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Equilibrium versions of variational principles in quasi-metric spaces and the robust trap problem

Abstract : Using a pre-order principle in [Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904–937], we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective bimap is defined on the product of left-complete quasi-metric spaces and taking values in a quasi-order linear space, and the perturbation consists of the quasi-metric and a positive vector . Here, the ordering is only to be -closed, which is strictly weaker than to be topologically closed. From the general equilibrium version, we deduce a number of particular equilibrium versions of EVP with set-valued bimaps or with vector-valued bimap. As applications of the equilibrium versions of EVP, we present several interesting results on equilibrium problems, vector optimization and fixed point theory in the setting of quasi-metric spaces. These results extend and improve the related known results. Using the obtained EVPs, we further study the existence and the robustness of traps in Behavioural Sciences.
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Jing-Hui Qiu, Fei He, Antoine Soubeyran. Equilibrium versions of variational principles in quasi-metric spaces and the robust trap problem. Optimization, Taylor & Francis, 2017, 67 (1), pp.25-53. ⟨10.1080/02331934.2017.1387257⟩. ⟨hal-02084464⟩



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