and Variational Analysis, Springer-Verlag, Berlin, 2005.
Peng D T, New existence theorem for vector equilibrium problem and its equivalent version with their applications, Acta Math. Sinica, Chinese Ser., 2009, 52(3): 441–450.
Yang H and Yu J, Essential components of the set of weakly Pareto-Nash equilibrium points, Appl. Math. Letter, 2002, 15: 553–560.
Yu J, Game Theory and Nonlinear Analysis, Science Press, Beijing, 2008 (in Chinese).
Fan K, A generalization of Tychonoff’s fixed-point theorem, Math. Ann., 1961, 142: 305–310.
Han J Y, Huang Z H, and Fang S C, Solvability of variational inequality problems, J. Opt. Theory Appl., 2004, 122(3): 501–520.
Han J Y, Xiu N H, and Qi H D, Theory and Algorithms for Nonlinear Complementary Problems, Shanghai Science and Technology Press, Shanghai, 2006 (in Chinese).
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Science and Technology Foundation of Guizhou Province under Grant No. 20102133.
This paper was recommended for publication by Editor WANG Shouyang.
Rights and permissions
About this article
Cite this article
Yu, J., Peng, D. Solvability of vector Ky Fan inequalities with applications. J Syst Sci Complex 26, 978–990 (2013). https://doi.org/10.1007/s11424-013-0284-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-013-0284-9