Coercivity Conditions for Equilibrium Problems

  • M. Bianchi
  • R. Pini


The study of the existence of solutions of equilibrium problems on unbounded domains involves usually the same sufficient assumptions as for bounded domains together with a coercivity condition. We focus on two different conditions: the first is obtained assuming the existence of a bounded set such that no elements outside is a candidate for a solution; the second allows the solution set to be unbounded. Our results exploit the generalized monotonicity properties of the function f defining the equilibrium problem. It turns out that, in both the pseudomonotone and the quasimonotone setting, an equivalence can be stated between the nonemptyness and boundedness of the solution set and these coercivity conditions. In the pseudomonotone case, we compare our coercivity conditions with various coercivity conditions that appeared in the literature.


Equilibrium problems generalized monotonicity generalized convexity coercivity conditions 


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  1. Harker, T., Pang, J.S. 1990Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survery of Theory, Algorithms, and ApplicationsMathematical Programming48161220Google Scholar
  2. Cottle, R.W., Yao, J.C. 1992Pseudomonotone Complementarity Problems in Hilbert SpaceJournal of Optimization Theory and Applications75281295Google Scholar
  3. Daniilidis, A., Hadjisavvas, N. 1999Coercivity Conditions and Variational InequalitiesMathematical Programming86433438Google Scholar
  4. Crouzeix, J.P. 1997Pseudomonotone Variational Inequality Problems: Existence of SolutionsMathematical Programming78305314Google Scholar
  5. Bianchi, M., Hadjisavvas, N., Schaible, S. 2004Minimal Coercivity Conditions for Variational Inequalities–Application to Exceptional Families of ElementsJournal of Optimization Theory and Applications122117Google Scholar
  6. Aussel, D., Hadjisavvas, N. 2004On Quasimonotone Variational InequalitiesJournal of Optimization Theory and Applications121445450Google Scholar
  7. Flores-bazán, F. 2000Existence Theorems for Generalized Noncoercive Equilibrium Problems: The Quasiconvex CaseSIAM Journal on Optimization11675690Google Scholar
  8. Hadjisavvas, N. 2003Continuity and Maximality Properties of Pseudomonotone OperatorsJournal of Convex Analysis10465475Google Scholar
  9. Karamardian, S. 1971Generalized Complementarity ProblemJournal of Optimization Theory and Applications8161168Google Scholar
  10. Bianchi, M., Pini, R. 2001A Note on Equilibrium Problems for Properly Quasimonotone BifunctionsJournal of Global Optimization206776Google Scholar
  11. Bianchi, M., Schaible, S. 1996Generalized Monotone Bifunctions and Equilibrium ProblemsJournal of Optimization Theory and Applications903143Google Scholar
  12. Zhao, Y.B., Han, J.Y., Qi, H.D. 1999Exceptional Families and Existence Theorems for Variational Inequalities ProblemsJournal of Optimization Theory and Applications101475495Google Scholar
  13. Brezis, H., Nirenberg, L., Stampacchia, G. 1972A Remark on Fan’s Minimax PrincipleBollettino dell’Unione Matematica Italiana6293300Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • M. Bianchi
    • 1
  • R. Pini
    • 2
  1. 1.Istituto di Econometria e Matematica per le Applicazioni Economiche, Finanziarie e AttuarialiUniversità Cattolica del Sacro CuoreMilanoItaly
  2. 2.Dipartimento di Metodi Quantitativi per l’EconomiaUniversità di Milano- BicoccaMilanoItaly

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