IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i4p564-d1586660.html
   My bibliography  Save this article

On the Existence Theorems of Equilibrium and Quasi-Equilibrium Problems in Different Spaces

Author

Listed:
  • Ali Farajzadeh

    (Department of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, Iran)

  • Mahmood Ghobadi

    (Department of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, Iran)

  • Jen-Chih Yao

    (Center for General Education, China Medical University, Taichung 404327, Taiwan
    UNEC Mathematical Modeling and Optimization Research Center, Azerbaijan State University of Economics (UNEC), Istiqlaliyyat Str. 6, Baku 1001, Azerbaijan)

Abstract

In this article, the solvability of the equilibrium problem (EP), Minty equilibrium problem (MEP), and quasi-equilibrium problem (QEP) by using the notions of cyclically monotone and cyclically antimonotone in the setting of topological vector spaces and metric spaces is investigated. Also, the concepts transfer lower continuity, transfer weakly lower continuity, lower semicontinuity, from above, and sequentially weakly lower semicontinous which are weaker notions than the lower semicontinuity for establishing the existence results for EP and QEP, and the other forms of them are applied. Moreover, by using the famous results for the minimum points of a function, some existence theorems, by using the triangle property, of solutions for EP and QEP are given when the domains of bifunctions are compact and not compact. The results of this paper can be viewed as new versions of the corresponding published results with new and mild assumptions.

Suggested Citation

  • Ali Farajzadeh & Mahmood Ghobadi & Jen-Chih Yao, 2025. "On the Existence Theorems of Equilibrium and Quasi-Equilibrium Problems in Different Spaces," Mathematics, MDPI, vol. 13(4), pages 1-20, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:4:p:564-:d:1586660
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/4/564/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/4/564/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Tian, Guoqiang & Zhou, Jianxin, 1995. "Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 281-303.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Subiza, Begona & Peris, Josep E., 1997. "Numerical representation for lower quasi-continuous preferences," Mathematical Social Sciences, Elsevier, vol. 33(2), pages 149-156, April.
    2. Prokopovych, Pavlo & Yannelis, Nicholas C., 2017. "On strategic complementarities in discontinuous games with totally ordered strategies," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 147-153.
    3. Rabia Nessah & Guoqiang Tian, 2008. "The Existence of Equilibria in Discontinuous and Nonconvex Games," Working Papers 2008-ECO-14, IESEG School of Management, revised Mar 2010.
    4. Rabia Nessah & Guoqiang Tian, 2016. "On the existence of Nash equilibrium in discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(3), pages 515-540, March.
    5. Alcantud, Jose C.R., 2006. "Maximality with or without binariness: Transfer-type characterizations," Mathematical Social Sciences, Elsevier, vol. 51(2), pages 182-191, March.
    6. Tian, Guoqiang, 2015. "On the existence of equilibria in games with arbitrary strategy spaces and preferences," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 9-16.
    7. Guoqiang Tian, 2016. "On the existence of price equilibrium in economies with excess demand functions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 5-16, April.
    8. Vincenzo Scalzo, 2013. "Essential equilibria of discontinuous games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(1), pages 27-44, September.
    9. M. Ali Khan & Metin Uyanik, 2021. "The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 799-840, April.
    10. Francesco Ciardiello, 2007. "Convexity on Nash Equilibria without Linear Structure," Quaderni DSEMS 15-2007, Dipartimento di Scienze Economiche, Matematiche e Statistiche, Universita' di Foggia.
    11. Scalzo, Vincenzo, 2012. "Discontinuous stable games and efficient Nash equilibria," Economics Letters, Elsevier, vol. 115(3), pages 387-389.
    12. Orestes Bueno & John Cotrina, 2021. "Existence of Projected Solutions for Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 344-362, October.
    13. Gianni Bosi & Magalì E. Zuanon, 2017. "Maximal elements of quasi upper semicontinuous preorders on compact spaces," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 5(1), pages 109-117, April.
    14. Gianni Bosi & Magalì Zuanon, 2019. "Upper Semicontinuous Representability of Maximal Elements for Nontransitive Preferences," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 758-765, June.
    15. Tian, Guoqiang, 2012. "A Full Characterization on Fixed-Point Theorem, Minimax Inequality, Saddle Point, and KKM Theorem," MPRA Paper 57929, University Library of Munich, Germany, revised Jul 2014.
    16. Paulo Barelli & Idione Meneghel, 2013. "A Note on the Equilibrium Existence Problem in Discontinuous Games," Econometrica, Econometric Society, vol. 81(2), pages 813-824, March.
    17. Nosratabadi, Hassan, 2014. "Partially upper continuous preferences: Representation and maximal elements," Economics Letters, Elsevier, vol. 125(3), pages 408-410.
    18. Duggan, John, 2011. "General conditions for the existence of maximal elements via the uncovered set," Journal of Mathematical Economics, Elsevier, vol. 47(6), pages 755-759.
    19. Federico Quartieri, 2022. "On the Existence of Greatest Elements and Maximizers," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 375-389, November.
    20. Scalzo, Vincenzo, 2010. "Pareto efficient Nash equilibria in discontinuous games," Economics Letters, Elsevier, vol. 107(3), pages 364-365, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:4:p:564-:d:1586660. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.