Taiwanese Journal of Mathematics

EXISTENCE RESULTS FOR VECTOR SADDLE POINTS PROBLEMS

O. Chadli and H. Mahdioui

Full-text: Open access

Abstract

In this paper, we study the existence of solutions for vector saddle points problems in a general setting. Our approach, first, is based on the KKM lemma and a relaxation of the $C-$ lower semicontinuity notion introduced by T.Tanaka by means of an extension to the vector setting of a Brézis-Nirenberg-Stampacchia condition, and arguments from generalized convexity. This leads us to generalize and improve some new existence results on vector saddle points problems. In the second approach, we establish an existence result for vector saddle point problems under a paracompacity assumption.

Article information

Source
Taiwanese J. Math., Volume 16, Number 2 (2012), 429-444.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406594

Digital Object Identifier
doi:10.11650/twjm/1500406594

Mathematical Reviews number (MathSciNet)
MR2892891

Zentralblatt MATH identifier
1268.90124

Subjects
Primary: 90C47: Minimax problems [See also 49K35] 90C48: Programming in abstract spaces 49J35: Minimax problems

Keywords
vector saddle point C-lower semicontinuity C-quasiconvexity KKM lemma paracompacity

Citation

Chadli, O.; Mahdioui, H. EXISTENCE RESULTS FOR VECTOR SADDLE POINTS PROBLEMS. Taiwanese J. Math. 16 (2012), no. 2, 429--444. doi:10.11650/twjm/1500406594. https://projecteuclid.org/euclid.twjm/1500406594


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References

  • K. L. Chew, Maximal points with respect to cone dominance in banach spaces and their existence, J. Optim. Theory Appl., 44 (1984), 1-53.
  • P. L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl., 14 (1974), 319-377.
  • F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl., 60(1) (1989), 19-31.
  • K. R. Kazmi and S. Khan, Existence of solutions for a vector saddle point problem, Bull. Austral. Math. Soc., 61 (2000), 201-206.
  • K. Kimura, Existence results for cone saddle points by using Vector-variational-like inequalities, Nihonkai Math. J., 15(1) (2004), 23-32.
  • K. Kimura and T. Tanaka, Existence theorem of cone saddle-points applying a nonlinear scalarization, Taiwanese J. Math., 10(2) (2006), 563-571.
  • D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.
  • K. Kimura, On some types of vectorial saddle-point problems, Nihonkai Mathematical Journal, to appear.
  • W. Takahashi, Nonlinear Functional Analysis-Fixed Point Theory and its Applications, Yokohama-Publishers Yokohama, 2000.
  • P. Deguire, K. K. Tan and G. X.-Z. Yuan, The study of maximal elements, fixed points for $L_S$-majorized mappings and their applications to minimax and variational inequalities in product topological spaces, Nonlinear Analysis, 37(7) (1999), 933-951.
  • T. Tanaka, Generalized semicontinuity and existence theorems for cone saddle points, Appl. Math. Optim., 36 (1997), 313-322.
  • T. Tanaka, Cone-quasiconvexity of vector-valued functions, Sci. Rep. Hirosaki Univ., 42 (1995), 157-163.
  • T. Tanaka, Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued funtions, J. Optim. Theory Appl., 81 (1994), 355-357.
  • K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann., 142 (1961), 305-310.
  • H. Brézis, L. Nirenberg and G. Stampacchia, A ramark on Ky Fan's minmax principale, Boll. Un. Mat. Ital., 4 (1972), 293-300.
  • H. Brézis, Equations et inéquations non Linéaires dans les espaces vectoriels en Dualité, Ann. Inst. Fourier Grenoble, 18 (1968), 123-145.
  • O. Chadli, Y. Chiang and S. Huang, Topological pseudomonotonicity and vector-valued equilibrium problems, J. Math. Anal. Appl., 270 (2002), 435-450.
  • Q. H. Ansari and J. C. Yao, An existence result for the generalized vector equilibrium problem, Appl. Math. Lett., 128 (1999), 53-56.
  • O. Chadli, X. Q. Yang and J. C. Yao, On generalized vector pre-variational and pre-quasivariational inequalities, J. Math. Anal. Appl., 295 (2004), 392-403.
  • G. Chen, X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, 2005.
  • J. P. Aubin, Mathematical Methods of Game and Economic Theory, Studies in Mathematics and its Applications, Vol. 7, North Holland, 1979.
  • C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd. ed. Berlin, Springer-Verlag, 2006.
  • C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, 2002.
  • O. Chadli, Z. Chbani and H. Riahi, Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities, J. Optim. Theory Appl., 105 (2000), 299-323.