Journal of Optimization Theory and Applications

, Volume 142, Issue 3, pp 493–499 | Cite as

Feasibility-Solvability Theorem for a Generalized System

  • R. Hu
  • Y. P. FangEmail author


In this paper, we consider a generalized system in the framework of the formulation proposed by Blum and Oettli. The concepts of feasibility and strict feasibility are introduced for a generalized system and a feasibility-solvability theorem is obtained.


Generalized systems Strict feasibility Solvability Monotonicity 


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  1. 1.
    Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, San Diego (1972) Google Scholar
  2. 2.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Chang, K.C.: Variational methods for non-differentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Motreanu, D., Panagiotopoulos, P.D.: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Kluwer Academic, Dordrecht (1999) zbMATHGoogle Scholar
  5. 5.
    Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hadjisavvas, N., Schaible, S.: From scalar to vector equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 96, 297–309 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Oettli, W.: A remark on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam. 22, 213–221 (1997) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Martínez-Legaz, J.E., Sosa, W.: Duality for equilibrium problems. J. Glob. Optim. 35, 311–319 (2006) zbMATHCrossRefGoogle Scholar
  10. 10.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problems. Academic Press, San Diego (1992) Google Scholar
  11. 11.
    Isac, G.: Topological Methods in Complementarity Theory. Kluwer Academic, Dordrecht (2000) zbMATHGoogle Scholar
  12. 12.
    Fang, Y.P., Huang, N.J.: Equivalence of equilibrium problems and least element problems. J. Optim. Theory Appl. 132, 411–422 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsChengdu University of Information TechnologyChengduPeople’s Republic of China
  2. 2.Department of MathematicsSichuan UniversityChengduPeople’s Republic of China

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