Abstract

Quasi-equilibrium problems represent a general framework covering, in many situations, quasi-variational inequalities, complementarity problems and generalized Nash equilibrium problems. In this work, we provide necessary and sufficient conditions guaranteeing the existence of a new kind of solution for quasi-equilibrium problems defined on Hausdorff topological spaces with non-self constraint maps. The main machineries for proving our results are a Tian's fixed point theorem for the convex case, and the finite intersection property for the non-convex case. Furthermore, quasi-variational inequality problems and generalized Nash equilibrium problems are considered as applications.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Quasi-equilibrium problems
  • quasi-variational inequalities
  • generalized Nash equilibrium problems
  • non-self maps
  • finite intersection property

Mathematics Subject Classification

47J20, 49J53, 91A10

References

  1. E. Allevi, M. E. De Giuli, M. Milasi, D. Scopelliti: Quasi-variational problems with non-self map on Banach spaces: Existence and applications, Nonlinear Analysis, Real World Appl. 67 (2022), art.no. 103641, 19p.https://doi.org/10.1016/j.nonrwa.2022.103641Google Scholar
  2. K. J. Arrow, G. Debreu: Existence of an equilibrium for a competitive economy, Econometrica 22/3 (1954) 265–290.https://doi.org/10.2307/1907353Google Scholar
  3. D. Aussel, J. Cotrina, A. Iusem: An existence result for quasi-equilibrium problems, J. Convex Analysis 24/1 (2017) 55–66.https://doi.org/10.68381/jca24004Google Scholar
  4. D. Aussel, J. Dutta, T. Pandit: About the links between equilibrium problems and variational inequalities, in: Mathematical Programming and Game Theory, New Delhi 2017, S. K. Neogy et al. (eds.), Indian Statistial Institute Series, Springer, Singapore (2018) 115–130.https://doi.org/10.1007/978-981-13-3059-9_6Google Scholar
  5. D. Aussel, A. Sultana, V. Vetrivel: On the existence of projected solutions of quasi-variational inequalities and generalized Nash equilibrium problems, J. Optim. Theory Appl. 170/3 (2016) 818–837.https://doi.org/10.1007/s10957-016-0951-9Google Scholar
  6. I. Benedetti, S. Lorenzini: Projected solutions for generalized quasivariational inequalities without compact constraint multimap, Commun. Nonlinear Sci. Numer. Simul. 129 (2024), art.no. 107687, 19p.https://doi.org/10.1016/j.cnsns.2023.107687Google Scholar
  7. M. Bianchi, N. Hadjisavvas, S. Schaible: On pseudomonotone maps T for which −T is also pseudomonotone, J. Convex Analysis 10/1 (2003) 149–168.https://doi.org/10.68381/jca1007Google Scholar
  8. M. Bianchi, R. Pini: A note on equilibrium problems with properly quasimonotone bifunctions, J. Global Optim. 20 (2001) 67–76.https://doi.org/10.1023/A:1011234525151Google Scholar
  9. M. Bianchi, S. Schaible: An extension of pseudolinear functions and variational inequality problems, J. Optim. Theory Appl. 104 (2000) 59–71.https://doi.org/10.1023/A:1004672604998Google Scholar
  10. P. Bich, R. Laraki: On the existence of approximate equilibria and sharing rule solutions in discontinuous games, Theor. Econ. 12/1 (2017) 79–108.https://doi.org/10.3982/TE2081Google Scholar
  11. E. Blum, W. Oettli: From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1993) 1–23.Google Scholar
  12. O. Bueno, C. Calderón, J. Cotrina: A note on jointly convex Nash games, Optimization (2023) 13p.https://doi.org/10.1080/02331934.2023.2281639Google Scholar
  13. O. Bueno, J. Cotrina: Existence of projected solutions for generalized Nash equilibrium problems, J. Optim. Theory Appl. 191/1 (2021) 344–362.https://doi.org/10.1007/s10957-021-01941-9Google Scholar
  14. C. Calderón, J. Cotrina: Remarks on projected solutions for generalized Nash games, arXive: 2307.13171 (2023).Google Scholar
  15. M. Castellani, M. Giuli: On equivalent equilibrium problems, J. Optim. Theory Appl. 147 (2010) 157–168.https://doi.org/10.1007/s10957-010-9703-4Google Scholar
