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A Ky Fan Minimax Inequality for Quasiequilibria on Finite-Dimensional Spaces

Author

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  • Marco Castellani

    (Computer Science and Mathematics)

  • Massimiliano Giuli

    (Computer Science and Mathematics)

  • Massimo Pappalardo

    (University of Pisa)

Abstract

Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite-dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued maps which holds in finite-dimensional spaces. Furthermore, this result allows one to locate the position of a solution. Sufficient conditions, which are easier to verify, may be obtained by imposing restrictions either on the domain or on the bifunction. These facts make it possible to yield various existence results which reduce to the well-known Ky Fan minimax inequality when the constraint map is constant and the quasiequilibrium problem coincides with an equilibrium problem. Lastly, a comparison with other results from the literature is discussed.

Suggested Citation

  • Marco Castellani & Massimiliano Giuli & Massimo Pappalardo, 2018. "A Ky Fan Minimax Inequality for Quasiequilibria on Finite-Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 53-64, October.
    • Handle: RePEc:spr:joptap:v:179:y:2018:i:1:d:10.1007_s10957-018-1319-0
    DOI: 10.1007/s10957-018-1319-0
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    References listed on IDEAS

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    1. Boualem Alleche & Vicenţiu D. Rădulescu, 2016. "Solutions and Approximate Solutions of Quasi-Equilibrium Problems in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 629-649, August.
    2. M. Castellani & M. Giuli, 2016. "Approximate solutions of quasiequilibrium problems in Banach spaces," Journal of Global Optimization, Springer, vol. 64(3), pages 615-620, March.
    3. Bergstrom, Theodore C. & Parks, Robert P. & Rader, Trout, 1976. "Preferences which have open graphs," Journal of Mathematical Economics, Elsevier, vol. 3(3), pages 265-268, December.
    4. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
    5. M. Castellani & M. Giuli, 2016. "Approximate solutions of quasiequilibrium problems in Banach spaces," Journal of Global Optimization, Springer, vol. 64(3), pages 615-620, March.
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    Citations

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    Cited by:

    1. John Cotrina & Javier Zúñiga, 2018. "Time-Dependent Generalized Nash Equilibrium Problem," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 1054-1064, December.
    2. M. Bianchi & G. Kassay & R. Pini, 2022. "Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization," Journal of Global Optimization, Springer, vol. 82(3), pages 483-498, March.
    3. John Cotrina & Michel Théra & Javier Zúñiga, 2020. "An Existence Result for Quasi-equilibrium Problems via Ekeland’s Variational Principle," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 336-355, November.
    4. Marco Castellani & Massimiliano Giuli, 2021. "A Generalized Ky Fan Minimax Inequality on Finite-Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 343-357, August.
    5. Domenico Scopelliti, 2022. "On a Class of Multistage Stochastic Hierarchical Problems," Mathematics, MDPI, vol. 10(21), pages 1-13, October.
    6. Scalzo, Vincenzo, 2020. "Doubly Strong Equilibrium," MPRA Paper 99329, University Library of Munich, Germany.
    7. Tran Thang & Xuan Thanh Le, 2024. "Self-Adaptive Extragradient Algorithms for Quasi-Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2988-3013, December.
    8. M. Castellani & M. Giuli, 2019. "A coercivity condition for nonmonotone quasiequilibria on finite-dimensional spaces," Journal of Global Optimization, Springer, vol. 75(1), pages 163-176, September.
    9. John Cotrina & Javier Zúñiga, 2019. "Quasi-equilibrium problems with non-self constraint map," Journal of Global Optimization, Springer, vol. 75(1), pages 177-197, September.

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