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Variational principles, completeness and the existence of traps in behavioral sciences

Author

Listed:
  • T. Q. Bao

    (Northern Michigan University)

  • S. Cobzaş

    (Babeş-Bolyai University)

  • A. Soubeyran

    (Aix-Marseille University)

Abstract

In this paper, driven by Behavioral applications to human dynamics, we consider the characterization of completeness in pseudo-quasimetric spaces in term of a generalization of Ekeland’s variational principle in such spaces, and provide examples illustrating significant improvements to some previously obtained results, even in complete metric spaces. At the behavioral level, we show that the completeness of a space is equivalent to the existence of traps, rather easy to reach (in a worthwhile way), but difficult (not worthwhile to) to leave. We first establish new forward and backward versions of Ekeland’s variational principle for the class of strict-decreasingly forward (resp. backward)-lsc functions in pseudo-quasimetric spaces. We do not require that the space under consideration either be complete or to enjoy the limit uniqueness property since, in a pseudo-quasimetric space, the collections of forward-limits and backward ones of a sequence, in general, are not singletons.

Suggested Citation

  • T. Q. Bao & S. Cobzaş & A. Soubeyran, 2018. "Variational principles, completeness and the existence of traps in behavioral sciences," Annals of Operations Research, Springer, vol. 269(1), pages 53-79, October.
    • Handle: RePEc:spr:annopr:v:269:y:2018:i:1:d:10.1007_s10479-016-2368-0
    DOI: 10.1007/s10479-016-2368-0
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    References listed on IDEAS

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    1. Phan Khanh & Dinh Quy, 2013. "Versions of Ekeland’s variational principle involving set perturbations," Journal of Global Optimization, Springer, vol. 57(3), pages 951-968, November.
    2. Truong Q. Bao & Boris S. Mordukhovich & Antoine Soubeyran, 2015. "Minimal points, variational principles, and variable preferences in set optimization," Post-Print hal-01457319, HAL.
    3. Truong Bao & Phan Khanh & Antoine Soubeyran, 2016. "Variational principles with generalized distances and the modelization of organizational change," Post-Print hal-01690191, HAL.
    4. T. Q. Bao & B. S. Mordukhovich & A. Soubeyran, 2015. "Variational Analysis in Psychological Modeling," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 290-315, January.
    5. Jeong Sheok Ume, 2002. "A minimization theorem in quasi-metric spaces and its applications," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 31, pages 1-5, January.
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    Cited by:

    1. Majid Fakhar & Mohammadreza Khodakhah & Ali Mazyaki & Antoine Soubeyran & Jafar Zafarani, 2022. "Variational rationality, variational principles and the existence of traps in a changing environment," Journal of Global Optimization, Springer, vol. 82(1), pages 161-177, January.
    2. Mi Zhou & Naeem Saleem & Basit Ali & Misha Mohsin & Antonio Francisco Roldán López de Hierro, 2023. "Common Best Proximity Points and Completeness of ℱ−Metric Spaces," Mathematics, MDPI, vol. 11(2), pages 1-21, January.
    3. Le Phuoc Hai & Phan Quoc Khanh & Antoine Soubeyran, 2022. "General Versions of the Ekeland Variational Principle: Ekeland Points and Stop and Go Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 347-373, October.
    4. L. P. Hai & L. Huerga & P. Q. Khanh & V. Novo, 2019. "Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems," Journal of Global Optimization, Springer, vol. 74(2), pages 361-382, June.
    5. Le Phuoc Hai, 2021. "Ekeland variational principles involving set perturbations in vector equilibrium problems," Journal of Global Optimization, Springer, vol. 79(3), pages 733-756, March.
    6. Ştefan Cobzaş, 2024. "The Strong Ekeland Variational Principle in Quasi-Pseudometric Spaces," Mathematics, MDPI, vol. 12(3), pages 1-13, February.

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