  16. M. Castellani, M. Giuli: Refinements of existence results for relaxed quasimonotone equilibrium problems, J. Global Optim. 57 (2013) 1213–1227.https://doi.org/10.1007/s10898-012-0021-2Google Scholar
  17. M. Castellani, M. Giuli: An existence result for quasiequilibrium problems in separable Banach spaces, J. Math. Analysis Appl. 425 (2015) 85–95.https://doi.org/10.1016/j.jmaa.2014.12.022Google Scholar
  18. M. Castellani, M. Giuli, M. Milasi, D. Scopelliti: Projected solutions of generalized quasivariational problems in Banach spaces, Nonlinear Analysis, Real World Appl. 76 (2024), art.no. 104021.https://doi.org/10.1016/j.nonrwa.2023.104021Google Scholar
  19. M. Castellani, M. Giuli, M. Pappalardo: A Ky Fan minimax inequality for quasiequilibria on finite-dimensional spaces. J. Optim. Theory Appl. 179 (2018) 53–64.https://doi.org/10.1007/s10957-018-1319-0Google Scholar
  20. S. Chang, B. Lee, X. Wu, Y. Cho, G. Lee: On the generalized quasivariational inequality problems, J. Math. Analysis Appl. 203 (1996) 686–711.https://doi.org/10.1006/jmaa.1996.0406Google Scholar
  21. J. Cotrina: Remarks on pseudo-continuity, Minimax Theory Appl. 9/1 (2024) 41–54.Google Scholar
  22. J. Cotrina, Y. García: Equilibrium problems: Existence results and applications, Set-Valued Var. Analysis 26/1 (2018) 159–177.https://doi.org/10.1007/s11228-017-0451-6Google Scholar
  23. J. Cotrina, A. Hantoute, A. Svensson: Existence of quasi-equilibria on unbounded constraint sets, Optimization 71/2 (2022) 337–354.https://doi.org/10.1080/02331934.2020.1778690Google Scholar
  24. J. Cotrina, A. Svensson: Finite intersection property for equilibrium problems, J. Global Optimization 79 (2021) 941–957.https://doi.org/10.1007/s10898-020-00961-5Google Scholar
  25. J. Cotrina, J. Zúñiga: A note on quasi-equilibrium problems, Oper. Res. Letters 46 (2018) 138–140.https://doi.org/10.1016/j.orl.2017.12.002Google Scholar
  26. J. Cotrina, J. Zúñiga: Quasi-equilibrium problems with non-self constraint map, J. Global Optimization 75 (2019) 177–197.https://doi.org/10.1007/s10898-019-00762-5Google Scholar
  27. C. Ewerhart, P. J. Reny: Corrigendum to “On the existence of pure and mixed strategy nash equilibrium in discontinuous games”, Econometrica 90/6 (2022) 1–2.https://doi.org/10.3982/ECTA20900Google Scholar
  28. C. J. Himmelberg: Fixed points of compact multifunctions, J. Math. Analysis Appl. 38 (1972) 205–207.https://doi.org/10.1016/0022-247X(72)90128-XGoogle Scholar
  29. J. Morgan, V. Scalzo: Pseudocontinuous functions and existence of Nash equilibria, J. Math. Economics 43/2 (2007) 174–183.https://doi.org/10.1016/j.jmateco.2006.10.004Google Scholar
  30. J. Nash: Non-cooperative games, Annals Math. 54/2 (1951) 286–295.https://doi.org/10.2307/1969529Google Scholar
  31. R. Nessah, G. Tian: Existence of solution of minimax inequalities, equilibria in games and fixed points without convexity and compactness assumptions, J. Optim. Theory Appl. 157 (2013) 756–795.https://doi.org/10.1007/s10957-012-0176-5Google Scholar
  32. H. Nikaidô, K. Isoda: Note on noncooperative convex games. Pacific J. Math. 52 (1955) 807–815.https://doi.org/10.2140/pjm.1955.5.807Google Scholar
  33. P. J. Reny: On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67/5 (1999) 1029–1056.https://doi.org/10.1111/1468-0262.00069Google Scholar
  34. G. Q. Tian: Fixed points theorems for mappings with non-compact and non-convex domains, J. Math. Analysis Appl. 158 (1991) 161–167.https://doi.org/10.1016/0022-247X(91)90274-4Google Scholar
  35. G. Q. Tian: Generalization of the KKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementary, J. Math. Analysis Appl. 170 (1992) 457–471.https://doi.org/10.1016/0022-247X(92)90030-HGoogle Scholar

Suggested citation

D. Aussel, J. Cotrina. “Quasi-Equilibrium Problems with Non-Self Constraint Maps in Topological Spaces. Necessary and Sufficient Conditions for the Existence of Solutions.” Journal of Convex Analysis 32 (2025), No. 1, 71–89. https://doi.org/10.68381/jca32006

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2025. Rights now held by Banach Press.