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Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 68))

Abstract

In this paper, we consider the vector quasi-equilibrium problem and prove some existence results for its solution with or without generalized pseudomonotonicity assumption. As consequences of our results, we also derive some existence results for a solution to the vector quasi-optimization problem, vector quasi-saddle point problem and vector quasi-variational inequality problem.

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References

  1. Q.H. Ansari, Vector Equilibrium Problems and Vector Variational Inequalities, In [19], pp. 1–16, 2000.

    Google Scholar 

  2. Q.H. Ansari, W. Oettli and D. Schläger, A Generalization of Vectorial Equilibria, Mathematical Methods of Operations Research, Vol. 46, pp. 147–152, 1997.

    Article  MathSciNet  Google Scholar 

  3. Q.H. Ansari, N.-C. Wong and J.-C. Yao, The Existence of Nonlinear Inequalities, Applied Mathematics Letters, Vol. 12, pp. 89–92, 1999.

    Article  MathSciNet  Google Scholar 

  4. Q.H. Ansari and J.-C. Yao, A Fixed Point Theorem and Its Applications to the System of Variational Inequalities, Bulletin of the Australian Mathematical Society, Vol. 59, pp. 433–442, 1999.

    Article  MathSciNet  Google Scholar 

  5. A.S. Antipin, On Convergence of Proximal Methods to Fixed Point of Extremal Mappings and Estimates of Their Rate of Convergence, Computational Mathematics and Mathematical Physics, Vol. 35, pp. 539–551, 1995.

    MathSciNet  Google Scholar 

  6. J.-P. Aubin, L’Analyse Non Linéaire et Ses Motivations Économiques, Masson, Paris, 1984.

    Google Scholar 

  7. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, John Wiley & Sons, New York, 1984.

    Google Scholar 

  8. M. Bianchi, N. Hadjisavvas and S. Schaible, Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 527–542, 1997.

    Article  MathSciNet  Google Scholar 

  9. M. Bianchi and S. Schaible, Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31–43, 1996.

    Article  MathSciNet  Google Scholar 

  10. E. Blum and W. Oettli, From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Student, Vol. 63, pp. 123–145, 1994.

    MathSciNet  Google Scholar 

  11. H. Brézis, L. Nirenberg and G. Stampacchia, A Remark on Ky Fan’s Minimax Principle, Bolletin Uni. Mat. Italiana, Vol. 6(4), pp. 293–300, 1972.

    Google Scholar 

  12. K.C. Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, U.K., 1985.

    Book  Google Scholar 

  13. O. Chadli, Z. Chbani and H. Riahi, Recession Methods for Equilibrium Problems and Applications to Variational and Hemivariational Inequalities, Discret and Continuous Dynamical Systems, Vol. 5, pp. 185–195, 1999.

    MathSciNet  Google Scholar 

  14. O. Chadli, Z. Chbani and H. Riahi, Equilibrium Problems and Non-coercive Variational Inequalities, Optimization, Vol. 49, pp. 1–12, 1999.

    Google Scholar 

  15. O. Chadli, Z. Chbani and H. Riahi, Equilibrium problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 105, pp. 299–323, 2000.

    Article  MathSciNet  Google Scholar 

  16. N.H. Dien, Some Remarks on Variational-like and Quasivariationallike Inequalities, Bulletin of the Australian Mathematical Society, Vol. 46, pp. 335–342, 1992.

    Article  MathSciNet  Google Scholar 

  17. X.P. Ding, Existence of Solutions for Quasi-Equilibrium Problems in Noncompact Topological Spaces, Computers and Mathematics with Applications, Vol. 39, pp. 13–21, 2000.

    Article  Google Scholar 

  18. F. Giannessi, Theorems of the Alternative, Quadratic Programs and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R.W. Cottle, F. Giannessi, and J.L. Lions, John Wiley and Sons, New York, pp.151–186, 1980.

    Google Scholar 

  19. F. Giannessi, (Ed.), Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, Kluwer Academic Publishers, Dordrecht-Boston-London, 2000.

    Google Scholar 

  20. N. Hadjisavvas and S. Schaible, From Scalar to Vector Equilibrium Problems in the Quasimonotone Case, Journal of Optimization Theory and Applications, Vol. 96, pp. 297–309, 1998.

    Article  MathSciNet  Google Scholar 

  21. T. Husain and E. Tarafdar, Simultaneous Variational Inequalities, Minimization Problems and Related Results, Mathematica Japonica, Vol. 39, pp. 221–231, 1994.

    MathSciNet  Google Scholar 

  22. W.K. Kim and K.-K. Tan, On generalized Vector Quasi-Variational Inequalities, Optimization, Vol. 46, pp. 185–198, 1999.

    Article  MathSciNet  Google Scholar 

  23. I.V. Konnov, A General Approach to Finding Stationary Point and the Solution of Related Problems, Computational Mathematics and Mathematical Physics, Vol. 36, pp. 585–593, 1996.

    MathSciNet  Google Scholar 

  24. G.M. Lee, D.S. Kim and B.S. Lee, On Noncooperative Vector Equilibrium, Indian Journal of Pure and Applied Mathematics, Vol. 27, pp. 735–739, 1996.

    MathSciNet  Google Scholar 

  25. G.M. Lee, D.S. Kim and B.S. Lee, On Vector Quasivariational-like Inequality, Bulletin of the Korean Mathematical Society, Vol. 33, pp. 45–55, 1996.

    MathSciNet  Google Scholar 

  26. L.-J. Lin and S. Park, On Some Generalized Quasi-Equilibrium Problems, Journal of Mathematical Analysis and Applications, Vol. 224, pp. 167–181, 1998.

    Article  MathSciNet  Google Scholar 

  27. L.-J. Lin and Z.-T. Yu, Fixed-point Theorems and Equilibrium Problems, To appear in Nonlinear Analysis, Theory, Methods and Applications, 2000.

    Google Scholar 

  28. D.T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, Vol. 319, 1989.

    Google Scholar 

  29. W. Oettli, A Remark on Vector-Valued Equilibria and Generalized Monotonicity, Acta Mathematica Vietnamica, Vol. 22, pp. 213–221, 1997.

    MathSciNet  Google Scholar 

  30. A.H. Siddiqi, Q.H. Ansari and A. Khaliq, On Vector Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 84, pp. 171–180, 1995.

    Article  MathSciNet  Google Scholar 

  31. N.X. Tan and P.N. Tinh, On the Existence of Equilibrium Points of Vector Functions, Numerical Functional Analysis and Optimization, Vol. 19, pp. 141–156, 1998.

    Article  MathSciNet  Google Scholar 

  32. E. Tarafdar and G. X.-Z. Yuan, Generalized Variational Inequalities and Its Applications, Nonlinear Analysis, Theory, Methods and Applications, Vol. 30, pp. 4171–4181, 1997.

    MathSciNet  Google Scholar 

  33. G. X.-Z. Yuan, G. Isac, K. K. Tan, and J. Yu, The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibra, Acta Applicandae Mathematicae, Vol. 54, pp. 135–166, 1998.

    Article  MathSciNet  Google Scholar 

  34. G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, Inc., New York, Basel, 1999.

    Google Scholar 

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Ansari, Q.H., Yao, JC. (2003). On Vector Quasi—Equilibrium Problems. In: Daniele, P., Giannessi, F., Maugeri, A. (eds) Equilibrium Problems and Variational Models. Nonconvex Optimization and Its Applications, vol 68. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0239-1_1

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  • DOI: https://doi.org/10.1007/978-1-4613-0239-1_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7955-3

  • Online ISBN: 978-1-4613-0239-1

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