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.

Keywords:

• Pre-order principle
• equilibrium version of Ekeland variational principle
• quasi-metric space
• Caristi’s fixed point theorem
• robust trap problem

Acknowledgements

The authors are grateful to the reviewers and the editors for their valuable comments and suggestions, which greatly improved the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first two authors were supported by the National Natural Science Foundation of China [grant number 11471236], [grant number 11561049].