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Coercivity Conditions for Equilibrium Problems

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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.

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  • T. Harker J.S. Pang (1990) ArticleTitleFinite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survery of Theory, Algorithms, and Applications Mathematical Programming 48 161–220

    Google Scholar 

  • R.W. Cottle J.C. Yao (1992) ArticleTitlePseudomonotone Complementarity Problems in Hilbert Space Journal of Optimization Theory and Applications 75 281–295

    Google Scholar 

  • A. Daniilidis N. Hadjisavvas (1999) ArticleTitleCoercivity Conditions and Variational Inequalities Mathematical Programming 86 433–438

    Google Scholar 

  • J.P. Crouzeix (1997) ArticleTitlePseudomonotone Variational Inequality Problems: Existence of Solutions Mathematical Programming 78 305–314

    Google Scholar 

  • M. Bianchi N. Hadjisavvas S. Schaible (2004) ArticleTitleMinimal Coercivity Conditions for Variational Inequalities–Application to Exceptional Families of Elements Journal of Optimization Theory and Applications 122 1–17

    Google Scholar 

  • D. Aussel N. Hadjisavvas (2004) ArticleTitleOn Quasimonotone Variational Inequalities Journal of Optimization Theory and Applications 121 445–450

    Google Scholar 

  • F. Flores-bazán (2000) ArticleTitleExistence Theorems for Generalized Noncoercive Equilibrium Problems: The Quasiconvex Case SIAM Journal on Optimization 11 675–690

    Google Scholar 

  • N. Hadjisavvas (2003) ArticleTitleContinuity and Maximality Properties of Pseudomonotone Operators Journal of Convex Analysis 10 465–475

    Google Scholar 

  • S. Karamardian (1971) ArticleTitleGeneralized Complementarity Problem Journal of Optimization Theory and Applications 8 161–168

    Google Scholar 

  • M. Bianchi R. Pini (2001) ArticleTitleA Note on Equilibrium Problems for Properly Quasimonotone Bifunctions Journal of Global Optimization 20 67–76

    Google Scholar 

  • M. Bianchi S. Schaible (1996) ArticleTitleGeneralized Monotone Bifunctions and Equilibrium Problems Journal of Optimization Theory and Applications 90 31–43

    Google Scholar 

  • Y.B. Zhao J.Y. Han H.D. Qi (1999) ArticleTitleExceptional Families and Existence Theorems for Variational Inequalities Problems Journal of Optimization Theory and Applications 101 475–495

    Google Scholar 

  • H. Brezis L. Nirenberg G. Stampacchia (1972) ArticleTitleA Remark on Fan’s Minimax Principle Bollettino dell’Unione Matematica Italiana 6 293–300

    Google Scholar 

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We thank an anonymous referee for valuable comments and suggestions.

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Bianchi, M., Pini, R. Coercivity Conditions for Equilibrium Problems. J Optim Theory Appl 124, 79–92 (2005).

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