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

  • RAOTA-2016
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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.

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Notes

  1. In Bao (2015), Bao et al. (2015a, b, c, 2016), Bao and Théra (2015) and the references therein, by a pseudo-quasimetric space one understands a quasimetric space.

  2. Kelly (1963, p. 71) “The notion of a bitopological space used in relation to semi-continuous functions restores sufficient symmetry to enable one to use some of the existing techniques of continuous functions.”

  3. It has been also defined a sequence as being bi-convergent if it is both forward- and backward-convergent, (i.e. if it convergent with respect to the pseudo-metric \(q^s\)).

  4. Known also as left-Cauchy (Bao et al. 2015a, b, c), q-Cauchy Kelly (1963), and left K-Cauchy (Reilly and Subrahmanyam 1982)

  5. Known also as \({\overline{q}}\)-Cauchy (Kelly 1963), and right K-Cauchy (Reilly and Subrahmanyam 1982)

  6. Known also as bi-Cauchy, or Cauchy in two topologies

  7. Recall that in Cobzaş (2013) a quasimetric satisfies the condition: \(q(x,y)=q(y,x)=0\) implies \(x=y\).

  8. Left-complete

  9. A quasimetric in the sense of Cobzaş (2011), Cobzaş is a pseuso-quasimetric satisfying \(q(x,y) = q(y,x) = 0 \Longrightarrow x=y\).

  10. There are two versions of Ekeland’s variational principle with respect to each kind of completeness.

  11. A quasimetric space enjoys the forward Hausdorff property, i.e. a sequence being forward-convergent has a unique forward-limit.

  12. The Picard–Lindelöf theorem, which shows that ordinary differential equations have solutions, is essentially an application of the Banach fixed point theorem to a special sequence of functions which forms a fixed point iteration, constructing the solution to the equation. Solving an ODE in this way is called Picard iteration, Picard’s method, or the Picard iterative process.

  13. The space in question is a metric space and thus there is no difference between two topologies.

  14. The conjugate quasimetric of the one studied in Lin et al. (2011, Example 3.16).

  15. Known as ‘left Hausdorff’ in the cited papers.

  16. The topology \(\tau ({\overline{q}})\).

  17. Cobzaş (2011, Remark 1.4) pointed out that the topology \(\tau (q)\) is regular under the assumptions made in the cited paper.

  18. A quasimetric in the sense of Karapinar et al. is a pseudo-quasimetric defined in Definition 2 satisfying that \(x = y\) if and only if \(q(x,y) = q(y,x) = 0\).

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Acknowledgements

The contribution of the first author was completed while visiting Aix-Marseille University with a GREQAM-AMSE grant.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI, USA

    T. Q. Bao

  2. Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania

    S. Cobzaş

  3. Aix-Marseille School of Economics, CNRS and EHESS, Aix-Marseille University, Marseille, France

    A. Soubeyran

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  1. T. Q. Bao
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  2. S. Cobzaş
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  3. A. Soubeyran
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Correspondence to T. Q. Bao.

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Bao, T.Q., Cobzaş, S. & Soubeyran, A. Variational principles, completeness and the existence of traps in behavioral sciences. Ann Oper Res 269, 53–79 (2018). https://doi.org/10.1007/s10479-016-2368-0

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